Affine complete varieties are congruence distributive

AbstractThe main results of the paper are the following: 1. Every locally finite affine complete variety admits a near unanimity term; 2. A locally finite congruence distributive variety is affine complete if and only if all its algebras with no proper subalgebras are affine complete and the variety is generated by one of such algebras. The first of these results sharpens a result of McKenzie asserting that all locally finite affine complete varieties are congruence distributive. The second one generalizes the result by Kaarli and Pixley that characterizes arithmetical affine complete varieties.

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CONGRUENCE FD-MAXIMAL VARIETIES OF ALGEBRAS

International Journal of Algebra and Computation ◽

10.1142/s0218196712500531 ◽

2012 ◽

Vol 22 (06) ◽

pp. 1250053 ◽

Cited By ~ 1

Author(s):

PIERRE GILLIBERT ◽

MIROSLAV PLOŠČICA

Keyword(s):

Necessary Condition ◽

Finitely Generated ◽

Distributive Variety ◽

Finite Lattices ◽

Congruence Lattices ◽

Congruence Distributive Variety ◽

Congruence Distributive ◽

Special Congruence

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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Another characterization of congruence distributive varieties

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2020.57.3.1471 ◽

2020 ◽

Vol 57 (3) ◽

pp. 284-289

Author(s):

Paolo Lipparini

Keyword(s):

Natural Number ◽

Congruence Identity ◽

Congruence Distributive

AbstractWe provide a Maltsev characterization of congruence distributive varieties by showing that a variety 𝓥 is congruence distributive if and only if the congruence identity … (k factors) holds in 𝓥, for some natural number k.

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Affine complete Stone algebras

Acta Mathematica Academiae Scientiarum Hungaricae ◽

10.1007/bf01895229 ◽

1982 ◽

Vol 39 (1-3) ◽

pp. 169-174 ◽

Cited By ~ 4

Author(s):

R. Beazer

Keyword(s):

Affine Complete

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Affine Complete Algebras

Springer Proceedings in Mathematics & Statistics - Mathematical Sciences with Multidisciplinary Applications ◽

10.1007/978-3-319-31323-8_11 ◽

2016 ◽

pp. 235-252

Author(s):

Gérard Kientega

Keyword(s):

Affine Complete

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Strictly locally order affine complete lattices

Order ◽

10.1007/bf01110547 ◽

1993 ◽

Vol 10 (3) ◽

pp. 261-270 ◽

Cited By ~ 1

Author(s):

Kalle Kaarli ◽

Karin T�ht

Keyword(s):

Complete Lattices ◽

Affine Complete

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On Hagemann's and Herrmann's characterization of strictly affine complete algebras

Algebra Universalis ◽

10.1007/s000120050173 ◽

2000 ◽

Vol 44 (1-2) ◽

pp. 105-121 ◽

Cited By ~ 9

Author(s):

Erhard Aichinger

Keyword(s):

Affine Complete

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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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