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Olson, J. S.; Raftery, J. G.

Residuated structures, concentric sums and finiteness conditions. (English) Zbl 1298.03117

Commun. Algebra 36, No. 10, 3632-3670 (2008).
Summary: This article is motivated by a concern with finiteness conditions on varieties of residuated structures – particularly residuated meet semilattice-ordered commutative monoids. A “concentric sum” construction is developed and is used to prove, among other results, a local finiteness theorem for a class that encompasses all \(n\)-potent hoops and all idempotent subdirect products of residuated chains. This in turn implies that a range of residuated lattice-based varieties have the finite embeddability property, whence their quasi-equational theories are decidable. Applications to substructural logics are discussed.

Cited in 2 Documents

MSC:

03G25 Other algebras related to logic
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
06D99 Distributive lattices
06F05 Ordered semigroups and monoids
08C15 Quasivarieties

Keywords:

decidability; finite embeddability property; hoop; locally finite variety; natural order; residuation; substructural logics
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