2012
DOI: 10.1142/s0218196711006716
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Clones of Algebras With Parallelogram Terms

Abstract: Abstract. We describe a manageable set of relations that generates the finitary relational clone of an algebra with a parallelogram term. This result applies to any algebra with a Maltsev term and to any algebra with a near unanimity term. One consequence of the main result is that on any finite set and for any finite k there are only finitely many clones of algebras with a k-ary parallelogram term which generate residually small varieties.

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Cited by 29 publications
(45 citation statements)
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“…In addition, [13,Theorem 3.5] [13]; [9]). Every variety with a parallelogram term is congruence modular.…”
Section: 3mentioning
confidence: 99%
See 1 more Smart Citation
“…In addition, [13,Theorem 3.5] [13]; [9]). Every variety with a parallelogram term is congruence modular.…”
Section: 3mentioning
confidence: 99%
“…Theorem 2.6 (From [13]). Let C be an n-ary critical relation of a finite algebra A with a k-parallelogram term, and let C be its reduced representation.…”
Section: 3mentioning
confidence: 99%
“…It has been noted, see for example [19], that term conditions similar to those listed in Definition 1.5 can sometimes be conveniently described using matrices over the variables x and y. If t is an n-ary term and A = (a ij ) is an m × n matrix of variables, then the expression t[A] can be interpreted as the column vector whose ith entry is the term t(a i1 , a i2 , .…”
Section: Matrix Presentations Of Some Maltsev Conditionsmentioning
confidence: 99%
“…. , x n ∈ {x, y} and x i = x. Cube terms were independently discovered in [12] and [4,10], in the latter two papers in connection with the property that algebras in the variety have few subpowers and with algorithms for the CSP generalizing Gaussian elimination.…”
Section: Cube Termmentioning
confidence: 99%