991 research outputs found
Implicative-orthomodular lattices
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.Comment: arXiv admin note: text overlap with arXiv:2401.0414
Bell-type inequalities for bivariate maps on orthomodular lattices
Bell-type inequalities on orthomodular lattices, in which conjunctions of
propositions are not modeled by meets but by maps for simultaneous measurements
(s-maps), are studied. It is shown that the most simple of these inequalities,
that involves only two propositions, is always satisfied, contrary to what
happens in the case of traditional version of this inequality in which
conjunctions of propositions are modeled by meets. Equivalence of various
Bell-type inequalities formulated with the aid of bivariate maps on
orthomodular lattices is studied. Our invesigations shed new light on the
interpretation of various multivariate maps defined on orthomodular lattices
already studied in the literature. The paper is concluded by showing the
possibility of using s-maps and j-maps to represent counterfactual conjunctions
and disjunctions of non-compatible propositions about quantum systems.Comment: 14 pages, no figure
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