Automatic semigroup

In mathematics, an automatic semigroup is a finitely generated semigroup equipped with several regular languages over an alphabet representing a generating set. One of these languages determines "canonical forms" for the elements of the semigroup, the other languages determine if two canonical forms represent elements that differ by multiplication by a generator.

Formally, let ${\displaystyle S}$ be a semigroup and ${\displaystyle A}$ be a finite set of generators. Then an automatic structure for ${\displaystyle S}$ with respect to ${\displaystyle A}$ consists of a regular language ${\displaystyle L}$ over ${\displaystyle A}$ such that every element of ${\displaystyle S}$ has at least one representative in ${\displaystyle L}$ and such that for each ${\displaystyle a\in A\cup \{\varepsilon \}}$, the relation consisting of pairs ${\displaystyle (u,v)}$ with ${\displaystyle ua=v}$ is regular.

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