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Encyclopedia > Algebraic structure

In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... Operations is that unit (be it a division or department) of an organization that carries out the actual execution of the core operating functions. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...


Abstractly, an "algebraic structure," is the collection of all possible models of a given set of axioms. More concretely, an algebraic structure is any particular model of some set of axioms. For example, the monster group both "is" an algebraic structure in the concrete sense, and abstractly, "has" the group structure in common with all other groups. This article employs both meanings of "structure." In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... In mathematics, the Monster group M is a group of order    246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 · 1053. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...


This definition of an algebraic structure should not be taken as restrictive. Anything that satisfies the axioms defining a structure is an instance of that structure, regardless of how many other axioms that instance happens to have. For example, all groups are also semigroups and magmas. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...

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Structures whose axioms are all identities

If the axioms defining a structure are all identities, the structure is a variety (not to be confused with algebraic variety in the sense of algebraic geometry). Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part of universal algebra. In mathematics, an identity is an equality that remains true regardless of the values of any variables that appear within it. ... In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satifying a given set of identities. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ... In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... Look up Relation in Wiktionary, the free dictionary In mathematics, a relation is a generalization of arithmetic relations, such as = and <, which occur in statements, such as 5 < 6 or 2 + 2 = 4. See relation (mathematics), binary relation (of set theory and logic) and relational algebra. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ...


All structures in this section are varieties. Some of these structures are most naturally axiomatized using one or more nonidentities, but are nevertheless varieties because there exists an equivalent axiomatization, one perhaps less perspicuous, composed solely of identities. Algebraic structures that are not varieties are described in the following section, and differ from varieties in their metamathematical properties. In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satifying a given set of identities. ... In general, metamathematics or meta-mathematics is reflection about mathematics seen as an entity/object in human consciousness and culture. ...


In this section and the following one, structures are listed in approximate order of increasing complexity, operationalized as follows:

  • Simple structures requiring but one set, the universe S, are listed before composite ones requiring two sets;
  • Structures having the same number of required sets are then ordered by the number of binary operations (0 to 4) they require. Incidentally, no structure mentioned in this entry requires an operation whose arity exceeds 2;
  • Let A and B be the two sets that make up a composite structure. Then a composite structure may include 1 or 2 functions of the form AxAB or AxBA;
  • Structures having the same number and kinds of binary operations and functions are more or less ordered by the number of required unary and 0-ary (distinguished elements) operations, 0 to 2 in both cases.

The indentation structure employed in this section and the one following is intended to convey information. If structure B is under structure A and more indented, then all theorems of A are theorems of B; the converse does not hold. In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics and computer programming the arity of a function or an operator is the number of arguments or operands it takes (arity is sometimes referred to as valency, although that actually refers to another meaning of valency in mathematics). ... Partial plot of a function f. ... In mathematics, a unary operation is an operation with only one operand. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... Converse, (pronounced kŏnvûrs), is an American shoe company which has been making shoes since the early 20th century. ...


Ringoids and lattices can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the distributive law; in the case of lattices, they are linked by the absorption law. Ringoids also tend to have numerical models, while lattices tend to have set-theoretic models. The ordinary meaning of lattice is the basis for several technical usages A cherry lattice pastry A mathematical lattice that is a type of partially ordered set. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In algebra, the absorption law is an identity linking a pair of binary operations. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...


Simple structures: No binary operation: In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

  • Set: a degenerate algebraic structure having no operations.
  • Pointed set: S has one or more distinguished elements, often 0, 1, or both.
  • Unary system: S and a single unary operation over S.
  • Pointed unary system: a unary system with S a pointed set.

Group-like structures: In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a pointed space is a topological space X with a distinguised basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i. ... In mathematics, a unary operation is an operation with only one operand. ... In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic. ... A successor function is the label in the literature for what is actually an operation. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satifying a given set of identities. ... In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ... Look up induction in Wiktionary, the free dictionary. ...


One binary operation, denoted by concatenation. For monoids, boundary algebras, and sloops, S is a pointed set. In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In formal language theory (and therefore in programming languages), concatenation is the operation of joining two character strings end to end. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... The phrase Laws of Form refers to either of two things: The book, hereinafter abbreviated LoF by G. Spencer-Brown. ... In mathematics, a pointed space is a topological space X with a distinguised basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i. ...

