Implications and equivalences in orthomodular lattices

• Autorzy: Hyčko M.
• Języki publikacji: EN

Abstrakty

EN

The present article describes a method for checking the validity of implications or equivalences in the free orthomodular lattice on two generators and in the F(a, b, c1,..., cn), which is the free orthomodular lattice generated by the elements a, 6, ci,... Cn, where the elements ci, i = 1,..., n are central in it. The structure of the previous lattices is described in [3] and [1]. The method presented is based on comparing the elements that are assigned to each expression on both sides of an implication or an equivalence. It gives a necessary condition for the implication or equivalence of arbitrary positive statements (a combination of identities and logical connectives AND and OR) to hold. When the conclusion part is an identity or a conjunction of identities, these conditions become also sufficient.

Słowa kluczowe

EN

ideal; free lattice; orthomodular lattice; congruence; equivalence

PL

ideał; kraty ortomodularne; kraty wolne; kongruencja

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2005
• Tom: Vol. 38, nr 4
• Strony: 777--792
• Opis fizyczny: Bibliogr. 4 poz.

Twórcy

autor

Hyčko M.

• Mathematical Institute, Slovak Academy of Sciences, Stefanikova 49, SK-814 73 Bratislava, Slovakia

Bibliografia

• [1] L. Beran, Orthomodular Lattices, Algebraic Approach, Academia, Prague 1984.
• [2] G. Kalmbach, Orthomodular Lattices, Academic Press, London, 1983.
• [3] M. Navara, On generating finite orthomodular sublattices, Tatra. Mt. Math. Publ. 10 (1997), 109-117.
• [4] M. Pavičić, N. D. Megill, Quantum and classical implication algebras with primitive implications, Inter. J. Theor. Physics, 37 (1998), 2091-2098.