Semigroup

In mathematics, a semigroup is a set with an associative binary operation on it.

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There is some disagreement on whether the empty set should be admitted as a

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semigroup. Many authors insist that a semigroup should be non-empty, and some even require an identity element. In this article, we shall assume that a semigroup may be empty and need not have an identity.

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A semigroup with an identity element is called 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 ∪ {e}.

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Some examples of semigroups:

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• 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.
• Matrix units form a 0-simple semigroup.
• 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: 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).

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.

Semigroup: Examples of semigroups ►



 

Set: :This article is about sets in mathematics. For other senses, see set (disambiguation)....

Associative: REDIRECT Associativity...

Binary operation: In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities. Binary operations are sometimes called dyadic operations in order to avoid confusion with the binary numeral system. Examples include the familiar arithmetic operations of addition, subtraction, ...