Finitely generated congruence distributive quasivarieties of algebras

We study the class of finite lattices that are isomorphic to the congruence lattices of algebras from a given finitely generated congruence-distributive variety. If this class is as large as allowed by an obvious necessary condition, the variety is called congruence FD-maximal. The main results of this paper characterize some special congruence FD-maximal varieties.

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CRITICAL POINTS OF PAIRS OF VARIETIES OF ALGEBRAS

International Journal of Algebra and Computation ◽

10.1142/s0218196709004932 ◽

2009 ◽

Vol 19 (01) ◽

pp. 1-40 ◽

Cited By ~ 11

Author(s):

PIERRE GILLIBERT

Keyword(s):

Natural Number ◽

Critical Points ◽

Modular Lattice ◽

General Theorem ◽

Finitely Generated ◽

Congruence Distributive

For a class [Formula: see text] of algebras, denote by Conc[Formula: see text] the class of all (∨, 0)-semilattices isomorphic to the semilattice ConcA of all compact congruences of A, for some A in [Formula: see text]. For classes [Formula: see text] and [Formula: see text] of algebras, we denote by [Formula: see text] the smallest cardinality of a (∨, 0)-semilattices in Conc[Formula: see text] which is not in Conc[Formula: see text] if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties [Formula: see text] and [Formula: see text], [Formula: see text] is either finite, or ℵn for some natural number n, or ∞. We also find two finitely generated modular lattice varieties [Formula: see text] and [Formula: see text] such that [Formula: see text], thus answering a question by J. Tůma and F. Wehrung.

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Finite bases for finitely generated, relatively congruence distributive quasivarieties

Algebra Universalis ◽

10.1007/bf01191083 ◽

1991 ◽

Vol 28 (3) ◽

pp. 303-323 ◽

Cited By ~ 5

Author(s):

Wies?aw Dziobiak

Keyword(s):

Finitely Generated ◽

Congruence Distributive

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CONGRUENCE LIFTING OF SEMILATTICE DIAGRAMS

International Journal of Algebra and Computation ◽

10.1142/s021819670900541x ◽

2009 ◽

Vol 19 (07) ◽

pp. 911-924 ◽

Cited By ~ 1

Author(s):

MIROSLAV PLOŠČICA

Keyword(s):

Graph Theory ◽

The Other ◽

Finitely Generated ◽

Four Color Theorem ◽

Congruence Lattices ◽

Congruence Distributive ◽

Distributive Semilattices

We consider the problem, whether the algebras in two finitely generated congruence-distributive varieties have isomorphic congruence lattices. According to the results of P. Gillibert, this problem is closely connected with the question, which diagrams of finite distributive semilattices can be represented by the congruence lattices of algebras in a given variety. We study this question for varieties of bounded lattices, generated by different nondistributive lattices of length 2 (denoted Mn). For each pair from this family of varieties we construct a diagram indexed by the product of three finite chains, which is liftable in one variety and nonliftable in the other one. We also discover an interesting link to the four-color theorem of graph theory.

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Iterative separation in distributive congruence lattices

Mathematica Slovaca ◽

10.2478/s12175-009-0119-2 ◽

2009 ◽

Vol 59 (2) ◽

Cited By ~ 1

Author(s):

Miroslav Ploščica

Keyword(s):

Close Connection ◽

Subdirectly Irreducible ◽

Finitely Generated ◽

Distributive Variety ◽

Algebraic Lattices ◽

Congruence Lattices ◽

Congruence Distributive Variety ◽

Congruence Distributive ◽

Subdirectly Irreducible Algebras

AbstractIn [PLOŠČICA, M.: Separation in distributive congruence lattices, Algebra Universalis 49 (2003), 1–12] we defined separable sets in algebraic lattices and showed a close connection between the types of non-separable sets in congruence lattices of algebras in a finitely generated congruence distributive variety and the structure of subdirectly irreducible algebras in . Now we generalize these results using the concept of separable mappings (defined on some trees) and apply them to some lattice varieties.

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Finite algebras that generate an injectively complete modular variety

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700029750 ◽

1991 ◽

Vol 44 (2) ◽

pp. 303-324 ◽

Cited By ~ 1

Author(s):

Keith A. Kearnes

Keyword(s):

Finite Type ◽

Finite Algebra ◽

Finitely Generated ◽

Modular Variety ◽

Finite Algebras ◽

Congruence Modular Variety ◽

Congruence Distributive

We extend Kollár's result on finitely generated, injectively complete congruence distributive varieties to the congruence modular setting. By doing so we show that, given any finite algebra A of finite type, there is an algorithm to decide whether V(A) is an injectively complete, congruence modular variety.

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PRINCIPAL AND SYNTACTIC CONGRUENCES IN CONGRUENCE-DISTRIBUTIVE AND CONGRUENCE-PERMUTABLE VARIETIES

Journal of the Australian Mathematical Society ◽

10.1017/s144678870800061x ◽

2008 ◽

Vol 85 (1) ◽

pp. 59-74 ◽

Cited By ~ 7

Author(s):

BRIAN A. DAVEY ◽

MARCEL JACKSON ◽

MIKLÓS MARÓTI ◽

RALPH N. MCKENZIE

Keyword(s):

Additional Assumption ◽

Finitely Generated ◽

Distributive Variety ◽

Congruence Distributive Variety ◽

Congruence Permutable Varieties ◽

Congruence Distributive

AbstractWe give a new proof that a finitely generated congruence-distributive variety has finitely determined syntactic congruences (or, equivalently, term finite principal congruences), and show that the same does not hold for finitely generated congruence-permutable varieties, even under the additional assumption that the variety is residually very finite.

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Automorphisms of finite order of nilpotent groups II

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.4.1297 ◽

2014 ◽

Vol 51 (4) ◽

pp. 547-555 ◽

Cited By ~ 1

Author(s):

B. Wehrfritz

Keyword(s):

Nilpotent Group ◽

Finite Order ◽

Finite Index ◽

Nilpotent Groups ◽

Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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