Reducibility of pointlike problems

TL;DR

This paper investigates the reducibility of pointlike problems in finite semigroup pseudovarieties, establishing conditions under which these problems are reducible or omega-reducible, with implications for aperiodic and idempotent semigroups.

Contribution

It introduces new reducibility results for pointlike problems across various pseudovarieties, extending understanding of their computational complexity and structural properties.

Findings

Pointlike and idempotent pointlike problems are reducible in specific pseudovarieties.

Omega-reducibility established for aperiodic semigroups and semigroups with idempotent regular elements.

Results connect reducibility properties with structural features of semigroups.

Abstract

We show that the pointlike and the idempotent pointlike problems are reducible with respect to natural signatures in the following cases: the pseudovariety of all finite semigroups in which the order of every subgroup is a product of elements of a fixed set of primes; the pseudovariety of all finite semigroups in which every regular J-class is the product of a rectangular band by a group from a fixed pseudovariety of groups that is reducible for the pointlike problem, respectively graph reducible. Allowing only trivial groups, we obtain omega-reducibility of the pointlike and idempotent pointlike problems, respectively for the pseudovarieties of all finite aperiodic semigroups (A) and of all finite semigroups in which all regular elements are idempotents (DA).