Semiconic idempotent residuated structures
type: Item
instance of: scholarly article
title: Semiconic idempotent residuated structures
zbMATH Open document ID: 1196.06009
author: Ai-ni Hsieh
author: J. G. Raftery
published in: Algebra Universalis
publication date: 2010-01-04T00:00:00Z
review text: A class \(\mathcal{K}\) of similar algebras is said to have the finite embeddability property (briefly, the FEP) if every finite subset of an algebra in \( \mathcal{K}\) can be extended to a finite algebra in \( \mathcal{K}\). If a finitely axiomatized variety or quasivariety of finite type has the FEP, then its universal first-order theory is decidable, hence its equational and quasi-equational theories are decidable as well. A residuated partially ordered monoid is said to be idempotent if its monoid operation is idempotent. In this case, the partial order is equationally definable, so the structures can be treated as pure algebras. Such an algebra is said to be conic if each of its elements lies above or below the monoid identity; it is semiconic if it is a subdirect product of conic algebras. In this paper, it is proved that the class SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite, i.e., every finitely generated member of this class is a finite algebra. It turns out that SCIP is a quasivariety; it is not a variety. The lattice-ordered members of SCIP form a variety SCIL, provided that we add the lattice operations \(\wedge, \vee\) to the similar type. The variety SCIL is not locally finite (it contains all Brouwerian lattices which are not locally finite, as was proved by \textit{C. G. McKay} [J. Symb. Log. 33, 258--264 (1968; Zbl 0175.27103)]). But the local finiteness of SCIP facilitates a proof that SCIL has the FEP. In fact, it is shown here that for every relative subvariety \(\mathcal{K}\) of SCIP, the lattice-ordered members of \(\mathcal{K}\) form a variety with the FEP. It is also shown that SCIL has a continuum of semisimple subvarieties. The results here give a unified explanation of the strong finite model property for many extensions of these and other systems. They partially generalize the main theorem of [\textit{J. G. Raftery}, Trans. Am. Math. Soc. 359, No.~9, 4405--4427 (2007; Zbl 1117.03070)], which showed…
reviewed by: Florentina Chirteş
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 03B47
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 08A50
Mathematics Subject Classification ID: 08C15
zbMATH DE Number: 5652292
zbMATH Keywords: residuated lattice
zbMATH Keywords: finite embeddability property
zbMATH Keywords: locally finite
MaRDI profile type: Publication
full work available at URL: s00012-009-0020-4
OpenAlex ID: W2080811444
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DOI: 10.1007/S00012-009-0020-4
description: scientific article; zbMATH DE number 5652292