Galatos, N.Raftery, James G.2015-03-052015-03-052015Galatos, N & Raftery, JG 2015, 'Idempotent residuated structures : some category equivalences and their applications',Transactions of the American Mathematical Society, vol. 367, no. 5, pp. 3189-3223.0002-9947 (print)1088-6850 (online)http://hdl.handle.net/2263/43872This 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.enFirst published in Transactions of the American Mathematical Society in vol 36, no. 4. 2015, published by the American Mathematical Society.IdempotentResiduationSemilinearRepresentableNucleusSugihara monoidRelative Stone algebraCategory equivalenceEpimorphismAmalgamationBeth de nabilityInterpolationR-mingleArticle