Categories of partial algebras for critical points between varieties of algebras
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- 作者：Pierre Gillibert (1)
- 关键词：Primary ; 08A55 ; Secondary ; 08A30 ; 06A12 ; 03C05 ; partial algebra ; congruence relation ; gamp ; pregamp ; variety of algebras ; critical point ; Condensate Lifting Lemma ; lattice ; congruence ; permutable
- 刊名：Algebra Universalis
- 出版年：2014
- 出版时间：June 2014
- 年：2014
- 卷：71
- 期：4
- 页码：299-357
- 全文大小：
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1. LIAFA, Université Paris Diderot, Paris 7, Case 7014, F-75205, Paris Cedex 13, France - ISSN：1420-8911
文摘
We denote by Conc A the \({(\vee, 0)}\) -semilattice of all finitely generated congruences of an algebra A. A lifting of a \({(\vee, 0)}\) -semilattice S is an algebra A such that \({S \cong {\rm Con}_{\rm c} A}\) . The assignment Conc can be extended to a functor. The notion of lifting is generalized to diagrams of \({(\vee, 0)}\) -semilattices. A gamp is a partial algebra endowed with a partial subalgebra together with a semilattice-valued distance; gamps form a category that lends itself to a universal algebraic-type study. The raison d’être of gamps is that any algebra can be approximated by its finite subgamps, even in case it is not locally finite. Let \({\mathcal{V}}\) and \({\mathcal{W}}\) be varieties of algebras (on finite, possibly distinct, similarity types). Let P be a finite lattice. We assume the existence of a combinatorial object, called an \({\aleph_0}\) -lifter of P, of infinite cardinality \({\lambda}\) . Let \({\vec{A}}\) be a P-indexed diagram of finite algebras in \({\mathcal{V}}\) . If \({{\rm Con}_{\rm c} \circ \vec{A}}\) has no partial lifting in the category of gamps of \({\mathcal{W}}\) , then there is an algebra \({A \in \mathcal{V}}\) of cardinality \({\lambda}\) such that Conc A is not isomorphic to Conc B for any \({B \in \mathcal{W}}\) . This makes it possible to generalize several known results. In particular, we prove the following theorem, without assuming that \({\mathcal{W}}\) is locally finite. Let \({\mathcal{V}}\) be locally finite and let \({\mathcal{W}}\) be congruence-proper (i.e., congruence lattices of infinite members of \({\mathcal{W}}\) are infinite). The following equivalence holds. Every countable \({(\vee, 0)}\) -semilattice with a lifting in \({\mathcal{V}}\) has a lifting in \({\mathcal{W}}\) if and only if every \({\omega}\) -indexed diagram of finite \({(\vee, 0)}\) -semilattices with a lifting in \({\mathcal{V}}\) has a lifting in \({\mathcal{W}}\) . Gamps are also applied to the study of congruence-preserving extensions. Let \({\mathcal{V}}\) be a non-semidistributive variety of lattices and let n ≥? be an integer. There is a bounded lattice \({A \in \mathcal{V}}\) of cardinality \({\aleph_1}\) with no congruence n-permutable, congruence-preserving extension. The lattice A is constructed as a condensate of a square-indexed diagram of lattices.