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Categorical abstract algebraic logic: The largest theory system included in a theory family
Author(s) -
Voutsadakis George
Publication year - 2006
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.200510034
Subject(s) - mathematics , categorical variable , algebraic theory , category theory , algebraic number , class (philosophy) , pure mathematics , sign (mathematics) , model theory , property (philosophy) , algebra over a field , discrete mathematics , calculus (dental) , epistemology , mathematical analysis , philosophy , statistics , medicine , dentistry
In this note, it is shown that, given a π ‐institution ℐ = 〈 Sign , SEN, C 〉, with N a category of natural transformations on SEN, every theory family T of ℐ includes a unique largest theory system $ \overleftarrow T $ of ℐ. $ \overleftarrow T $ satisfies the important property that its N ‐Leibniz congruence system always includes that of T . As a consequence, it is shown, on the one hand, that the relation Ω N ( $ \overleftarrow T $ ) = Ω N ( T ) characterizes N ‐protoalgebraicity inside the class of N ‐prealgebraic π ‐institutions and, on the other, that all N ‐Leibniz theory families associated with theory families of a protoalgebraic π ‐institution ℐ are in fact N ‐Leibniz theory systems. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)