Idempotent residuated structures : some category equivalences and their applications

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Galatos, N.
Raftery, James G.

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American Mathematical Society

Abstract

This paper concerns residuated lattice-ordered idempotent commutative monoids that are subdirect products of chains. An algebra of this kind is a generalized Sugihara monoid (GSM) if it is generated by the lower bounds of the monoid identity; it is a Sugihara monoid if it has a compatible involution :. Our main theorem establishes a category equivalence between GSMs and relative Stone algebras with a nucleus (i.e., a closure operator preserving the lattice operations). An analogous result is obtained for Sugihara monoids. Among other applications, it is shown that Sugihara monoids are strongly amalgamable, and that the relevance logic RMt has the projective Beth de nability property for deduction.

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Keywords

Idempotent, Residuation, Semilinear, Representable, Nucleus, Sugihara monoid, Relative Stone algebra, Category equivalence, Epimorphism, Amalgamation, Beth de nability, Interpolation, R-mingle

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Citation

Galatos, N & Raftery, JG 2015, 'Idempotent residuated structures : some category equivalences and their applications',Transactions of the American Mathematical Society, vol. 367, no. 5, pp. 3189-3223.