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Operators Max L and Min U and duals of Boolean posets

Ivan Chajda, Miroslav Kolařík, Helmut Länger

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.

Operators Max L and Min U and duals of Boolean posets

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.
Paper Structure (5 sections, 11 theorems, 29 equations)

This paper contains 5 sections, 11 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Basic concepts
  3. Axiomatization of the operators $\mathop{\mathrm{Max}}\nolimits L$ and $\mathop{\mathrm{Min}}\nolimits U$
  4. Symmetric difference and the Sheffer operator
  5. Duals of Boolean posets

Key Result

Lemma 2.1

Let $\mathbf P=(P,\le)$ be a poset satisfying the ACC and the DCC and let $A,B\subseteq P$. Then the following holds:

Theorems & Definitions (28)

  • Lemma 2.1
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 3.5
  • ...and 18 more