MV-pairs and state operators

Flaminio and Montagna (2008) enlarged the language of MV-algebras by a unary operation ?, called internal state or state operator, equationally defined so as to preserve the basic properties of a state in its usual meaning. The resulting class of MV-algebras is called state MV-algebras. Jenca (2007) and Vetterlein (2008), using different approaches, represented MV-algebras through the quotient of a Boolean algebra B by a suitable subgroup G of the group of all automorphisms of B. Such a couple ( B , G ) is called an MV-pair. We introduce the notion of a state MV-pair as a triple ( B , G , ? ) , where ( B , G ) is an MV-pair and ? is a state operator on B, and show that there are relations between state MV-pairs and state MV-algebras similar to the relations between MV-pairs and MV-algebras. We also give a characterization of those MV-pairs, resp. state MV-pairs, that induce subdirectly irreducible MV-algebras, resp. state MV-algebras.

[1]  Anatolij Dvurecenskij,et al.  Loomis-Sikorski theorem and Stone duality for effect algebras with internal state , 2010, Fuzzy Sets Syst..

[2]  Sylvia Pulmannová,et al.  Ideals in MV-pairs , 2008, Soft Comput..

[3]  Antonio di Nola,et al.  The category of MV-pairs , 2009, Log. J. IGPL.

[4]  Sylvia Pulmannová.  MV-pairs and states , 2009, Soft Comput..

[5]  Anatolij Dvurecenskij,et al.  Erratum to "State-morphism MV-algebras" [Ann. Pure Appl. Logic 161 (2009) 161-173] , 2010, Ann. Pure Appl. Log..

[6]  Gejza Jenca.  Boolean algebras R-generated by MV-effect algebras , 2004, Fuzzy Sets Syst..

[7]  Anatolij Dvurecenskij,et al.  State-morphism MV-algebras , 2009, Ann. Pure Appl. Log..

[8]  C. Chang,et al.  Algebraic analysis of many valued logics , 1958 .

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

[11]  Anatolij Dvurecenskij,et al.  On varieties of MV-algebras with internal states , 2010, Int. J. Approx. Reason..

[12]  Gejza Jenca.  A Representation Theorem for MV-algebras , 2007, Soft Comput..

[13]  Hernán de la Vega.  Normal and complete Boolean ambiguity algebras and MV-pairs , 2012, Log. J. IGPL.

[14]  Franco Montagna,et al.  MV-algebras with internal states and probabilistic fuzzy logics , 2009, Int. J. Approx. Reason..

[15]  DANIELE MUNDICI,et al.  Averaging the truth-value in Łukasiewicz logic , 1995, Stud Logica.

[16]  G. Grätzer.  General Lattice Theory , 1978 .

[17]  A. Nola,et al.  ON THE LOOMIS–SIKORSKI THEOREM FOR MV-ALGEBRAS WITH INTERNAL STATE , 2010, Journal of the Australian Mathematical Society.

[18]  Kerstin Vogler,et al.  Algebraic Foundations Of Many Valued Reasoning , 2016 .

[19]  Thomas Vetterlein,et al.  Boolean Algebras with an Automorphism Group: a Framework for Lukasiewicz Logic , 2008, J. Multiple Valued Log. Soft Comput..

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