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Encyclopedia > Universal algebra

Universal algebra (sometimes called General algebra) is the field of mathematics that studies the ideas common to all algebraic structures. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... 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. ...

## Contents

From the point of view of universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) can be represented simply as an element of A, or a constant, often denoted by a letter like a. A 1-ary operation (or unary operation) is simply a function from A to A, often denoted by a symbol placed in front of its argument, like ~x. A 2-ary operation (or binary operation) is often denoted by a symbol placed between its arguments, like x * y. Operations of higher or unspecified arity are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like f(x,y,z) or f(x1,...,xn). Some researchers allow infinitary operations, such as $bigwedge_{alphain J} x_alpha$ where J is an infinite index set, thus leading into the algebraic theory of complete lattices. One way of talking about an algebra, then, is by referring to it as an algebra of a certain type Ω, where Ω is an ordered sequence of natural numbers representing the arity of the operations of the algebra. This article is about sets in mathematics. ... The mathematical term arity sprang from words like unary, binary, ternary, etc. ... In logic and mathematics, an operation Ï‰ is a function of the form Ï‰ : X1 Ã— â€¦ Ã— Xk â†’ Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... Partial plot of a function f. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... In mathematics, a unary operation is an operation with only one operand. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics or logic, a finitary operation is one, like those of arithmetic, that take a number of input values to produce an output. ... In mathematics, an index set is another name for a function domain. ... In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ...

After the operations have been specified, the nature of the algebra can be further limited by axioms, which in universal algebra often take the form of equational laws. An example is the associative axiom for a binary operation, which is given by the equation x * (y * z) = (x * y) * z. The axiom is intended to hold for all elements x, y, and z of the set A. This article does not adequately cite its references or sources. ... In mathematics, associativity is a property that a binary operation can have. ...

Universal algebra can be seen as a special branch of model theory, in which we are dealing with structures having operations only (i.e., no relations except for equality), and in which the language used to talk about these structures uses equations only. On the other hand the structures are such that they can be defined in any category which has finite products. In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... In mathematics, a finitary relation is defined by one of the formal definitions given below. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

## Examples

Most of the usual algebraic systems of mathematics are examples of universal algebras, but not always in an obvious way.

### Groups

To see how this works, let's consider the definition of a group. Normally a group is defined in terms of a single binary operation *, subject to these axioms: This picture illustrates how the hours in a clock form a group. ...

• Associativity (as in the previous paragraph): x * (y * z)  =  (x * y) * z.
• Identity element: There exists an element e such that e * x  =  x  =  x * e.
• Inverse element: For each x, there exists an element i such that x * i  =  e  =  i * x.

(Sometimes you will also see an axiom called "closure", stating that x * y belongs to the set A whenever x and y do. But from a universal algebraist's point of view, that is already implied when you call * a binary operation.) In mathematics, associativity is a property that a binary operation can have. ... 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 inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...

Now, this definition of a group is problematic from the point of view of universal algebra. The reason is that the axioms of the identity element and inversion are not stated purely in terms of equational laws but also have clauses involving the phrase "there exists ... such that ...". This is inconvenient; the list of group properties can be simplified to universally quantified equations if we add a nullary operation e and a unary operation ~ in addition to the binary operation *, then list the axioms for these three operations as follows:

• Associativity: x * (y * z)  =  (x * y) * z.
• Identity element: e * x  =  x  =  x * e.
• Inverse element: x * (~x)  =  e  =  (~x) * x.

(Of course, we usually write "x -1" instead of "~x", which shows that the notation for operations of low arity is not always as given in the second paragraph.) The mathematical term arity sprang from words like unary, binary, ternary, etc. ...

It's important to check that this really does capture the definition of a group. The reason that it might not is that specifying one of these universal groups might give more information than specifying one of the usual kind of group. After all, nothing in the usual definition said that the identity element e was unique; if there is another identity element e', then it's ambiguous which one should be the value of the nullary operator e. However, this is not a problem, because identity elements can be proved to be always unique. The same thing is true of inverse elements. So the universal algebraist's definition of a group really is equivalent to the usual definition. 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 inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...

## Basic constructions

We assume that the type, Ω, has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.

