Semiconic idempotent residuated structures.

*(English)*Zbl 1196.06009A 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 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 [J. G. Raftery, Trans. Am. Math. Soc. 359, No. 9, 4405–4427 (2007; Zbl 1117.03070)], which showed that the variety generated by all idempotent commutative residuated chains is locally finite. It is shown here that the involutive algebras in SCIL are subdirect products of chains.

Finally, it is shown that there are just \(2^{\aleph_{0}}\) semisimple varieties of semiconic idempotent commutative residuated lattices and every such lattice is distributive, and therefore a Sugihara monoid. Thus, we gain no finiteness results by imposing involution on the algebras in SCIL. Moreover, since the subdirectly irreducible Sugihara monoids are totally ordered, most of the algebras in SCIL (or in SCIP) cannot be embedded into involutive algebras in SCIL.

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 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 [J. G. Raftery, Trans. Am. Math. Soc. 359, No. 9, 4405–4427 (2007; Zbl 1117.03070)], which showed that the variety generated by all idempotent commutative residuated chains is locally finite. It is shown here that the involutive algebras in SCIL are subdirect products of chains.

Finally, it is shown that there are just \(2^{\aleph_{0}}\) semisimple varieties of semiconic idempotent commutative residuated lattices and every such lattice is distributive, and therefore a Sugihara monoid. Thus, we gain no finiteness results by imposing involution on the algebras in SCIL. Moreover, since the subdirectly irreducible Sugihara monoids are totally ordered, most of the algebras in SCIL (or in SCIP) cannot be embedded into involutive algebras in SCIL.

Reviewer: Florentina Chirteş (Craiova)

##### MSC:

06F05 | Ordered semigroups and monoids |

03B47 | Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) |

03G25 | Other algebras related to logic |

08A50 | Word problems (aspects of algebraic structures) |

08C15 | Quasivarieties |

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\textit{A.-n. Hsieh} and \textit{J. G. Raftery}, Algebra Univers. 61, No. 3--4, 413--430 (2009; Zbl 1196.06009)

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