Material implications in lattice effect algebras

Abstract In this paper we show that it is not acceptable using the minimal conditions of implication introduced for orthomodular lattices as minimal conditions of a good implication for lattice effect algebras. Hence, we first define the concept of a partial t-norm on bounded lattices. Then we obtain the concept of a pt-implication on bounded involutive lattices by introducing some conditions using partial t-norms. Furthermore, we conclude that the Sasaki arrow which is a very important implication in lattice effect algebras is the best pt-implication on them. Finally, applying the Sasaki arrow, we construct a fuzzy implication on a lattice effect algebra and also we prove that it is a weak pt-implication as well.

[1]  Ivan Chajda.  Basic algebras, logics, trends and applications , 2015 .

[2]  Gary M. Hardegree.  Stalnaker conditionals and quantum logic , 1975, J. Philos. Log..

[3]  Gary M. Hardegree.  Material implication in orthomodular (and Boolean) lattices , 1981, Notre Dame J. Formal Log..

[4]  Ivan Chajda,et al.  Every effect algebra can be made into a total algebra , 2009 .

[5]  Renata Reiser,et al.  (S,N)-Implications on Bounded Lattices , 2013 .

[6]  David J. Foulis,et al.  Filters and supports in orthoalgebras , 1992 .

[7]  Radomír Halas,et al.  Commutative basic algebras and non-associative fuzzy logics , 2009, Arch. Math. Log..

[8]  Ivan Chajda.  Effect-like algebras induced by means of basic algebras , 2009 .

[9]  Sylvia Pulmannová,et al.  New trends in quantum structures , 2000 .

[10]  Radomír Halas,et al.  Effect algebras are conditionally residuated structures , 2011, Soft Comput..

[11]  David J. Foulis,et al.  Logical Connectives on Lattice Effect Algebras , 2012, Stud Logica.

[12]  Roberto Giuntini,et al.  Reasoning in Quantum Theory: Sharp and Unsharp Quantum Logics , 2010 .

[13]  Benjamín R. C. Bedregal,et al.  T-Norms on Bounded Lattices: t-norm Morphisms and Operators , 2006, 2006 IEEE International Conference on Fuzzy Systems.

[14]  Ivan Chajda,et al.  A note on residuated po-groupoids and lattices with antitone involutions , 2017 .

[15]  Ivan Chajda,et al.  Residuation in orthomodular lattices , 2017 .

[16]  David J. Foulis,et al.  Phi-symmetric effect algebras , 1995 .

[17]  V. Varadarajan.  Geometry of quantum theory , 1968 .

[18]  J. M. Jauch,et al.  WHAT IS ''QUANTUM-LOGIC''. , 1970 .

[19]  Guo-Jun Wang,et al.  Residuated lattices and lattice effect algebras , 2007, Fuzzy Sets Syst..

[20]  D. Foulis,et al.  Effect algebras and unsharp quantum logics , 1994 .

[21]  Robert Piziak,et al.  Implication connectives in orthomodular lattices , 1975, Notre Dame J. Formal Log..

[22]  Yongjian Xie,et al.  Atomic Effect Algebras with the Riesz Decomposition Property , 2012 .

[23]  Gary M. Hardegree.  An axiom system for orthomodular quantum logic , 1981 .

[24]  Ladislav Beran,et al.  Orthomodular Lattices: Algebraic Approach , 1985 .

[25]  Karl Svozil,et al.  Quantum Logic , 1998, Discrete mathematics and theoretical computer science.