Characterization of Cross varieties of $J$-trivial monoids

Abstract

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.

Figures (6)

• Figure 1: The lattice $\mathfrak{L}(\mathbf{H})$
• Figure 2: The lattice $\mathfrak{L}(\mathbf{P})$
• Figure 3: The lattice $\mathfrak{L}(\mathbf{I})$
• Figure 4: The lattice $\mathfrak{L}(\mathbf{K})$
• Figure 5: The lattices $\mathfrak{L}(\mathbf{Y}_1)$ and $\mathfrak{L}(\mathbf{Y}_2)$

Theorems & Definitions (56)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4: Jackson Jac05
• Lemma 2.5
• proof
• Lemma 2.6