## On finite presentations of inverse semigroups with zero having polynomial growth

### L.M. Shneerson and D. Easdown

#### Abstract

We study growth of inverse semigroups defined by finite presentations. Let $$S$$ be a finitely presented Rees quotient of a free inverse semigroup given by an irredundant presentation with $$n$$ generators and $$m$$ relators. We show that if $$S$$ has polynomial growth, then $$m\ge n^2-1$$ and this estimate is sharp. For any positive integer $$n$$, we also find, up to isomorphism, syntactic descriptions of all presentations that achieve this sharp lower bound. As part of the process, we describe all irredundant presentations of finite Rees quotients of free inverse semigroups having rank $$n$$, with the smallest number, namely $$n^2$$, of relators.

Keywords: free inverse semigroup, Rees quotient, inverse semigroup presentation, growth.

: Primary 20M18.

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 Wednesday, January 23, 2019