Unary enhancements of inherently nonfinitely based semigroups

TL;DR

This paper identifies conditions under which finite involutary semigroups retain their inherently nonfinitely based property when considering unary operations, providing new examples and an algorithmic characterization.

Contribution

It introduces a simple condition linking the nonfinite basis property of semigroup reducts to their unary extensions, with applications to regular semigroups.

Findings

Established a sufficient and necessary condition for finite regular semigroups.

Provided new examples of inherently nonfinitely based involutory semigroups.

Developed an algorithmic method to identify such semigroups.

Abstract

We exhibit a simple condition under which a finite involutary semigroup whose semigroup reduct is inherently nonfinitely based is also inherently nonfinitely based as a unary semigroup. As applications, we get already known as well as new examples of inherently nonfinitely based involutory semigroups. We also show that for finite regular semigroups, our condition is not only sufficient but also necessary for the property of being inherently nonfinitely based to persist. This leads to an algorithmic description of regular inherently nonfinitely based involutory semigroups.