Käänteisalkioksi

Käänteisalkioksi, also known as an inverse semigroup, is a mathematical structure in abstract algebra. It is a generalization of a semigroup. A semigroup is a set equipped with an associative binary operation. An inverse semigroup adds a further condition related to the existence of "inverses" for its elements.

Formally, an inverse semigroup is a semigroup S where for every element a in S, there exists a unique element a⁻¹ in S such that aa⁻¹a = a and a⁻¹aa⁻¹ = a⁻¹. This element a⁻¹ is called the inverse of a. The conditions aa⁻¹a = a and a⁻¹aa⁻¹ = a⁻¹ are the defining properties of an inverse element in this context.

A key property of inverse semigroups is that their idempotents (elements e such that ee = e) form a commutative subsemigroup. This means that if e and f are idempotents in an inverse semigroup, then ef = fe. This commutativity of idempotents is a significant characteristic that distinguishes inverse semigroups from general semigroups.

The study of inverse semigroups has connections to various areas of mathematics, including formal language theory, automata theory, and category theory. They provide a framework for understanding structures with a form of "cancellability" and the existence of inverses, albeit not in the strict sense of group theory where every element has an inverse such that aa⁻¹ = a⁻¹a = e. Examples of inverse semigroups include the set of partial bijections on a set with the operation of composition, and the semigroup of relations on a set.