Quasi-modal operators on distributive nearlattices

Ismael Calomino,Luciano J González,Sergio A Celani

doi:10.33044/revuma.v61n2a10

Abstract

We introduce the notion of quasi-modal operator in the variety of distributive nearlattices, which turns out to be a generalization of the necessity modal operator studied in [S. Celani and I. Calomino, Math. Slovaca 69 (2019), no. 1, 35–52]. We show that there is a one to one correspondence between a particular class of quasi-modal operators on a distributive nearlattice and the class of possibility modal operators on the distributive lattice of its finitely generated filters. Finally, we consider the concept of quasi-modal congruence, and we show that the lattice of quasi-modal congruences of a quasi-modal distributive nearlattice is isomorphic to the lattice of congruences of the lattice of finitely generated filters with a possibility modal operator.

Highlights

Introduction and preliminariesImplication algebras, called Tarski algebras, were introduced and studied by Abbott in [1, 2]
A particular class of nearlattices is the class of distributive nearlattices, i.e., join-semilattices with greatest element in which every principal filter is a bounded distributive lattice
We show a correspondence between finite quasi-modal operators on a distributive nearlattice A and possibility modal operators on the distributive lattice of its finitely generated filters Fif (A)

Summary

Introduction

Introduction and preliminariesImplication algebras, called Tarski algebras, were introduced and studied by Abbott in [1, 2]. The main aim of this article is to study quasi-operators on distributive nearlattices, and prove that they are in one to one correspondence with possibility modal operators on the distributive lattice of its finitely generated filters. We show a correspondence between finite quasi-modal operators on a distributive nearlattice A and possibility modal operators on the distributive lattice of its finitely generated filters Fif (A).

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Conclusion