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Encyclopedia > Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. A semigroup is, in effect, an associative groupoid. Euclid, detail from The School of Athens by Raphael. ... In higher mathematics, algebraic structure is a loosely-defined phrase referring to the mathematical objects traditionally studied in the field of abstract algebra: sets with operations. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...


Juxtaposition suffices to denote the semigroup operation. That is, xy denotes the result of applying the semigroup operation to the ordered pair (xy). Juxtaposition (noun) is an act or instance of placing two things close together or side by side. ...


A semigroup with an identity element is a monoid. Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining es = s = se for all sS ∪ {e}. Some require that a semigroup have an identity element, which would render semigroups identical to monoids. Moreover, Not all agree that S should be nonempty. This entry assumes that a semigroup may be empty, and need not have an identity. 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 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 abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...


Examples of semigroups

  • The positive integers with addition.
  • Any monoid, and therefore any group.
  • Any ideal of a ring, with the operation of multiplication. (Thus, any ring, including the integers, rational, real, complex or quaternionic numbers, functions with values in a ring (including sequences), polynomials and matrices.)
  • Any subset of a semigroup which is closed under the semigroup operation.
  • The set of all finite strings over some fixed alphabet Σ, with string concatenation as operation. If the empty string is included, then this is actually a monoid, called the "free monoid over Σ"; if it is excluded, then we have a semigroup, called the "free semigroup over Σ".
  • The bicyclic semigroup.
  • C0-semigroups.
  • Matrices:
  • A semigroup that has an idempotent operation is a band.
  • A semigroup that has a commutative idempotent operation is a semilattice.
  • A transformation semigroup : any finite semigroup S can be represented by transformations of a (state-) set Q of at most |S|+1 states. Each element x of S then maps Q into itself x: QQ and sequence xy is defined by q(xy) = (qx)y for each q in Q. Sequencing clearly is an associative operation, here equivalent to function composition. This representation is basic for any automaton or finite state machine (FSM).
  • All subsets of a group that contain the identity form a semigroup with elementwise multiplication

The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... 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 integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... In computer programming and some branches of mathematics, strings are sequences of various simple objects. ... In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from A, with the binary operation of concatenation. ... In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from A, with the binary operation of concatenation. ... In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. ... In mathematics, a C0-semigroup is a continuous morphism from (R+,+) into a topological monoid, usually L(H), the algebra of linear continuous operators on some Hilbert space H. Thus, strictly speaking, not the C0-semigroup, but rather its image, is a semigroup. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ... In mathematics, a matrix unit is an idealisation of the concept of a matrix, with a focus on the algebraic properties of matrix multiplication. ... A nonnegative matrix is a matrix where all the elements are equal to or above zero A positive matrix is defined similarly. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... 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 mathematical order theory, a semilattice is a partially ordered set (poset) within which either all binary sets have a supremum (join) or all binary sets have an infimum (meet). ... In mathematics, a transformation semigroup is a collection of mappings on a set X closed under composition. ... A drummer automaton An automaton (plural: automata) is a self-operating machine. ... Fig. ...

Structure of semigroups

This section sets out concepts useful for understanding the structure of semigroups. Two semigroups S and T are said to be isomorphic if there is a bijection f : ST with the property that, for any elements a, b in S, f(ab) = f(a)f(b). In this case, T and S are also isomorphic, and for the purposes of semigroup theory, the two semigroups are identical. 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. ... A bijective function. ...


If A and B are subsets of some semigroup, then AB denotes the set { ab | a in A and b in B }. A subset A of a semigroup S is called a subsemigroup if it is closed under the semigroup operation, that is, AA is a subset of A. If A is nonempty then A is called a right ideal if AS is a subset of A, and a left ideal if SA is a subset of A. If A is both a left ideal and a right ideal then it is called an ideal (or a two-sided ideal). The intersection of two ideals is also an ideal, so a semigroup can have at most one minimal ideal. An example of semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a commutative semigroup, when it exists, is a group. 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. ...


Green's relations are important tools for analysing the ideals of a semigroup, and related notions of structure. In mathematics, Greens relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. ...


If S is a semigroup, then the intersection of any collection of subsemigroups of S is also a subsemigroup of S. So the subsemigroups of S form a complete lattice. For any subset A of S there is a smallest subsemigroup T of S which contains A, and we say that A generates T. A single element x of S generates the subsemigroup { xn | n is a positive integer }. If this is finite, then x is said to be of finite order, otherwise it is of infinite order. A semigroup is said to be periodic if all of its elements are of finite order. A semigroup generated by a single element is said to be monogenic (or cyclic). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive integers with the operation of addition. If it is finite and nonempty, then it must contain at least one idempotent. It follows that every nonempty periodic semigroup has at least one idempotent. In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...


A subsemigroup which is also a group is called a subgroup. There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each idempotent e of the semigroup there is a unique maximal subgroup containing e. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here the term maximal subgroup differs from its standard use in group theory. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. ...


More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimal ideal and at least one idempotent. For more on the structure of finite semigroups, see Krohn-Rhodes theory. In mathematics, the term ideal has multiple meanings. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... Krohn-Rhodes theory is an approach to the study of finite semigroups and automata, which seeks to decompose them in terms of finite aperiodic semigroups and finite groups. ...


  Results from FactBites:
 
NationMaster - Encyclopedia: Bicyclic semigroup (1125 words)
In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups.
The bicyclic semigroup has the property that the image of any morphism φ from B to another semigroup S is either cyclic, or it is an isomorphic copy of B.
A semigroup whose operation is idempotent and commutative is a semilattice.
NationMaster - Encyclopedia: Free semigroup (1181 words)
As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups.
A semigroup with an identity element is a monoid.
Each free semigroup (or monoid) S has exactly one set of free generators, the cardinality of which is called the rank of S. As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups.
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