On some admissible lattices

TL;DR

This work uses Freese's technique to study which lattices can arise as congruence sublattices within algebras of a variety, and to characterize key algebraic classes via lattice omissions. By analyzing $N_5$, $D_{13}$, and $D_2$ through the Freese framework, the authors derive new lattice-omission characterizations for congruence modular varieties, Taylor varieties, and varieties with a non-trivial congruence identity, respectively, and construct explicit lattice patterns that realize or exclude these properties. The results include a complete description of the sublattice generated by certain congruences in $ ext{Con}( extbf{A}(eta))$, a Taylor-variety criterion via omission of $D_{13}$, and a D2-based characterization using infinite families of lattices, illustrating both uniqueness and multiplicity phenomena in omission patterns. Collectively, the paper advances the program of linking Mal’cev-type conditions to lattice-omission phenomena and provides concrete tools for identifying when a given lattice can or must be omitted in the congruence lattice of a class of algebras.

Abstract

This paper explores applications of the so-called Freese's technique, a classical approach to study the congruence variety of a given algebra. We leverage this tool to investigate lattices that are admissible as congruence sublattice of a given algebra. In particular, we present a novel characterization of congruence modular varieties, Taylor varieties, and varieties satisfying a non-trivial congruence identity by means of lattice omission.

Figures (9)

• Figure 1: Respectively ${\textbf{\upshape N}}_5$, ${\textbf{\upshape D}}_1$ and ${\textbf{\upshape D}}_2$
• Figure 2: ${\textbf{\upshape L}}_{14}$
• Figure 3: Respectively, the lattices ${\textbf{\upshape M}}_1$ and ${\textbf{\upshape K}}$
• Figure 4: ${\textbf{\upshape K}}_{i+1}$ and ${\textbf{\upshape M}}_{i+1}$
• Figure 5: ${\textbf{\upshape K}}_{\infty}$

Theorems & Definitions (31)

• Theorem 2.1
• Theorem 3.1: Theorem 6.1 in Freese2024
• Lemma 3.2
• Lemma 3.3: Lemma 3.2 of ABF
• Theorem 3.4: Theorem 1.1 ABF
• Lemma 4.1: Lemma 3.4 of ABF
• proof
• Lemma 4.2
• proof
• Lemma 4.3