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On covers of quasivarieties of p-algebras

Zalán Gyenis

Abstract

This paper characterizes the covers of varieties of p-algebras in the lattice of quasivarieties of p-algebras. In particular, it is shown that every such variety has exactly one cover in the lattice of subquasivarieties. This answers a problem of Kowalski and Słomczyńska.

On covers of quasivarieties of p-algebras

Abstract

This paper characterizes the covers of varieties of p-algebras in the lattice of quasivarieties of p-algebras. In particular, it is shown that every such variety has exactly one cover in the lattice of subquasivarieties. This answers a problem of Kowalski and Słomczyńska.
Paper Structure (3 sections, 12 theorems, 42 equations, 4 figures)

This paper contains 3 sections, 12 theorems, 42 equations, 4 figures.

Table of Contents

  1. Overview
  2. Subvariety covers of $\mathsf{Pa}_m$ for $m\geq 2$
  3. Illustration: the case $m=2$

Key Result

Lemma 1.1

Let $\mathcal{P}$ be a family of finite posets, and let $X$ be a finite poset. Then $\varepsilon(X)\in \mathbf{Q}(\{\varepsilon(P): P\in\mathcal{P}\})$ if and only if for some $P_1, \ldots, P_n\in\mathcal{P}$ there is a surjective pp-morphism $P_1\uplus\cdots\uplus P_n\twoheadrightarrow X$.

Figures (4)

  • Figure 1: The dual posets $\delta(\bar{\mathbf{B}}_m)$ of the p-algebras $\bar{\mathbf{B}}_m$ generating $\mathsf{Pa}_m$.
  • Figure 2: The three reduced posets $P(M, \mathcal{F})$ with $M=\{1,2,3\}$, and $\mathcal{F}\neq\emptyset$, up to isomorphisms.
  • Figure 3: Surjective pp-morphisms $h:P\uplus P\twoheadrightarrow Q$ and $k:Q\uplus Q\twoheadrightarrow R$. Points with the same labels are identified.
  • Figure 4: No pp-morphisms $h:Q\twoheadrightarrow P$ and $k:R\twoheadrightarrow Q$ mapping bottoms onto bottoms: $h(M(e))\neq M(h(e))$, $k(M(f))\neq M(k(f))$.

Theorems & Definitions (17)

  • Lemma 1.1: Lemma 1.7 in KowSlom
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Claim 2.8
  • Claim 2.9
  • ...and 7 more