Implication algebras

Ivan Chajda

ArticleOriginal scientific text

Title

Implication algebras

Authors ¹

Affiliations

Department of Algebra and Geometry, Palacký University of Olomouc, Tomkova 40, 779 00 Olomouc, Czech Republic

Abstract

We introduce the concepts of pre-implication algebra and implication algebra based on orthosemilattices which generalize the concepts of implication algebra, orthoimplication algebra defined by J.C. Abbott [2] and orthomodular implication algebra introduced by the author with his collaborators. For our algebras we get new axiom systems compatible with that of an implication algebra. This unified approach enables us to compare the mentioned algebras and apply a unified treatment of congruence properties.

Keywords

ENG

implication algebra, pre-implication algebra, orthoimplication algebra, orthosemilattice, congruence kernel

J.C. Abbott, Semi-Boolean algebra, Matem. Vestnik 4 (1967), 177-198.
J.C. Abbott, Orthoimplication Algebras, Studia Logica 35 (1976), 173-177.
L. Beran, Orthomodular Lattices, Algebraic Approach, Mathematic and its Applications, D. Reidel Publ. Comp., 1985.
I. Chajda and R. Halaš, An Implication in Orthologic, submitted to Intern. J. Theor. Phys. 44 (2006), 735-744.
I. Chajda, R. Halaš and H. Länger, Orthomodular implication algebras, Intern. J. Theor. Phys. 40 (2001), 1875-1884.
G.M. Hardegree, Quasi-implication algebras, Part I: Elementary theory, Algebra Universalis 12 (1981), 30-47.
G.M. Hardegree, Quasi-implication algebras, Part II: Sructure theory, Algebra Universalis 12 (1981), 48-65.
J. Hedliková, Relatively orthomodular lattices, Discrete Math., 234 (2001), 17-38.
M.F. Janowitz, A note on generalized orthomodular lattices, J. Natural Sci. Math. 8 (1968), 89-94.
N.D. Megill and M. Pavičić, Quantum implication algebras, Intern. J. Theor. Phys. 48 (2003), 2825-2840.

Title

Implication algebras

Affiliations

Abstract

Keywords

Bibliography