Getting Larders from Congruence Lattices of First-Order Structures

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• 作者：Pierre Gillibert (1) pgillibert@yahoo.fr
  Friedrich Wehrung (2) wehrung@math.unicaen.fr
• 刊名：Lecture Notes in Mathematics
• 出版年：2011
• 出版时间：2011
• 年：2011
• 卷：2029
• 期：1
• 页码：81-116
• 全文大小：615.0 KB
• 作者单位：1. Department of Mathematics, Charles University in Prague, Sokolovská 83, 18600 Prague Praha, Czech Republic2. Department of Mathematics, University of Caen, LMNO, CNRS UMR 6139, 14032 Caen, Cedex, France
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Dynamical Systems and Ergodic Theory
  Mathematical Biology
  Partial Differential Equations
  Functional Analysis
  Abstract Harmonic Analysis
  Group Theory and Generalizations
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1617-9692

文摘

One of the main origins of our work is the first author’s paper Gillibert (Int. J. Algebra Comput. 19(1):1–40, 2009), where it is proved, in particular, that the critical point crit(A;B) between a locally finite variety A and a finitely generated congruence-distributive variety B such that Con_cA \nsubseteq Con_cB {\rm Con_c}A \nsubseteq {\rm Con_c}B is always less than \mathfrakN_w \mathfrak{N}_w . One of the goals of the present chapter is to show how routine categorical verifications about algebraic systems make it possible, using CLL, to extend this result to relative compact congruence semilattices of quasivarieties of algebraic systems (i.e., the languages now have relations as well as operations, and we are dealing with quasivarieties rather than varieties). That particular extension is stated and proved in Theorem 4.9.4. We also obtain a version of Gr?tzer–Schmidt’s Theorem for poset-indexed diagrams of (V, 0)-semilattices and (V, 0)-homomorphisms in Theorem 4.7.2.With further potential applications in view, most of Chap. 4 is designed to build up a framework for being able to easily verify larderhood of many structures arising from (generalized) quasivarieties of algebraic systems. Although we included in this chapter, for convenience sake, a number of already known or folklore results, it also contains results which, although they could be in principle obtained from already published results, could not be so in a straightforward fashion. Such results are Proposition 4.2.3 (description of some weakly k-presented structures in MInd) or Theorem4.4.1 (preservation of all small directed colimits by the relative compact congruence semilattice functor within a given generalized quasivariety). The structures studied in Chap. 4 will be called monotone-indexed struc- tures. They form a category, that we shall denote by MInd. The objects of MInd are just the first-order structures. For first-order structures A and B, a morphism from A to B in MInd can exist only if the language of A is contained in the language of B, and then it is defined as a homomorphism (in the usual sense) from A to the reduct of B to the language of A.