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Foulis-Holland theorem for implicative-orthomodular lattices

Lavinia Corina Ciungu

Abstract

We introduce the notion of distributivity for implicative-orthomodular lattices, proving an analogue result of the Foulis-Holland theorem. Based on this result, we characterize the distributive implicative-orthomodular lattices. Moreover, we define the center of an implicative-orthomodular lattice as the set of all elements that commute with all other elements, and we prove that the center is an implicative-Boolean algebra. Additionally, we give new characterizations of implicative-orthomodular lattices.

Foulis-Holland theorem for implicative-orthomodular lattices

Abstract

We introduce the notion of distributivity for implicative-orthomodular lattices, proving an analogue result of the Foulis-Holland theorem. Based on this result, we characterize the distributive implicative-orthomodular lattices. Moreover, we define the center of an implicative-orthomodular lattice as the set of all elements that commute with all other elements, and we prove that the center is an implicative-Boolean algebra. Additionally, we give new characterizations of implicative-orthomodular lattices.
Paper Structure (6 sections, 39 theorems, 1 equation)

This paper contains 6 sections, 39 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On implicative-orthomodular lattices
  4. Foulis-Holland theorem for implicative-orthomodular lattices
  5. Implicative-Boolean center of implicative-orthomodular lattices
  6. Applications of Foulis-Holland theorem

Key Result

Lemma 2.1

$\rm($Ciu83$\rm)$ Let $(X,\rightarrow ,1)$ be a BE algebra. The following hold for all $x,y,z\in X$: $(1)$$x\rightarrow (y\rightarrow x)=1;$$(2)$$x\le (x\rightarrow y)\rightarrow y$. If $X$ is bounded, then: $(3)$$x\rightarrow y^*=y\rightarrow x^*;$$(4)$$x\le x^{**}$. If $X$ is involutive, the

Theorems & Definitions (85)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 75 more