Three binary operations. Quasigroups are listed here, despite their having 3 binary operations, because they are (nonassociative) magmas. Quasigroups feature 3 binary operations only because establishing the quasigroup cancellation property by means of identities alone requires two binary operations in addition to the group operation. In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In mathematics, associativity is a property that a binary operation can have. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a unary operation is an operation with only one operand. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... A semilattice is a mathematical concept with two definitions, one as a type of ordered set, the other as an algebraic structure. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... See: JOIN, join command in SQL, a relational database keyword. ... The phrase Laws of Form refers to either of two things: The book, hereinafter abbreviated LoF by G. Spencer-Brown. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, a unary operation is an operation with only one operand. ... In the mathematical discipline of order theory, and in particular, in lattice theory, a complemented lattice is a bounded lattice in which each element x has a complement, defined as a unique element ~ x such that and A Boolean algebra may be defined as a complemented distributive lattice. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a * b = a * c always implies b = c. ...

  • Quasigroup: a cancellative magma. Equivalently, ∀x,yS, ∃!a,bS, such that xa = y and bx = y.
    • Loop: a unital quasigroup with a unary operation, inverse.
      • Moufang loop: a loop in which a weakened form of associativity, (zx)(yz) = z(xy)z, holds.
        • Group: an associative loop.

Lattice: Two or more binary operations, including meet and join, connected by the absorption law. S is both a meet and join semilattice, and is a pointed set if and only if S is bounded. Lattices often have no unary operations. Every true statement has a dual, obtained by replacing every instance of meet with join, and vice versa. In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a * b = a * c always implies b = c. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... See: JOIN, join command in SQL, a relational database keyword. ... In algebra, the absorption law is an identity linking a pair of binary operations. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... See: JOIN, join command in SQL, a relational database keyword. ... In mathematics, a pointed space is a topological space X with a distinguised basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i. ... The word duality has a variety of different meanings in different contexts: In several spiritual, religious, and philosophical doctrines, duality refers to a two-fold division also called dualism. ...

  • Bounded lattice: S has two distinguished elements, the greatest lower bound and the least upper bound. Dualizing requires replacing every instance of one bound by the other, and vice versa.
  • Modular lattice: a lattice in which the modular identity holds.
    • Distributive lattice: a lattice in which each of meet and join distributes over the other. Distributive lattices are modular, but the converse does not hold.
      • Kleene algebra: a bounded distributive lattice with a unary operation whose identities are x"=x, (x+y)'=x'y', and (x+x')yy'=yy'. See "ring-like structures" for another structure having the same name.
      • Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.
      • Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix " ' ", and governed by the axioms x'x=1, x(x'y) = xy, x'(yz) = (x'y)(x'z), (xy)'z = (x'z)(y'z).

Ringoids: Two binary operations, addition and multiplication, with multiplication distributing over addition. Semirings are pointed sets. See lattice for other mathematical as well as non-mathematical meanings of the term. ... In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... The word duality has a variety of different meanings in different contexts: In several spiritual, religious, and philosophical doctrines, duality refers to a two-fold division also called dualism. ... In the mathematical discipline of order theory, and in particular, in lattice theory, a complemented lattice is a bounded lattice in which each element x has a complement, defined as a unique element ~ x such that and A Boolean algebra may be defined as a complemented distributive lattice. ... In the mathematical discipline of order theory, and in particular, in lattice theory, a complemented lattice is a bounded lattice in which each element x has a complement, defined as a unique element ~ x such that and A Boolean algebra may be defined as a complemented distributive lattice. ... Reverse Polish notation (RPN), also known as postfix notation, was invented by Australian philosopher and computer scientist Charles Hamblin in the mid-1950s, to enable zero-address memory stores. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. ... In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. ... In mathematics, a Kleene algebra (named after Stephen Cole Kleene, pronounced clay-knee) is either of two different things: A bounded distributive lattice with an involution satisfying De Morgans laws, and the inequality x&#8743;&#8722;x &#8804; y&#8744;&#8722;y. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI &#8804; x xII = xI (xy)I = xIyI 1I = 1 xI is called... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ... Reverse Polish notation (RPN), also known as postfix notation, was invented by Australian philosopher and computer scientist Charles Hamblin in the mid-1950s, to enable zero-address memory stores. ... In mathematics, relation algebra (RA) is an algebraic structure, a proper extension of the two-element Boolean algebra 2 intended to capture the mathematical properties of binary relations. ... In logic and mathematics, the inverse relation of a binary relation is the binary relation defined by . ... In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. X × Y = { (x, y) | x ∈ X and y... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In logic and mathematics, the composition of relations is the generalization of the composition of functions. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ... Infix has similar meanings in linguistics and mathematics. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, a pointed space is a topological space X with a distinguised basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i. ...

