Categorical Abstract Algebraic Logic: More on Protoalgebraicity

Protoalgebraic logics are characterized by the monotonicity of the Leibniz operator on their theory lattices and are at the lower end of the Leibniz hierarchy of abstract algebraic logic. They have been shown to be the most primitive among those logics with a strong enough algebraic character to be amenable to algebraic study techniques. Protoalgebraic π -institutions were introduced recently as an analog of protoalgebraic sentential logics with the goal of extending the Leibniz hierarchy from the sentential framework to the π -institution framework. Many properties of protoalgebraic logics, studied in the sentential logic framework by Blok and Pigozzi, Czelakowski, and Font and Jansana, among others, have already been adapted in previous work by the author to the categorical level. This work aims at further advancing that study by exploring in this new level some more properties of protoalgebraic sentential logics.

[1]  Willem J. Blok,et al.  Protoalgebraic logics , 1986, Stud Logica.

[2]  J. Czelakowski.  Filter distributive logics , 1984 .

[3]  Burghard Herrmann.  Characterizing Equivalential and Algebraizable Logics by the Leibniz Operator , 1997, Stud Logica.

[4]  George Voutsadakis,et al.  Categorical Abstract Algebraic Logic: Equivalent Institutions , 2003, Stud Logica.

[5]  Josep Maria Font,et al.  A Survey of Abstract Algebraic Logic , 2003, Stud Logica.

[6]  T. Prucnal,et al.  An algebraic characterization of the notion of structural completeness , 1974 .

[7]  G. Voutsadakis.  Categorical Abstract Algebraic Logic , 1998 .

[8]  George Voutsadakis.  Categorical Abstract Algebraic Logic: Prealgebraicity and Protoalgebraicity , 2007, Stud Logica.

[9]  GEORGE VOUTSADAKIS,et al.  Categorical Abstract Algebraic Logic: Algebraizable Institutions , 2002, Appl. Categorical Struct..

[10]  Hans-Jörg Kreowski,et al.  Recent Trends in Data Type Specification , 1985, Informatik-Fachberichte.

[11]  Michael Barr,et al.  Category theory for computing science , 1995, Prentice Hall International Series in Computer Science.

[12]  George Voutsadakis.  Categorical Abstract Algebraic Logic: Full Models, Frege Systems and Metalogical Properties , 2006, Reports Math. Log..

[13]  F. Borceux.  Handbook Of Categorical Algebra 1 Basic Category Theory , 2008 .

[14]  George Voutsadakis.  Categorical Abstract Algebraic Logic: (ℐ,N)-Algebraic Systems , 2005, Appl. Categorical Struct..

[15]  Joseph A. Goguen,et al.  Introducing Institutions , 1983, Logic of Programs.

[16]  G. Voutsadakis.  CATEGORICAL ABSTRACT ALGEBRAIC LOGIC: GENTZEN (π)-INSTITUTIONS , 2007 .

[17]  George Voutsadakis.  Categorical Abstract Algebraic Logic Metalogical Properties , 2003, Stud Logica.

[18]  Janusz Czelakowski,et al.  Weakly Algebraizable Logics , 2000, J. Symb. Log..

[19]  José Luiz Fiadeiro,et al.  Structuring Theories on Consequence , 1988, ADT.

[20]  B. Herrmann.  Equivalential and algebraizable logics , 1996 .

[21]  Sally Popkorn,et al.  A Handbook of Categorical Algebra , 2009 .

[22]  Joseph A. Goguen,et al.  Institutions: abstract model theory for specification and programming , 1992, JACM.

[23]  George Voutsadakis.  Categorical Abstract Algebraic Logic: Models of π-Institutions , 2005, Notre Dame J. Formal Log..

[24]  W. Dziobiak,et al.  A deduction theorem schema for deductive systems of propositional logics , 1991, Stud Logica.

[25]  S. Lane.  Categories for the Working Mathematician , 1971 .

[26]  W. Blok.  Algebraic Semantics for Universal Horn Logic Without Equality , 1992 .

[27]  J. Czelakowski.  Equivalential logics (II) , 1981 .