On generic properties of finitely presented monoids and semigroups

TL;DR

This paper investigates the typical properties of finitely presented monoids and semigroups, showing that generically they satisfy certain small overlap conditions, are torsion-free, J-trivial, and have efficiently solvable word problems.

Contribution

It establishes that generic finitely presented monoids and semigroups satisfy the small overlap condition C(m), leading to linear-time solutions for their word problems and regular normal forms.

Findings

Generic monoids satisfy small overlap condition C(m)

Word problem solvable in linear time for generic monoids

Normal forms form a regular language

Abstract

We study the generic properties of finitely presented monoids and semigroups. We show that for positive integers a > 1, k and m, the generic a-generator k-relation monoid and semigroup presentation (defined in any of several definite statistical senses) satisfy the small overlap condition C(m). It follows that the generic monoid is torsion-free and J-trivial and, by a recent result of the author, admits a linear time solution to its word problem and a regular language of unique normal forms for its elements. Moreover, the uniform word problem for finitely presented monoids is generically solvable in time linear in the word lengths and quadratic in the presentation size. We also prove some technical results about generic sets which may be of independent interest.