Semigroup

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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 magma.

Juxtaposition suffices to denote the semigroup operation. That is, xy denotes the result of applying the semigroup operation to the ordered pair (x, y).

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 s ∈ S ∪ . Some require that a semigroup have an identity element, which would render semigroups identical to monoids. This does not assume that S is nonempty. If it is nonempty, its members need not include an identity.

Examples of semigroups

The positive integers with addition.
Any monoid, and therefore any group.
Any subset of a semigroup closed under the semigroup operation.
All subsets of a group that contain the identity form a semigroup with elementwise multiplication.
A semigroup whose operation is idempotent is a band.
A semigroup whose operation is idempotent and commutative is a semilattice.
Any ideal of a ring, given multiplication. Thus any ring including the integers, rational, real, complex or quaternionic numbers, functions with values in a ring (including sequences), polynomials and matrices.
* Matrix units form a 0-simple semigroup.
* Square nonnegative matrices with matrix multiplication.
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".
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: Q → Q 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).
The bicyclic semigroup.
C₀-semigroups.
Rectangular bands

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 : S ↔ T 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.

If A and B are subsets of some semigroup, then AB denotes the set . 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.

Green's relations are important tools for analysing the ideals of a semigroup, and related notions of structure.

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 . 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.

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.

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.

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