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Subfitness in distributive (semi)lattices

Guram Bezhanishvili, James Madden, M. Andrew Moshier, Marcus Tressl, Joanne Walters-Wayland

TL;DR

This work investigates whether the set of subfit elements, specifically join-subfit elements, forms an ideal in distributive structures. It proves a positive result when the semilattice arises from a distributive lattice, but provides a counterexample for general distributive semilattices, showing the property can fail without lattice structure. A distributive lattice envelope (via injective hulls) preserves subfitness, explaining why subfitness interacts well with envelopes even when the ideal property does not hold in the semilattice. A topological lens via frame theory and spectral spaces ties the algebraic results to patch topology and Grätzer–Stone duality, giving constructive proofs and a unifying perspective across algebra and topology.

Abstract

We investigate whether the set of subfit elements of a distributive semilattice is an ideal. This question was raised by the second author at the BLAST conference in 2022. We show that in general it has a negative solution, however if the semilattice is a lattice, then the solution is positive. This is somewhat unexpected since, as we show, a semilattice is subfit if and only if so is its distributive lattice envelope.

Subfitness in distributive (semi)lattices

TL;DR

This work investigates whether the set of subfit elements, specifically join-subfit elements, forms an ideal in distributive structures. It proves a positive result when the semilattice arises from a distributive lattice, but provides a counterexample for general distributive semilattices, showing the property can fail without lattice structure. A distributive lattice envelope (via injective hulls) preserves subfitness, explaining why subfitness interacts well with envelopes even when the ideal property does not hold in the semilattice. A topological lens via frame theory and spectral spaces ties the algebraic results to patch topology and Grätzer–Stone duality, giving constructive proofs and a unifying perspective across algebra and topology.

Abstract

We investigate whether the set of subfit elements of a distributive semilattice is an ideal. This question was raised by the second author at the BLAST conference in 2022. We show that in general it has a negative solution, however if the semilattice is a lattice, then the solution is positive. This is somewhat unexpected since, as we show, a semilattice is subfit if and only if so is its distributive lattice envelope.
Paper Structure (5 sections, 6 theorems, 14 equations, 1 figure)

This paper contains 5 sections, 6 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Subfit elements in distributive lattices are directed
  3. Subfit elements in distributive semilattices are not directed
  4. Distributive envelopes and subfitness
  5. Topological perspective

Key Result

Theorem 2.1

If $A$ is a bounded distributive lattice, then the set $S$ of join-subfit elements of $A$ is an ideal of $A$.

Figures (1)

  • Figure 1: The meet-semilattice $A$ (black) within its distributive lattice envelope (black and red)

Theorems & Definitions (21)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Corollary 2.4
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 11 more