The weakest nontrivial idempotent equations

Olšák, Miroslav

doi:10.1112/blms.12097

Cited by 28 publications

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“…Then we simply refer to the abovementioned existence of a weakest condition of width 3. The identities we obtain are not the same as the identities obtained in ; our identities are also of width 3, but have four variables.…”

Section: Introductionmentioning

confidence: 74%

“…This includes the above criterion for non‐triviality of finite idempotent algebras, and more generally, a locally finite variety (this includes varieties generated by a single algebra) is non‐trivial if and only if it satisfies either of the Siggers' identities , or equivalently, weak near unanimity identities . Moreover, a finite idempotent algebra is non‐trivial if and only if satisfies a cyclic identity ; and an arbitrary (possibly infinite) idempotent algebra (or locally finite variety) is non‐trivial if and only if it satisfies Olšák's identities , that is the set of identities

o (x, y, y, y, x, x) \approx o (y, x, y, x, y, x) \approx o (y, y, x, x, x, y) .

…”

Section: Introductionmentioning

confidence: 99%

“…The first result of this paper is showing that for each fixed width

m ⩾ 2

, there is a weakest non‐trivial loop condition. This approach allows us to provide a short and, compared to the original one in , elementary proof of the existence of a weakest non‐trivial strong Mal'cev condition for idempotent algebras. We perform a purely syntactic composition in order to obtain, from a Taylor term (which exists in any non‐trivial idempotent algebra by the classical result due to Taylor ), a term which satisfies some loop condition of width 3 (but of unknown arity).…”

Section: Introductionmentioning

confidence: 99%

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Pseudo‐loop conditions

Gillibert

Jonušas

Pinsker

2019

Bull. London Math. Soc.

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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.

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Section: Introductionmentioning

confidence: 74%

o (x, y, y, y, x, x) \approx o (y, x, y, x, y, x) \approx o (y, y, x, x, x, y) .

…”

Section: Introductionmentioning

confidence: 99%

“…The first result of this paper is showing that for each fixed width

m ⩾ 2

Section: Introductionmentioning

confidence: 99%

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Pseudo‐loop conditions

Gillibert

Jonušas

Pinsker

2019

Bull. London Math. Soc.

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“…Taylor varieties are essential in universal algebra, especially in tame congruence theory and Maltsev conditions. The fact that locally finite Taylor algebras can be characterized by such a simple condition was utterly unexpected in universal algebra, and a similar condition was later found for infinite Taylor algebras [10].…”

Section: Introductionmentioning

confidence: 70%

“…Another loop lemma based on absorption, which was used for the proof that there are the weakest non-trivial idempotent equations [10] and which drops the finiteness assumption, has the following form. Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

The local loop lemma

Olšák

2020

Algebra Univers.

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We prove that an idempotent operation generates a loop from a strongly connected digraph containing directed cycles of all lengths under very mild (local) algebraic assumptions. Using the result, we reprove the existence of a weakest non-trivial idempotent equations, and that a strongly connected digraph with algebraic length 1 compatible with a Taylor term has a loop.Proposition 2.1 (Periodicity lemma). Let a, b be positive integers and x be a word of length at least a + b − gcd(a, b). If x is both a-periodic and b-periodic, it is also gcd(a, b)-periodic.Corollary 2.2. Let x be a word and k ≥ 2 be the shortest period of x. If y is a subword of x such that |y| ≥ 2k − 2. Then k is the shortest period of y.Proof. The word y is k-periodic. To obtain a contradiction, let k ′ ≤ k − 1 be another period of y. Since |y| ≥ k ′ + k − 1, the word y is gcd(k, k ′ )-periodic. Since |y| ≥ k and y is a subword of the k-periodic word x, also x is gcd(k, k ′ )periodic, which contradicts the minimality of k.Corollary 5.4. Let G be a strongly connected digraph with algebraic length 1 compatible with a Taylor operation. Then G has a loop.Proof. Let us denote the Taylor operation as t, and the vertex set of G as A.Since the digraph G is strongly connected and has algebraic length 1, it remains

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