Modular elements in the lattice of monoid varieties

TL;DR

This work resolves the modular elements problem for the lattice $\mathbb{M\text{ON}}$ of all monoid varieties, providing a complete classification. It shows that modular monoid varieties are precisely either $\mathbf M(\text{ON})$ or arise from explicit word-sets $\mathcal W$ via $\mathbf M(\mathcal W)$, characterized by identities including $x^2\approx x^3$, $x^2y\approx yx^2$, certain linear-balanced swaps, and the families $\mathbf a_{n,m}[\rho]\approx \mathbf a'_{n,m}[\rho]$ for all $n,m,\rho$. The authors establish the equivalence of several formulations of modularity, prove the necessary and sufficient conditions, and demonstrate that the modular elements form a sublattice with proper modular elements forming an order ideal. The results extend the understanding of modularity from semigroup varieties to monoid varieties, with exact, infinite families of modular varieties identified and a robust proof framework built on isoterms and Deduction techniques.

Abstract

An element $x$ of a lattice $L$ is modular if $L$ has no five-element sublattice isomorphic to the pentagon in which $x$ would correspond to the lonely midpoint. In the present work, we classify all modular elements of the lattice of all monoid varieties.

Figures (2)

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Theorems & Definitions (11)

• Theorem 1
• Proposition 2: Birkhoff's Completeness Theorem for Equational Logic; see Burris-Sankappanavar-81
• Lemma 3: Jackson-05
• Lemma 4: Gusev-Vernikov-21
• Lemma 5
• Claim 1
• proof
• Claim 2
• proof
• Claim 3