Inverse semigroup

Summary

An inverse semigroup is a semigroup in which each element $a$ has exactly one element $a^{-1}$ with $a a^{-1} a = a$ and $a^{-1} a a^{-1} = a^{-1}$. The model example is the semigroup of partial bijections of a set, just as the model group is a group of permutations. By the Wagner–Preston theorem, every inverse semigroup embeds in a semigroup of partial bijections. Inverse semigroups describe partial symmetries, and they appear in the theory of tilings and of operator algebras.