Characterization of Cross varieties of $J$-trivial monoids

A finitely based, finitely generated variety with finitely many subvarieties is a Cross variety. In the present article, it is shown that a variety of $J$-trivial monoids is Cross if and only if it excludes as subvarieties a certain list of 14 almost Cross varieties. Consequently, the list of 14 varieties exhausts all almost Cross varieties of $J$-trivial monoids.

💡 Research Summary

The paper investigates the landscape of varieties of J‑trivial monoids with respect to the notion of a Cross variety. A Cross variety, following Higman’s terminology, is one that simultaneously satisfies three finiteness conditions: it is finitely based (its equational theory can be axiomatized by finitely many identities), finitely generated (it is generated by a finite monoid), and small (its lattice of subvarieties is finite). While many classical algebraic classes (groups, associative rings, Lie rings, lattices) yield only Cross varieties, the situation for monoids is far richer; there exist non‑Cross varieties with highly complex subvariety lattices.

The authors focus on the class J of all J‑trivial monoids, a class of central importance in automata theory because finite J‑trivial monoids correspond exactly to piecewise‑testable languages. Within J, several “almost‑Cross” varieties are already known: these are non‑Cross varieties whose every proper subvariety is Cross. Prior work identified twelve such varieties, namely the nine almost‑Cross varieties from the aperiodic commutative class A​com (including the duals of F, I, P) together with the dual pairs K/←K and Z. The present article expands this list to a total of fourteen by introducing a new almost‑Cross variety H and its dual ←H.

The paper proceeds in a systematic fashion. Section 2 collects necessary preliminaries: the formalism of words, identities, isoterms, Rees quotients Rq W, and basic lemmas about local finiteness and the relationship between complete regularity and the subvarieties Jₙ (where Jₙ = var{aⁿ, cⁿ} with aⁿ = xⁿ⁺¹≈xⁿ and cⁿ = (xy)ⁿ≈(yx)ⁿ). Lemma 2.4 shows that a Rees quotient belongs to a variety precisely when each word in its defining set is an isoterm for that variety, a tool used repeatedly to locate varieties inside J.

Section 3 classifies the non‑finitely generated almost‑Cross varieties. Subsection 3.1 revisits the known pair F and its dual, which are characterized by the identity xhx≈x²h. Subsection 3.2 introduces H, defined by the simultaneous satisfaction of xhx≈x²h and the eventual commutativity identities (xy)ⁿ≈(yx)ⁿ, and proves that H is minimal non‑Cross: any proper subvariety either collapses to a Cross variety or forces commutativity. Subsection 3.3 treats the varieties L, P and their duals, which arise from Rees quotients Rq {xy} and Rq {xⁿ} and are shown to be almost‑Cross via explicit identity bases.

Section 4 turns to the finitely generated almost‑Cross varieties. Subsection 4.1 handles I and its dual, which satisfy xⁿ⁺¹≈xⁿ together with xhx≈hx². Subsection 4.2 analyses K and its dual, showing they lie inside the intersection O∩J₂, where O is defined by two specific identities involving a distinguished letter h. The authors prove that any non‑commutative subvariety of O∩J₂ can be described by a finite set of identities drawn from a restricted schema (Proposition 2.8). Subsection 4.3 revisits the previously discovered varieties Y₁ and Y₂, which belong to the central‑idempotent subclass A​cen. Finally, Subsection 4.4 introduces Z, another finitely generated almost‑Cross variety defined by xhx≈hx² together with the eventual commutativity identities.

The technical heart of the paper lies in the careful analysis of isoterms and Rees quotients, the use of Lemma 2.5 to restrict the possible identities when a particular Rees quotient is excluded, and the construction of identity bases that separate each almost‑Cross variety from all Cross varieties. The authors also exploit duality (the operation of reversing multiplication) to double many of the examples, ensuring that the list of fourteen varieties is closed under taking duals.

Section 5 culminates in the main theorem: a subvariety V of J is Cross if and only if it does not contain any of the fourteen almost‑Cross varieties listed above. The proof proceeds by Zorn’s Lemma: assuming V is non‑Cross, one extracts a minimal non‑Cross subvariety, which must be one of the fourteen by the exhaustive analysis of Sections 3 and 4. Conversely, if V excludes all fourteen, any proper subvariety of V must be Cross, and the three finiteness conditions follow from Lemma 2.1 and Lemma 2.3, establishing V as Cross.

In summary, the article delivers a complete characterization of Cross subvarieties within the class of J‑trivial monoids. By identifying and proving the minimality of exactly fourteen almost‑Cross varieties, the authors provide a clean, verifiable criterion for Crossness in this important algebraic setting, extending earlier work on aperiodic and commutative monoids and offering a template for similar classifications in other varieties.