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Implicative-orthomodular lattices

Lavinia Corina Ciungu

Abstract

Based on implicative involutive BE algebras, we redefine the orthomodular lattices, by introducing the notion of implicative-orthomodular lattices, and we study their properties. We characterize these algebras, proving that the implicative-orthomodular lattices are quantum-Wajsberg algebras. We also define and characterize the implicative-modular algebras as a subclass of implicative-orthomodular lattices. The orthomodular softlattices and orthomodular widelattices are also redefined, by introducing the notions of implicative-orthomodular softlattices and implicative-orthomodular widelattices. Finally, we prove that the implicative-orthomodular softlattices are equivalent to implicative-orthomodular lattices and that the implicative-orthomodular widelattices are special cases of quantum-Wajsberg algebras.

Implicative-orthomodular lattices

Abstract

Based on implicative involutive BE algebras, we redefine the orthomodular lattices, by introducing the notion of implicative-orthomodular lattices, and we study their properties. We characterize these algebras, proving that the implicative-orthomodular lattices are quantum-Wajsberg algebras. We also define and characterize the implicative-modular algebras as a subclass of implicative-orthomodular lattices. The orthomodular softlattices and orthomodular widelattices are also redefined, by introducing the notions of implicative-orthomodular softlattices and implicative-orthomodular widelattices. Finally, we prove that the implicative-orthomodular softlattices are equivalent to implicative-orthomodular lattices and that the implicative-orthomodular widelattices are special cases of quantum-Wajsberg algebras.
Paper Structure (6 sections, 35 theorems, 3 equations)

This paper contains 6 sections, 35 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Implicative BE algebras
  4. Implicative-orthomodular lattices
  5. Implicative-orthomodular softlattices and widelattices
  6. Concluding remarks

Key Result

Lemma 2.3

$\rm($Ciu78$\rm)$ 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 (82)

  • Definition 2.1
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 72 more