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Residuated lattices arising from equivalence relations on Boolean and Brouwerian algebras
Author(s) -
Vetterlein Thomas
Publication year - 2008
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.200710048
Subject(s) - mathematics , quotient algebra , quotient , pure mathematics , dual (grammatical number) , algebraic structure , equivalence (formal languages) , equivalence relation , free boolean algebra , interior algebra , stone's representation theorem for boolean algebras , algebra over a field , set (abstract data type) , heyting algebra , two element boolean algebra , algebra representation , jordan algebra , computer science , art , literature , programming language
Abstract Logics designed to deal with vague statements typically allow algebraic semantics such that propositions are interpreted by elements of residuated lattices. The structure of these algebras is in general still unknown, and in the cases that a detailed description is available, to understand its significance for logics can be difficult. So the question seems interesting under which circumstances residuated lattices arise from simpler algebras in some natural way. A possible construction is described in this paper. Namely, we consider pairs consisting of a Brouwerian algebra (i. e. a dual Heyting algebra) and an equivalence relation. The latter is assumed to be in a certain sense compatible with the partial order, with the formation of differences, and with the formation of suprema of pseudoorthogonal elements; we then call it an s‐equivalence relation. We consider operations which, under a suitable additional assumption, naturally arise on the quotient set. The result is that the quotient set bears the structure of a residuated lattice. Further postulates lead to dual BL‐algebras. In the case that we begin with Boolean algebras instead, we arrive at dual MV‐algebras. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)