Mathematical Logic Quarterly 54 (4):350-367 (2008)

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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 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
Keywords s‐equivalence relation  divisible residuated lattice  BL‐algebra  Heyting algebra  MV‐algebra  Boolean algebra  Brouwerian algebra
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DOI 10.1002/malq.200710048
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Effect Algebras and Unsharp Quantum Logics.D. J. Foulis & M. K. Bennett - 1994 - Foundations of Physics 24 (10):1331-1352.
A Non-Classical Logic for Physics.Robin Giles - 1974 - Studia Logica 33 (4):397 - 415.
Representation Theorems for Quantales.Silvio Valentini - 1994 - Mathematical Logic Quarterly 40 (2):182-190.

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