N.B. The above definition of ring does not command universal assent. Some authorities employ "ring" to denote what is here called a rng, and refer to a ring in the above sense as a "ring with identity." In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In abstract algebra a rng is an algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists of idempotent elements. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... In mathematics, a Kleene algebra (named after Stephen Cole Kleene, pronounced clay-knee) is either of two different things: A bounded distributive lattice with an involution satisfying De Morgans laws, and the inequality x&#8743;&#8722;x &#8804; y&#8744;&#8722;y. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In mathematical logic and computer science, the Kleene star (or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. ... Reverse Polish notation (RPN), also known as postfix notation, was invented by Australian philosopher and computer scientist Charles Hamblin in the mid-1950s, to enable zero-address memory stores. ...


Modules: Composite Systems Defined over Two Sets, M and R: The members of: In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...

  1. R are scalars, denoted by Greek letters. R is a ring under the binary operations of scalar addition and multiplication;
  2. M are module elements (often but not necessarily vectors), denoted by Latin letters. M is an abelian group under addition. There may be other binary operations.

The scalar product of scalars and module elements is a function RxMM which commutes, associates (∀r,sR, ∀xM, r(sx) = (rs)x ), has 1 as identity element, and distributes over module and scalar addition. If only the pre(post)multiplication of module elements by scalars is defined, the result is a left (right) module. In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V &#8594; F, where V is a vector space and F its underlying field. ...

  • Free module: a module having a free basis, {e1, ... en}⊂M, where the positive integer n is the dimension of the free module. For every vM, there exist κ1, ..., κnR such that v = κ1e1 + ... + κnen. Let 0 and 0 be the respective identity elements for module and scalar addition. If r1e1 + ... + rnen = 0, then r1 = ... = rn = 0.
  • Algebra over a ring (also R-algebra): a (free) module where R is a commutative ring. There is a second binary operation over M, called multiplication and denoted by concatenation, which distributes over module addition and is bilinear: α(xy) = (αx)y = xy).
  • Jordan ring: an algebra over a ring whose module multiplication commutes, does not associate, and respects the Jordan identity.

Vector spaces, closely related to modules, are defined in the next section. In mathematics, a free module is a module having a free basis. ... In mathematics, a basis or set of generators is a collection of objects that can be systematically combined to produce a larger collection of objects. ... :For other senses of this word, see dimension (disambiguation). ... In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, a Jordan algebra is defined in abstract algebra as an algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ... In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ... In mathematics, a Jordan algebra is defined in abstract algebra as a (usually nonassociative) algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...

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Structures with some axioms that are not identities

The structures in this section are not varieties because they cannot be axiomatized with identities alone. Nearly all of the nonidentities below are one of two very elementary kinds: In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satifying a given set of identities. ...

  1. The starting point for all structures in this section is a "nontrivial" ring, namely one such that S≠{0}, 0 being the additive identity element. The nearest thing to an identity implying S≠{0} is the nonidentity 0≠1, which requires that the additive and multiplicative identities be distinct.
  2. Nearly all structures described in this section include identities that hold for all members of S except 0. In order for an algebraic structure to be a variety, its operations must be defined for all members of S; there can be no partial operations.

Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and vector spaces. Moreover, much of theoretical physics can be recast as models of multilinear algebras. Although structures with nonidentities retain an undoubted algebraic flavor, they suffer from defects varieties do not have. For example, neither the product of integral domains nor a free field over any set exist. In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, multilinear algebra extends the methods of linear algebra. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...


Arithmetics: Two binary operations, addition and multiplication. S is an infinite set. Arithmetics are pointed unary systems, whose unary operation is injective successor, and with distinguished element 0. In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In set theory, an infinite set is a set that is not a finite set. ... In mathematics, a unary operation is an operation with only one operand. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... A successor function is the label in the literature for what is actually an operation. ...

  • Robinson arithmetic. Addition and multiplication are recursively defined by means of successor. 0 is the identity element for addition, and annihilates multiplication. Robinson arithmetic is listed here even though it is a variety, because of its closeness to Peano arithmetic.
    • Peano arithmetic. Robinson arithmetic with an axiom schema of induction. Most ring and field axioms bearing on the properties of addition and multiplication are theorems of Peano arithmetic or of proper extensions thereof.

