inverse semigroup

inverse semigroup (plural inverse semigroups)

1. (algebra, group theory) A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy.
   • 2002, Peter G. Trotter, “A.9: Regular Semigroups”, in Aleksandr Vasilʹevich Mikhalev, Günter Pilz, editors, The Concise Handbook of Algebra, page 35:

     Examples of regular semigroups include any band, inverse semigroup or completely regular semigroup (see sections A. 2, A. 11, and A. 10); in particular, any group is a regular semigroup.

   • 2008, Olexandr Ganyushkin, Volodymyr Mazorchuk, Classical Finite Transformation Semigroups: An Introduction, page v:

     Inverse semigroups form a class of semigroups which are closest (in some sense) to groups.

   • 2014, Christopher Hollings, Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, page 249:

     Inverse semigroups are central to modern semigroup theory: arguably, they form the most-studied class of semigroups.

• regular semigroup