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Quasicomplemented distributive nearlattices

Ismael Calomino

TL;DR

This work develops a comprehensive framework for quasicomplemented distributive nearlattices by linking $α$-filters and $α$-ideals to congruence kernels, and by analyzing how normality and quasicomplementation shape the lattice of annihilators and dense elements. It shows that, in normal and quasicomplemented settings, there is a tight one-to-one correspondence between $α$-filters and $α$-ideals, via $F\mapsto I_{F}$ and $I\mapsto F_{I}$, and that ideal-congruence kernels can be characterized through these correspondences. The paper also introduces Stone distributive nearlattices and provides a σ-filter-based characterization, establishing that Stoneness is equivalent to every $α$-filter being a $σ$-filter, and connects Stoneness with normality and quasicomplementation. Overall, the results integrate annihilator-based structure with filter–ideal–congruence theory and supply a Stone-type description via $σ$-filters in this generalized lattice setting.

Abstract

The aim of this paper is to study the class of quasicomplemented distributive nearlattices. We investigate $α$-filters and $α$-ideals in quasicomplemented distributive nearlattices and some results on ideals-congruence-kernels. Finally, we also study the notion of Stone distributive nearlattice and give a characterization by means $σ$-filters.

Quasicomplemented distributive nearlattices

TL;DR

This work develops a comprehensive framework for quasicomplemented distributive nearlattices by linking -filters and -ideals to congruence kernels, and by analyzing how normality and quasicomplementation shape the lattice of annihilators and dense elements. It shows that, in normal and quasicomplemented settings, there is a tight one-to-one correspondence between -filters and -ideals, via and , and that ideal-congruence kernels can be characterized through these correspondences. The paper also introduces Stone distributive nearlattices and provides a σ-filter-based characterization, establishing that Stoneness is equivalent to every -filter being a -filter, and connects Stoneness with normality and quasicomplementation. Overall, the results integrate annihilator-based structure with filter–ideal–congruence theory and supply a Stone-type description via -filters in this generalized lattice setting.

Abstract

The aim of this paper is to study the class of quasicomplemented distributive nearlattices. We investigate -filters and -ideals in quasicomplemented distributive nearlattices and some results on ideals-congruence-kernels. Finally, we also study the notion of Stone distributive nearlattice and give a characterization by means -filters.

Paper Structure

This paper contains 10 sections, 29 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distributive nearlattices
  4. Annihilators and normal distributive nearlattices
  5. $\alpha$-filters, $\alpha$-ideals and congruences
  6. Quasicomplemented distributive nearlattices
  7. Connection between $\alpha$-ideals and $\alpha$-filters
  8. Ideals-congruence-kernels
  9. Stone distributive nearlattices
  10. $\sigma$-filters

Key Result

Theorem 2.2

ArKin11 Let ${\bf{A}} = \langle A,\vee, 1 \rangle$ be a distributive nearlattice. Let $m \colon A^{3} \to A$ be the ternary operation given by $m(x,y,z) = (x \vee z) \wedge_{z} (y \vee z)$, where $\wedge_{z}$ denotes the meet in $[z)$. Then the structure ${\bf{A}}_{*} = \langle A, m, 1 \rangle$ sati Conversely, let ${\bf{A}} = \langle A, m, 1 \rangle$ be an algebra of type $(3,0)$ satisfying the i

Theorems & Definitions (64)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 54 more