Field-like structures: Two binary operations, addition and multiplication. S is nontrivial, i.e., S≠{0}. In mathematics, Robinson arithmetic, or Q, is a fragment of the theory of the natural numbers, set out in R. M. Robinson (1950). ... See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page — a list of pages that otherwise might share the same title. ... A successor function is the label in the literature for what is actually an operation. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ... Look up induction in Wiktionary, the free dictionary. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

The following structures are not varieties for reasons in addition to S≠{0}: In abstract algebra, a domain is the noncommutative analogue of an integral domain. ... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a*b = a*c always implies b = c. ... In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, an ordered field is a field (F,+,*) together with a total order &#8804; on F that is compatible with the algebraic operations in the following sense: if a &#8804; b then a + c &#8804; b + c if 0 &#8804; a and 0 &#8804; b then 0 &#8804; a... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... Please refer to Real vs. ... In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x &#8804; a, then x is in A as well) and B...

Composite Systems: Vector Spaces, and Algebras over Fields. Two Sets, M and R, and at least three binary operations. In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X... In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 &#8804; a2 &#8804; ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ...


The members of:

  1. M are vectors, denoted by lower case letters. M is at minimum an abelian group under vector addition, with distinguished member 0.
  2. R are scalars, denoted by Greek letters. R is a field, nearly always the real or complex field, with 0 and 1 as distinguished members.

Three binary operations. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Please refer to Real vs. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

Four binary operations. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a free module is a module having a free basis. ... :For other senses of this word, see dimension (disambiguation). ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... Link titleIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... The word norm coming from the latin word norma which means angle measure or (lawlike) rule, has a number of meanings: A social or sociological norm; see norm (sociology). ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... The word linear comes from the Latin word linearis, which means created by lines. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Please refer to Real vs. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, an orthonormal basis of an inner product space V(i. ... In mathematics, a free module is a module having a free basis. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Conjugate can be: in mathematics in terms of complex numbers, the complex conjugate; more generally see conjugate element (field theory). ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, a graded vector space is a vector space with an extra piece of structure, known as a grading. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ...

Composite Systems: Multilinear algebras. Two sets, V and K. Four binary operations: In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ... In mathematics, a Jordan algebra is defined in abstract algebra as an algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ... In mathematics, a Jordan algebra is defined in abstract algebra as an algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... A module is a self-contained component of a system, which has a well-defined interface to the other components; something is modular if it includes or uses modules which can be interchanged as units without disassembly of the module. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... Look up matrix in Wiktionary, the free dictionary. ... :For other senses of this word, see dimension (disambiguation). ... This article gives an overview of the various ways to multiply matrices. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, an orthonormal basis of an inner product space V(i. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules based on such rings; and of fields and their algebras. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, multilinear algebra extends the methods of linear algebra. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

  1. The members of V are multivectors (including vectors), denoted by lower case Latin letters. V is an abelian group under multivector addition, and a monoid under outer product. The outer product goes under various names, and is multilinear in principle but usually bilinear. The outer product defines the multivectors recursively starting from the vectors. Thus the members of V have a "degree" (see graded algebra below). Multivectors may have an inner product as well, denoted uv: V×VK, that is symmetric, linear, and positive definite; see inner product space above.
  2. The properties and notation of K are the same as those of R above, except that K may have -1 as a distinguished member. K is usually the real field, as multilinear algebras are designed to describe physical phenomena without complex numbers.
  3. The multiplication of scalars and multivectors, V×KV, has the same properties as the multiplication of scalars and module elements that is part of a module.
  • Graded algebra: an associative algebra with unital outer product. The members of V have a direct sum decomposition resulting in their having a "degree," with vectors having degree 1. If u and v have degree i and j, respectively, the outer product of u and v is of degree i+j. V also has a distinguished member 0 for each possible degree. Hence all members of V having the same degree form an Abelian group under addition.
    • Exterior algebra (also Grassmann algebra): a graded algebra whose anticommutative outer product, denoted by infix ∧, is called the exterior product. V has an orthonormal basis. v1v2 ∧ ... ∧ vk = 0 if and only if v1, ..., vk are linearly dependent. Multivectors also have an inner product.
      • Clifford algebra: an exterior algebra with a symmetric bilinear form Q: V×VK. The special case Q=0 yields an exterior algebra. The exterior product is written 〈u,v〉. Usually, 〈ei,ei〉 = -1 (usually) or 1 (otherwise).
      • Geometric algebra: an exterior algebra whose exterior (called geometric) product is denoted by concatenation. The geometric product of parallel multivectors commutes, that of orthogonal vectors anticommutes. The product of a scalar with a multivector commutes. vv yields a scalar.
        • Grassmann-Cayley algebra: a geometric algebra without an inner product.
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A multivector is an element of a geometric algebra , most generally a summation of elements of different or mixed grade such as the summation of a scalar, a vector, and a bivector although a few authors favour polyvector for mixed grade elements, reserving multivector for geoemetric algebra elements consisting of... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... A multivector is an element of a geometric algebra , most generally a summation of elements of different or mixed grade such as the summation of a scalar, a vector, and a bivector although a few authors favour polyvector for mixed grade elements, reserving multivector for geoemetric algebra elements consisting of... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ... In mathematics, the modern component-free approach to the theory of tensors views tensors initially as abstract objects, expressing some definite type of multi-linear concept. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... The word linear comes from the Latin word linearis, which means created by lines. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Please refer to Real vs. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... A module is a self-contained component of a system, which has a well-defined interface to the other components; something is modular if it includes or uses modules which can be interchanged as units without disassembly of the module. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... A mathematical operator (typically a binary operator, represented by *) is anticommutative iff it is true that x * y = &#8722;(y * x) for all x and y on the operators valid domain (e. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, an orthonormal basis of an inner product space V(i. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... Clifford algebras are a type of associative algebra in mathematics. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ... A geometric algebra is a multilinear algebra with a geometric interpretation. ... Grassmann-Cayley algebra is a form of modelling algebra for projective geometry, based on work by German mathematician Hermann Grassmann on exterior algebra, and, subsequently, by British mathematician Arthur Cayleys work on matrices and linear algebra. ...

