Dualizable algebras with parallelogram terms

Kearnes, Keith A.; Szendrei, Ágnes

doi:10.1007/s00012-016-0410-3

Cited by 6 publications

(7 citation statements)

References 13 publications

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“…Finally, Gillibert proved the dualizability of all finite Abelian algebras [4], answering a question from [1]. The same result was independently shown by Kearnes and Szendrei [8].…”

Section: Introductionmentioning

confidence: 64%

Finite aﬃne algebras are fully dualizable

Bentz

Gillibert

Sequeira

2017

Communications in Algebra

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We show that every finite Abelian algebra A from congruencepermutable varieties admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure A ∼ of finite type. We give an explicit bound on the arities of the partial and total operations appearing in A ∼ . In addition, we show that the enriched partial hom-clone of A is finitely generated as a clone.Given an algebra A, we denote by A its underlying set, and by Sub(A) the set of subalgebras of A. The variety Var(A) (respectively, the quasivariety QVar(A)) generated by A is the smallest classes of algebras, with the signature of A, that is closed under taking products, subalgebras, and homomorphic images (respectively, products, subalgebras, and isomorphic algebras).Given X ⊆ A we denote by X the subalgebra of A generated by X. For an arbitrary set X and a variety V, we denote by F V (X) the algebra freely generated by X in V.A subproduct algebra A ≤ Π i A i is called a subdirect product if π i (A) = A i for each projection π i . An algebra is subdirectly irreducible if whenever it is isomorphic to a subdirect product, it is already isomorphic to one of its factors.An algebra A is affine if there exists an Abelian group structure A; +, 0, − such that t(x, y, z) = x − y + z is both a term function of A and a homomorphism from A 3 to A. A class of algebras C is affine if all of its algebras are. In the case of an affine variety V, it is easy to see that we may choose one term t that witnesses the affinity simultaneously for all members of V (e.g. we could take the term witnessing the affinity of F V (ω)).An algebra A is Abelian if [1 A , 1 A ] = 0 A , where 1 A and 0 A are the universal and trivial relations on A, and [·, ·] denotes the binary commutator on the congruences of A (we refer to [3] for the definition of the commutator). A class of algebras is Abelian if all of its members are. As usual, when dealing with commutator theoretic conditions, we restrict to algebras that generate congruence-modular varieties. With this condition, Abelian algebras and varieties coincide with affine algebras and varieties [3, Corollary 5.9], and we will use the two notations interchangeably throughout the paper. Our results will rely exclusively on the defining property of affine algebras.We repeat several results about congruences of Abelian algebras from [4].Definition 2.1 ([4], Definition 3.1). Let A be an Abelian algebra and B ∈ Sub(A). The congruence generated by B, denoted by Θ B is the smallest congruence of A containing B 2 .We remark that not every congruence of A can be written in the form Θ B for some subalgebra B, and that we might have Θ B = Θ C with B = C.Lemma 2.2 ([4], Lemma 3.3). Let A be an Abelian algebra, let B ∈ Sub(A), and let t be a term witnessing the affinity of A. ThenNote that this result implies that B is a congruence class of Θ B . Lemma 2.3 ([4], Corollary 3.7). Let A be an Abelian algebra and let B ∈ Sub(A) such that B is meet irreducible in the semilattice Sub(A);...

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“…Finally, Gillibert proved the dualizability of all finite Abelian algebras [4], answering a question from [1]. The same result was independently shown by Kearnes and Szendrei [8].…”

Section: Introductionmentioning

confidence: 64%

Finite aﬃne algebras are fully dualizable

Bentz

Gillibert

Sequeira

2017

Communications in Algebra

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“…In [7], the split centralizer condition for a finite algebra A was defined to be the condition that, for Q := SP(A) and for any subalgebra B ≤ A, each relevant triple (δ, θ, ν) of B is split (relative to Q) by some triple (α, β, κ). Here we shall consider a modified version of this condition, which allows us to ignore the role of the quasivariety Q. Namely, we shall only consider the situation where relevant triples (δ, θ, ν) are split at 0.…”

Section: Introductionmentioning

confidence: 99%

“…If A A is the expansion of A by constants, then A A satisfies the split centralizer condition as defined in [7] if and only if A A has centralizers split at 0 as defined in Definition 1.3. But without expanding A by constants we do not have equivalence.…”

Section: Introductionmentioning

confidence: 99%

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Neutrabelian algebras

Kearnes,

Meredith,

Szendrei

2020

Preprint

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“…Terms of equal strength, called parallelogram terms, were discovered independently and at the same time in the study of finitely related clones, [9]. Cube terms and their equivalents have played roles in [8] in the study of constraint satisfaction problems, in [1,2,9] in the study of finitely related clones, in [10,13] in natural duality theory, and in [5] concerning the subpower membership problem.…”

Section: Introductionmentioning

confidence: 99%

Cube term blockers without finiteness

Kearnes

Szendrei

2017

Algebra Univers.

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Abstract. We show that an idempotent variety has a d-dimensional cube term if and only if its free algebra on two generators has no d-ary compatible cross. We employ Hall's Marriage Theorem to show that a variety of finite signature whose fundamental operations have arities n 1 , . . . , n k has a d-dimensional cube term if and only if it has one of dimension d = 1 + k i=1 (n i − 1). This lower bound on dimension is shown to be sharp. We show that a pure cyclic term variety has a cube term if and only if it contains no 2-element semilattice. We prove that the Maltsev condition "existence of a cube term" is join prime in the lattice of idempotent Maltsev conditions.

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Dualizable algebras with parallelogram terms

Cited by 6 publications

References 13 publications

Finite aﬃne algebras are fully dualizable

Finite aﬃne algebras are fully dualizable

Neutrabelian algebras

Cube term blockers without finiteness

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