Semiring and involution identities of powers of inverse semigroups

TL;DR

This paper investigates the algebraic identities of the involution semiring formed by subsets of finite inverse semigroups, revealing conditions under which these identities lack a finite basis, thus advancing understanding of their structural complexity.

Contribution

It establishes structural conditions on finite inverse semigroups that determine the non-existence of finite identity bases for their subset involution semirings.

Findings

Identifies conditions preventing finite identity bases

Shows the complexity of involution semiring identities

Advances structural understanding of inverse semigroups

Abstract

The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis.