Finite distributive nearlattices
Luciano J. Gonz\'alez, Ismael Calomino
TL;DR
This paper develops a new representation for finite distributive nearlattices using ordered structures, generalizing Birkhoff's representation for finite distributive lattices, and explores their properties related to boolean and complemented elements.
Contribution
It introduces a generalized representation for finite distributive nearlattices and analyzes their boolean, complemented, and irreducible elements.
Findings
Boolean elements form semi-boolean algebras
Complemented elements form semi-boolean algebras
Boolean elements in finite distributive lattices form a boolean lattice
Abstract
Our main goal is to develop a representation for finite distributive nearlattices through certain ordered structures. This representation generalizes the well-known representation given by Birkhoff for finite distributive lattices through finite posets. We also study finite distributive nearlattices through the concepts of dual atoms, boolean elements, complemented elements and irreducible elements. We prove that the sets of boolean elements and complemented elements form semi-boolean algebras. We show that the set of boolean elements of a finite distributive lattice is a boolean lattice.
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