Non-finitely based monoids

TL;DR

This paper introduces a powerful general method for proving that certain semigroups are non-finitely based, simplifying existing proofs and extending known results, including new examples of non-finitely based monoids.

Contribution

The paper develops a general, simplified method for establishing non-finite basis properties in semigroups, extending previous results and providing new examples.

Findings

Proved eleven new sufficient conditions for non-finite basis of monoids.

Generalized and simplified proofs of existing non-finite basis results.

Discovered infinitely many new examples of non-finitely based monoids via direct products.

Abstract

We present a general method for proving that a semigroup is non-finitely based. The method is strong enough to cover the non-finite basis arguments in articles [1,3,4,5,7,8, 11,14,16,21,27,31,36,37]. In particular, the method allows to generalize the results in [1,8,36,37] and to simplify their proofs. The method also allows to remove one of the requirements on the "special system of identities" used by P. Perkins in [16] to find the first two examples of finite non-finitely based semigroups. We use our method to prove eleven new sufficient conditions under which a monoid is non-finitely based. As an application, we find infinitely many new examples of finite finitely based aperiodic monoids whose direct product is non-finitely based.