Examples

Some recurring universes: N=natural numbers; Z=integers; Q=rational numbers; R=real numbers; C=complex numbers. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... The integers consist of the positive natural numbers (1, 2, 3, &#8230;) the negative natural numbers (&#8722;1, &#8722;2, &#8722;3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...


N is a pointed unary system, and under addition and multiplication, is both the standard interpretation of Peano arithmetic and a commutative semiring. In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ...


Boolean algebras are at once semigroups, lattices, and rings. They would even be Abelian groups if the identity and inverse elements were identical instead of complements. In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... The ordinary meaning of lattice is the basis for several technical usages A cherry lattice pastry A mathematical lattice that is a type of partially ordered set. ... Look up ring in Wiktionary, the free dictionary. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...


Group-like structures

  • Nonzero N under addition (+) is a magma.
  • N under addition is a magma with an identity.
  • Z under subtraction (−) is a quasigroup.
  • Nonzero Q under division (÷) is a quasigroup.
  • Every group is a loop, because a * x = b if and only if x = a−1 * b, and y * a = b if and only if y = b * a−1.
  • 2x2 matrices(of non-zero determinant) with matrix multiplication form a group.
  • Z under addition (+) is an Abelian group.
  • Nonzero Q under multiplication (×) is an Abelian group.
  • Every cyclic group G is Abelian, because if x, y are in G, then xy = aman = am+n = an+m = anam = yx. In particular, Z is an Abelian group under addition, as is the integers modulo n Z/nZ.
  • A monoid is a category with a single object, in which case the composition of morphisms and the identity morphism interpret monoid multiplication and identity element, respectively.
  • The Boolean algebra 2 is a boundary algebra.
  • More examples of groups and list of small groups.

Lattices 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... 5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, multiplication is an elementary arithmetic operation. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... -1... Some elementary examples of groups in mathematics are given on Group (mathematics). ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ... The ordinary meaning of lattice is the basis for several technical usages A cherry lattice pastry A mathematical lattice that is a type of partially ordered set. ...

Ring-like structures In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics a field of sets is a pair where is a set and is an algebra over i. ... In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ... It has been suggested that Predicate calculus be merged into this article or section. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ... A modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ... S4 can refer to: SATA International S4 algebra - a variety of modal logic, also called Interior algebra the Audi S4 car This is a disambiguation page — a list of pages that otherwise might share the same title. ... In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI &#8804; x xII = xI (xy)I = xIyI 1I = 1 xI is called... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... This article or section is in need of attention from an expert on the subject. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiom system for set theory designed to yield the same results as Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC), but with only a finite number of axioms, that is without axiom schemas. ... In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. ... In mathematics, relation algebra (RA) is an algebraic structure, a proper extension of the two-element Boolean algebra 2 intended to capture the mathematical properties of binary relations. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

  • The set R[X] of all polynomials over some coefficient ring R is a ring.
  • 2x2 matrices with matrix addition and multiplication form a ring.
  • If n is a positive integer, then the set Zn = Z/nZ of integers modulo n (the additive cyclic group of order n ) forms a ring having n elements (see modular arithmetic).