A homomorphism between two algebras A and B is a function h: A → B from the set A to the set B such that, for every operation f (of arity, say, n), h(fA(x1,...,xn)) = fB(h(x1),...,h(xn)). (Here, subscripts are placed on f to indicate whether it is the version of f in A or B. In theory, you could tell this from the context, so these subscripts are usually left off.) For example, if e is a constant (nullary operation), then h(eA) = eB. If ~ is a unary operation, then h(~x) = ~h(x). If * is a binary operation, then h(x * y) = h(x) * h(y). And so on. A few of the things that can be done with homomorphisms, as well as definitions of certain special kinds of homomorphisms, are listed under the entry Homomorphism. In particular, we can take the homomorphic image of an algebra, h(A). In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ... Partial plot of a function f. ... In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ...

A subalgebra of A is a subset of A that is closed under all the operations of A. A product of some set of algebraic structures is the cross product of the sets with the operations defined coordinatewise. In mathematics, the Cartesian product is a direct product of sets. ...

## Some Basic Theorems

In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ... This picture illustrates how the hours in a clock form a group. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... 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 universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satifying a given set of identities. ... In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satifying a given set of identities. ...

## Motivations and applications

In addition its unifying approach, Universal algebra also gives deep theorems and important examples and counterexamples. It provides a useful framework for those who intend to start the study new classes of algebras. It can enable the use of methods invented for some particular classes of algebras to other classes of algebras, by recasting the method in terms of universal algebra (if possible), and then interpreting these as applied to other classes. It has also provided conceptual clarification; as J.D. H. Smith puts it, "What looks messy and complicated in a particular framework may turn out to be simple and obvious in the proper general one."

In particular, universal algebra can be applied to the study of monoids, rings, and lattices. Before universal algebra came along, many theorems (most notably the isomorphism theorems) were proved separately in all of these fields, but with universal algebra, you can prove them once and for all for every kind of algebraic system. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ...

## Further issues

A more generalised program along these lines is carried out by category theory. Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category. Category theory applies to many situations where universal algebra does not, extending the reach of the theorems. Conversely, some theorems that hold in universal algebra do not generalise all the way to category theory. Thus both fields of study are useful. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

This is a list of important publications in mathematics, organized by field. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ... See lattice for other mathematical as well as non-mathematical meanings of the term. ... 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. ... In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satifying a given set of identities. ... Look up universal in Wiktionary, the free dictionary. ... Operad theory is a field of abstract algebra concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. ...

## References

• Bergman, George M.: An Invitation to General Algebra and Universal Constructions (pub. Henry Helson, 15 the Crescent, Berkeley CA, 94708) 1998, 398 pp. ISBN 0-9655211-4-1.
• Burris, Stanley N., and H.P. Sankappanavar: A Course in Universal Algebra Springer-Verlag, 1981 ISBN 3-540-90578-2. Free online edition.
• Cohn, Paul Moritz, 1981. Universal Algebra. Dordrecht , Netherlands: D.Reidel Publishing. ISBN 90-277-1213-1 (First published in 1965 by Harper & Row)
• Freese, Ralph, and Ralph McKenzie, 1987. Commutator Theory for Congruence Modular Varieties, 1st ed. London Mathematical Society Lecture Note Series, 125. Cambridge Univ. Press. ISBN 0-521-34832-3. Free online second edition.
• Hobby, David, and Ralph McKenzie, 1988. The Structure of Finite Algebras American Mathematical Society. ISBN 0-8218-3400-2. (Available free online as of May 31, 2006).
• Jipsen, Peter, and Henry Rose, 1992. Varieties of Lattices, Lecture Notes in Mathematics 1533. Springer Verlag. ISBN 0-387-56314-8. Free online edition.
• Smith, J.D.H. Mal'cev Varieties, Springer-Verlag, 1976.

Results from FactBites:

 Variety (universal algebra) - Wikipedia, the free encyclopedia (683 words) In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Those equations are statements from the predicate calculus involving universal quantifiers and equality only: each is a mathematical identity enforced in each model, for example the commutative law, or the absorption law. It is simple to see that the class of algebras satisfying a given set of equations will always be closed under the HSP operations, so the burden of Birkhoff's theorem is the converse: classes of algebras that satisfy those conditions must be equational.
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