A minimal nonfinitely based semigroup whose variety is polynomially recognizable

TL;DR

This paper presents a 6-element semigroup that, despite lacking a finite identity basis, generates a variety with a polynomial-time decidable finite membership problem.

Contribution

It introduces a minimal nonfinitely based semigroup with a polynomially recognizable variety, bridging a gap between algebraic complexity and computational tractability.

Findings

The semigroup has no finite identity basis.

The generated variety's finite membership problem is polynomially decidable.

The result demonstrates a separation between algebraic complexity and computational complexity.

Abstract

We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm.