Operators Max L and Min U and duals of Boolean posets

TL;DR

This paper introduces and axiomatizes Max L and Min U operators for non-lattice posets, explores their use in complemented posets, and establishes duality concepts for Boolean posets and their duals.

Contribution

It presents a novel axiomatization of Max L and Min U operators and develops duality theory for Boolean posets using these operators.

Findings

Axiomatization of Max L and Min U operators.

Introduction of symmetric difference and Sheffer operator in complemented posets.

Establishment of duality between Boolean posets and their duals.

Abstract

When working with posets which are not necessarily lattices, one has a lack of lattice operations which causes problems in algebraic constructions. This is the reason why we use the operators Max L and Min U substituting infimum and supremum, respectively. We axiomatize these operators. Two more operators, namely the so-called symmetric difference and the Sheffer operator, are introduced and studied in complemented posets by using the operators Max L and Min U. In Boolean algebras, the symmetric difference is used to construct its dual structure, the corresponding unitary Boolean ring. By generalizing this idea, we assign to each Boolean poset a so-called dual and prove that also, conversely, a Boolean poset can be derived from its dual.