University of Ostrava

Recently, a special algebra called EQ-algebra has been introduced by Vilem Novák in [31], which aims at becoming the algebra of truth values for fuzzy type theory. It has three binary operations – meet, multiplication and fuzzy equality – and a top element. The multiplication, in EQ-algebra, is assumed to be both commutative and associative. In this paper, we generalize the concept of EQ-algebra by excluding both the commutativity and the associativity of the multiplication showing that nothing is lost. We call such type of algebra a semicopula-based EQ-algebra. We show that all proved properties of EQ-algebras remain valid and applicable in semicopula-based EQalgebras and vice versa. Besides these main results, a lot of new and important properties concerning (semicopula-based) EQ-algebras and their special kinds are proved.

[1]  R. Mesiar,et al.  Aggregation operators: properties, classes and construction methods , 2002 .

[2]  Fabrizio Durante,et al.  Semicopulas: characterizations and applicability , 2006, Kybernetika.

[3]  S. Gottwald.  A Treatise on Many-Valued Logics , 2001 .

[4]  Andrei Popescu,et al.  Non-commutative fuzzy structures and pairs of weak negations , 2004, Fuzzy Sets Syst..

[5]  Peter B. Andrews.  An introduction to mathematical logic and type theory - to truth through proof , 1986, Computer science and applied mathematics.

[6]  J. Fodor.  Contrapositive symmetry of fuzzy implications , 1995 .

[7]  V. Novák.  Applications of Fuzzy Modeling EQ-algebras : primary concepts and properties , 2007 .

[8]  Bernard De Baets,et al.  EQ-algebras , 2009, Fuzzy Sets Syst..

[9]  Nehad N. Morsi,et al.  Propositional calculus under adjointness , 2002, Fuzzy Sets Syst..

[10]  Nehad N. Morsi,et al.  The logic of tied implications, part 2: Syntax , 2006, Fuzzy Sets Syst..

[11]  V. Em,et al.  On fuzzy type theory , 2004 .

[12]  C. Sempi,et al.  Semicopulæ , 2005, Kybernetika.

[13]  Radko Mesiar,et al.  On copulas, quasicopulas and fuzzy logic , 2008, Soft Comput..

[14]  Nehad N. Morsi,et al.  The logic of tied implications, part 1: Properties, applications and representation , 2006, Fuzzy Sets Syst..

[15]  Lluis Godo,et al.  Monoidal t-norm based logic: towards a logic for left-continuous t-norms , 2001, Fuzzy Sets Syst..

[16]  Nehad N. Morsi,et al.  Propositional Calculus for Adjointness Lattices , 2002 .

[17]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[18]  Nehad N. Morsi,et al.  Issues on adjointness in multiple-valued logics , 2006, Inf. Sci..

[19]  R. Mesiar,et al.  Logical, algebraic, analytic, and probabilistic aspects of triangular norms , 2005 .

[20]  T. Blyth.  Lattices and Ordered Algebraic Structures , 2005 .

[21]  Masaaki Miyakoshi,et al.  Composite Fuzzy Relational Equations with Non-Commutative Conjunctions , 1998, Inf. Sci..

[22]  F. García,et al.  Two families of fuzzy integrals , 1986 .

[23]  Vilém Novák,et al.  EQ-Algebras in Progress , 2007, IFSA.

[24]  Nehad N. Morsi,et al.  Associatively tied implications , 2003, Fuzzy Sets Syst..

[25]  Fabio Spizzichino,et al.  Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes , 2005 .

[26]  George Georgescu,et al.  Pseudo-t-norms and pseudo-BL algebras , 2001, Soft Comput..

[27]  R. Mesiar,et al.  Conjunctors and their Residual Implicators: Characterizations and Construction Methods , 2007 .

[28]  Christian Eitzinger,et al.  Triangular Norms , 2001, Künstliche Intell..

[29]  Petr Hájek.  Observations on non-commutative fuzzy logic , 2003, Soft Comput..

[30]  Ulrich Bodenhofer,et al.  A Similarity-Based Generalization of Fuzzy Orderings Preserving the Classical Axioms , 2000, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[31]  János C. Fodor,et al.  Nonstandard conjunctions and implications in fuzzy logic , 1995, Int. J. Approx. Reason..

[32]  S. Gottwald,et al.  Triangular norm-based mathematical fuzzy logics , 2005 .

[33]  U. Höhle.  Quotients with respect to similarity relations , 1988 .