Integral domains In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...

  • Z under addition and multiplication is an integral domain.
  • The p-adic integers.

Fields The p-adic number systems were first described by Kurt Hensel in 1897. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

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In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, an ordered field is a field (F,+,*) together with a total order &#8804; on F that is compatible with the algebraic operations in the following sense: if a &#8804; b then a + c &#8804; b + c if 0 &#8804; a and 0 &#8804; b then 0 &#8804; a... The categorical imperative is the philosophical concept central to the moral philosophy of Immanuel Kant and to modern deontological ethics. ... Please refer to Real vs. ... Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the subset of the real numbers consisting of the numbers which can be computed by a finite, terminating algorithm. ... A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (in the von Neumann universe V). ... In mathematics, an algebraic number field (or simply number field) is a finite-dimensional (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension, or degree, when considered as a vector space over Q. The study of... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

Allowing additional structure

Algebraic structures can also be defined on sets with added structure of a non-algebraic nature, such as a topology. The added structure must be compatible, in some sense, with the algebraic structure. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...

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In abstract algebra, an ordered group is a group G equipped with a partial order &#8804; which is translation-invariant; in other words, &#8804; has the property that, for all a, b, and g in G, if a &#8804; b then ag &#8804; bg and ga &#8804; gb. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, a linearly ordered group is both a group and a linearly ordered set, in which the group operation is in a certain sense compatible with the linear ordering. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, the Archimedean property of an ordered algebraic structure, such as a linearly ordered group, and in particular of the real numbers, is the property of having no (non-zero) infinitesimals. ... In mathematics, the Archimedean property of an ordered algebraic structure, such as a linearly ordered group, and in particular of the real numbers, is the property of having no (non-zero) infinitesimals. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Link titleIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...

Category theory

The discussion above has been cast in terms of elementary abstract and universal algebra. Category theory is another way of reasoning about algebraic structures (see, for example, Mac Lane 1998). A category is a collection of objects with associated morphisms. Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This concrete category may be seen as a category of sets with added category-theoretic structure. Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms) is a category of topological spaces with extra structure. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In abstract algebra, a homomorphism is a structure-preserving map. ... Partial plot of a function f. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In mathematics, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... The category Top has topological spaces as objects and continuous maps as morphisms. ...


There are various concepts in category theory that try to capture the algebraic character of a context, for instance

  • algebraic
  • essentially algebraic
  • presentable
  • locally presentable
  • monadic functors and categories
  • universal property.
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In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

See also

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Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics and computer programming the arity of a function or an operator is the number of arguments or operands it takes (arity is sometimes referred to as valency, although that actually refers to another meaning of valency in mathematics). ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... The idea of a free object in mathematics is one of the basics of abstract algebra. ... In mathematical logic, a first-order theory is given by a set of axioms in some language. ... In mathematics, a signature for an algebraic structure A over a set S is a list of the operations that characterize A, along with their arities. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ... In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satifying a given set of identities. ...

References

  • Garrett Birkhoff and Saunders MacLane, 1999 (1967). Algebra, 2nd ed. Chelsea.
  • Michel, Anthony N., and Herget, Charles J., 1993 (1981). Applied Algebra and Functional Analysis. Dover.

A monograph available free online: Garrett Birkhoff (January 19, 1911, Princeton, New Jersey, USA - November 22, 1996, Water Mill, New York, USA) was an American mathematician. ... Saunders Mac Lane (born 4 August 1909) is a US mathematician. ...

Category theory:

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Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ... Categories for the Working Mathematician is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. ...

External links


  Results from FactBites:
 
Guide to the Mathematics Subject Classification Scheme (5525 words)
Algebra is principally concerned with symmetry, patterns, discrete sets, and the rules for manipulating arithmetic operations; one might think of this as the outgrowth of arithmetic and algebra classes in primary and secondary school.
55: Algebraic topology is the study of algebraic objects attached to topological spaces; the algebraic invariants illustrate some of the rigidity of the spaces.
One might characterize algebra and geometry as the search for elegant conclusions from small sets of axioms; in analysis on the other hand the measure of success is more frequently the ability to hone a tool which could be applied throughout science.
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