Small congruence distributive varieties: Retracts, injectives, equational compactness and amalgamation

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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Subdirect irreducibility and equational compactness in unary algebras

Archiv der Mathematik ◽

10.1007/bf01220912 ◽

1970 ◽

Vol 21 (1) ◽

pp. 256-264 ◽

Cited By ~ 17

Author(s):

G�nter H. Wenzel

Keyword(s):

Subdirect Irreducibility ◽

Equational Compactness

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Equational compactness in infinitary algebras

Colloquium Mathematicum ◽

10.4064/cm-27-2-197-205 ◽

1973 ◽

Vol 27 (2) ◽

pp. 197-205 ◽

Cited By ~ 2

Author(s):

Bernhard Banaschewski ◽

Evelyn Nelson

Keyword(s):

Equational Compactness

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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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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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