Idempotent residuated structures: Some category equivalences and their applications

Galatos, N.; Raftery, J.

doi:10.1090/S0002-9947-2014-06072-8

Idempotent residuated structures: Some category equivalences and their applications
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by N. Galatos and J. G. Raftery PDF

Trans. Amer. Math. Soc. 367 (2015), 3189-3223 Request permission

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 $\neg$. 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 $\mathbf {RM}^\mathbf {t}$ has the projective Beth definability property for deduction.

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