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Categorical Abstract Algebraic Logic: Algebraic Semantics for ( \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\bf{\pi }$\end{document} )‐Institutions
Author(s) -
Voutsadakis George
Publication year - 2013
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200920103
Subject(s) - mathematics , algebraic semantics , unary operation , algebra over a field , semantics (computer science) , algebraic number , context (archaeology) , finitary , algebraic extension , pure mathematics , discrete mathematics , computer science , programming language , mathematical analysis , paleontology , differential algebraic equation , ordinary differential equation , differential equation , biology
Abstract Various aspects of the work of Blok and Rebagliato on the algebraic semantics for deductive systems are studied in the context of logics formalized as π‐institutions. Three kinds of semantics are surveyed: institution, matrix (system) and algebraic (system) semantics, corresponding, respectively, to the generalized matrix, matrix and algebraic semantics of the theory of sentential logics. After some connections between matrix and algebraic semantics are revealed, it is shown that every (finitary) N ‐rule based extension of an N ‐rule based π‐institution possessing an algebraic semantics also possesses an algebraic semantics. This result abstracts one of the main theorems of Blok and Rebagliato. An attempt at a Blok‐Rebagliato‐style characterization of those π‐institutions with a mono‐unary category of natural transformations on their sentence functors having an algebraic semantics is also made. Finally, a necessary condition for a π‐institution to possess an algebraic semantics is provided.