Strongly nonfinitely based monoids

TL;DR

The paper identifies a finite monoid, the $42$-element $IC_4$, that is strongly nonfinitely based yet not inherently so, resolving a question about the existence of finite semigroups with this precise combination of properties. Building on the framework of DS and LDS pseudovarieties and the notion of sparse isoterms, the authors prove that any finite semigroup whose variety contains $IC_4$ is nonfinitely based, by combining two complementary arguments depending on the locality condition. They also provide a concrete application by embedding $IC_4$ into $T_4(2)$, and hence into $T_n(2)$ for all $n>3$, establishing that these upper triangular matrix semigroups are strongly nonfinitely based. This work advances understanding of the finite basis problem in semigroups and answers open questions about the relationship between strong and inherent nonfinite basing, with practical impact on a broad class of finite semigroups including triangular matrix varieties.

Abstract

We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite semigroup.

Theorems & Definitions (18)

• Lemma 1: Sapir-Volkov-22
• Proposition 2
• proof
• Lemma 3
• proof
• Lemma 4
• Corollary 5
• proof
• Proposition 6
• proof