Subsemigrupo

Subsemigrupo, also known as subsemigroup, is a subset of a semigroup that is itself a semigroup under the same operation. In other words, if S is a semigroup and T is a subset of S, then T is a subsemigroup of S if for all a, b in T, the product ab is also in T.

The concept of a subsemigroup is fundamental in semigroup theory and has applications in various areas of mathematics, computer science, and engineering. For instance, in formal language theory, the set of all words of even length over a given alphabet forms a subsemigroup of the free semigroup over that alphabet.

A subsemigroup T of a semigroup S is said to be trivial if T = {e}, where e is the identity element of S, or if T = S. Otherwise, T is said to be non-trivial. The study of subsemigroups includes the investigation of their properties, such as the existence of idempotents, and their relationship with other algebraic structures.

In the context of semigroups with zero, a subsemigroup that contains the zero element is called an ideal. Ideals play a crucial role in the structure theory of semigroups with zero. The intersection of all ideals containing a given subsemigroup is called the ideal generated by that subsemigroup.

The concept of a subsemigroup can be extended to other algebraic structures, such as groups and rings, leading to the notions of subgroup and subring, respectively. However, these extensions are not direct generalizations, as the properties of subsemigroups do not necessarily carry over to subgroups or subrings.