What are semigroups?
Semigroups and monoids can be viewed as both "groups without inverses" and "rings without addition". Their study typically involves borrowing methods from group theory and ring theory, as well as ideas bespoke to the semigroup case. Their relatively unconstrained structure makes them ubiquitous and highly applicable - wherever there are non-invertible functions, there are usually semigroups to be found - but also hard to study in complete generality. As a result, much of the theory is specialised to particular classes of semigroups which arise frequently and are sufficiently constrained to admit a powerful theory.
Perhaps chief among these are the inverse semigroups: those in which each element \(x\) admits a unique "weak inverse" \(x'\) such that \(x x' x = x\) and \(x' x x'=x'\). Inverse semigroups model partial symmetry (symmetries between parts of a structure) in the same way that groups model total symmetry. Basic examples include the monoid of bijections between subsets of a given set (the symmetric inverse monoid) and the bicyclic monoid generated by two elements \(p\) and \(q\) such that \(pq=1\) (but \(qp \neq 1\)!). Other major areas of research include classes of semigroups with even weaker notions of inverses and/or cancellativity conditions, semigroups defined by (typically finitely many) generators and relations, linear representations of semigroups, and numerous interactions with theoretical computer science.