Model Completions, r-Heyting Categories

Abstract Under some assumptions on an equational theoryS (EDPC and EDPM), we give a necessary and sufficient condition so thatS admits a model completion. These assumptions are often met by the equational theories arising from logic. They say that the dual of the category of finitely presentedS-algebras has some categorical stucture. The results of this paper combined with those of [7] show that all the 8 theories of amalgamable varieties of Heyting algebras [12] admit a model completion. Further applications to varieties of modal algebras are given in [8].

[1]  W. Blok,et al.  A finite basis theorem for quasivarieties , 1986 .

[2]  Gonzalo E. Reyes,et al.  Completeness Results for Intuitionistic and Modal Logic in a Categorical Setting , 1995, Ann. Pure Appl. Log..

[3]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[4]  L. L. Maksimova,et al.  An analog of Beth's theorem in normal extensions of the modal logic K4 , 1992 .

[5]  P. Köhler,et al.  Varieties with equationally definable principal congruences , 1980 .

[6]  Andrew M. Pitts,et al.  Conceptual Completeness for First-Order Intuitionistic Logic: An Application of Categorical Logic , 1989, Ann. Pure Appl. Log..

[7]  W. Wheeler.  Model-companions and definability in existentially complete structures , 1976 .

[8]  W. Blok,et al.  On the structure of varieties with equationally definable principal congruences IV , 1994 .

[9]  Michael Barr,et al.  Representation of categories , 1986 .

[10]  M. Makkai,et al.  First order categorical logic , 1977 .

[11]  Silvio Ghilardi,et al.  A Sheaf Representation and Duality for Finitely Presenting Heyting Algebras , 1995, J. Symb. Log..

[12]  L. Maksimova.  Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras , 1977 .

[13]  Andrew M. Pitts,et al.  On an interpretation of second order quantification in first order intuitionistic propositional logic , 1992, Journal of Symbolic Logic.

[14]  P. Gabriel,et al.  Lokal α-präsentierbare Kategorien , 1971 .