Finitely related algebras in congruence modular varieties have few subpowers

Barto, Libor

doi:10.4171/jems/790

Cited by 18 publications

(34 citation statements)

References 41 publications

(61 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The algebra A is a finitely related if A can be chosen to have only finitely many relations. Some of the most spectacular results (for example, [1,2]) have been proven (and are only true) under this assumption.…”

Section: Smallness Assumptionsmentioning

confidence: 99%

“…m(x, x, y) ≈ m(y, x, x) ≈ y (2) are satisfied in any group (A, +, −): one can simply assign to m the term operation (x, y, z) → x − y + z. A set of identities is non-trivial if it cannot be satisfied in any algebra on at least two-element domain whose only fundamental operations are projections.…”

Section: Mal'cev Conditionsmentioning

confidence: 99%

“…Historically, some of the earliest results of the type described above dealt with the shape of the lattice of congruence relations of an algebra. For example, an algebra A satisfies the identities (2) if and only if the variety generated by A is congruence permutable [24]; similarly sets of identities can be used to classify congruence distributive [21], congruence modular [18], and congruence meet-semidistributive [28] varieties. Another recent result in this direction is the characterisation of congruence distributivity by near unanimity identities in finitely related algebras [1].…”

Section: Mal'cev Conditionsmentioning

confidence: 99%

See 2 more Smart Citations

Pseudo‐loop conditions

Gillibert

Jonušas

Pinsker

2019

Bull. London Math. Soc.

View full text Add to dashboard Cite

About a decade ago, it was realised that the satisfaction of a given identity (or equation) of the form ffalse(x1,…,xnfalse)≈ffalse(y1,…,ynfalse) in an algebra is equivalent to the algebra forcing a loop into any graph on which it acts and which contains a certain finite subgraph associated with the identity. Such identities have since also been called loop conditions, and this characterisation has produced spectacular results in universal algebra, such as the satisfaction of a Siggers identity s(x,y,z,x)≈s(y,x,y,z) in any arbitrary non‐trivial finite idempotent algebra. We initiate, from this viewpoint, the systematic study of sets of identities of the form ffalse(x1,1,…,x1,nfalse)≈⋯≈ffalse(xm,1,…,xm,nfalse), which we call loop conditions of width m. We show that their satisfaction in an algebra is equivalent to any action of the algebra on a certain type of relation forcing a constant tuple into the relation. Proving that for each fixed width m there is a weakest loop condition (that is, one entailed by all others), we obtain a new and short proof of the recent celebrated result stating that there exists a concrete loop condition of width 3 which is entailed in any non‐trivial idempotent, possibly infinite, algebra. The framework of classical (width 2) loop conditions is insufficient for such proof. We then consider pseudo‐loop conditions of finite width, a generalisation suitable for non‐idempotent algebras; they are of the form u1∘ffalse(x1,1,…,x1,nfalse)≈⋯≈um∘ffalse(xm,1,…,xm,nfalse), and of central importance for the structure of algebras associated with ω‐categorical structures. We show that for the latter, satisfaction of a pseudo‐loop condition is characterised by pseudo‐loops, that is, loops modulo the action of the automorphism group, and that a weakest pseudo‐loop condition exists (for ω‐categorical cores). This way we obtain a new and short proof of the theorem that the satisfaction of any non‐trivial identities of height 1 in such algebras implies the satisfaction of a fixed single identity.

show abstract

Section: Smallness Assumptionsmentioning

confidence: 99%

Section: Mal'cev Conditionsmentioning

confidence: 99%

Section: Mal'cev Conditionsmentioning

confidence: 99%

See 1 more Smart Citation

Pseudo‐loop conditions

Gillibert

Jonušas

Pinsker

2019

Bull. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…For item (2), let us assume first that V has no cube term. Applying Theorem 2.5 (for δ = ω) we conclude that there is an ω-sequence, σ = (U 0 , U 1 , .…”

Section: Cube Terms and Crossesmentioning

confidence: 99%

“…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.

View full text Add to dashboard Cite

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.

show abstract