scholarly journals The Gumm level equals the alvin level in congruence distributive varieties

2020 ◽  
Vol 115 (4) ◽  
pp. 391-400 ◽  
Author(s):  
Paolo Lipparini
Keyword(s):  
Congruence Distributive

scholarly journals CONGRUENCE FD-MAXIMAL VARIETIES OF ALGEBRAS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250053 ◽  
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.

scholarly journals Another characterization of congruence distributive varieties

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.

scholarly journals CRITICAL POINTS OF PAIRS OF VARIETIES OF ALGEBRAS

2009 ◽  
Vol 19 (01) ◽  
pp. 1-40 ◽  
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.

Congruence—distributive polynomial reducts of lattices

1979 ◽  
Vol 9 (1) ◽  
pp. 142-145 ◽  
Author(s):  
Kirby A. Baker
Keyword(s):  
Congruence Distributive

scholarly journals Mildly distributive semilattices

1984 ◽  
Vol 36 (3) ◽  
pp. 287-315 ◽  
Author(s):  
Robert Hickman
Keyword(s):  
Congruence Distributive ◽  
Distributive Semilattices

AbstractThere is no single generalization of distributivity to semilattices. This paper investigates the class of mildly distributive semilattices, which lies between the two most commonly discussed classes in this area—weakly distributive semilattices and distributive semilattices. Particular attention is paid to describing and characterizing congruence distributive mildly distributive semilattices, in contrast to distributive semilattices, whose lattice of join partial congruences is badly behaved and which are difficult to describe.

scholarly journals Monotone clones, residual smallness and congruence distributivity

1990 ◽  
Vol 41 (2) ◽  
pp. 283-300 ◽  
Author(s):  
Ralph McKenzie
Keyword(s):  
Ordered Set ◽  
Ordered Sets ◽  
Congruence Distributivity ◽  
Congruence Modularity ◽  
Congruence Distributive

Corresponding to each ordered set there is a variety, determined up to equivalence, generated by an algebra whose term operations are all the monotone operations on the ordered set. We produce several characterisations of the finite bounded ordered sets for which the corresponding variety is congruence-distributive. In particular, we find that congruence-distributivity, congruence-modularity, and residual smallness are equivalent for these varieties.

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