subsemigrupp

A subsemigroup is a subset of a semigroup that itself forms 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 and only if T is closed under the operation *. This means that for any two elements a and b in T, the product a * b is also in T.

Subsemigroups are a fundamental concept in the study of semigroups, which are algebraic structures consisting of a set equipped with an associative binary operation. They play a crucial role in the classification and understanding of semigroups, as well as in various applications such as automata theory and formal language theory.

Every semigroup has at least two subsemigroups: the semigroup itself and the trivial subsemigroup consisting of a single element, which is the identity element if it exists. A subsemigroup that is also a group is called a subgroup. The study of subsemigroups and their properties is an active area of research in semigroup theory.