Taylor’s modularity conjecture holds for linear idempotent varieties

A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable ω-categorical relational structures have two algebraic counterparts for their polymorphism clones: a semantic one via the standard algebraic operators H, S, P, and a syntactic one via clone homomorphisms (capturing identities). We provide a similar characterization which incorporates all relational constructions relevant for CSPs, that is, homomorphic equivalence and adding singletons to cores in addition to pp-interpretations. For the semantic part we introduce a new construction, called reflection, and for the syntactic part we find an appropriate weakening of clone homomorphisms, called h1 clone homomorphisms (capturing identities of height 1).As a consequence, the complexity of the CSP of an at most countable ω-categorical structure depends only on the identities of height 1 satisfied in its polymorphism clone as well as the natural uniformity thereon. This allows us in turn to formulate a new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures.Finally, we reveal a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties.

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“…As a corollary, we obtain the following generalization of [4] where it was additionally assumed that the varieties are idempotent. Theorem 1.12.…”

Section: 3mentioning

confidence: 92%

The wonderland of reflections

Barto¹,

Opršal²,

Pinsker³

2017

Isr. J. Math.

182

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“…Several similar partial results on primeness of some Mal'cev filters have been obtained before. Bentz and Sequeira in [BS14] also proved for two linear idempotent varieties: if the join of the two varieties is congruence n-permutable for some n, then so is one of the two varieties; and similarly if their join satisfies a non-trivial congruence identities, then so does one of the two varieties. Stronger versions of these results also follow from the work of Valeriote and Willard [VW14], who proved that every idempotent variety that is not n-permutable for any n is interpretable in the variety of distributive lattices, and the work of Kearnes and Kiss [KK13], who proved that any idempotent variety which does not satisfy a non-trivial congruence identity is interpretable in the variety of semilattices.…”

Section: Introductionmentioning

confidence: 92%

“…In [BS14], Bentz and Sequeira proved that this is true if V and W are idempotent varieties that can be defined by linear identities (such varieties are called linear idempotent varieties), and later in [BOP15], Barto, Pinsker, and the author generalized their result to linear varieties that do not need to be idempotent. In this paper we generalize Bentz and Sequeira's result in a different direction.…”

Section: Introductionmentioning

confidence: 99%

Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

Opršal

2017

Order

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Abstract. We provide a partial result on Taylor's modularity conjecture, and several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and we prove an analogue for idempotent varieties with a cube term. Also, similar results are proved for linear varieties and the properties of congruence modularity, having a cube term, congruence n-permutability for a fixed n, and satisfying a non-trivial congruence identity.

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“…Moreover, consider a variable y ∈ Fm( i∈I i ) not occurring in Γ. 2 Again bearing in mind that i∈I i is the logic induced by the class i∈I Mod ≡ ( i ), it is clear that the following rule is valid in i∈I i :…”

Section: Truth-minimal Logics Definitionmentioning

confidence: 99%

“…Affirmative answers to these questions can then be interpreted as stating that the Leibniz classes under consideration capture primitive or fundamental concepts. Similar problems were studied in the setting of the Maltsev hierarchy for instance in [21] (see also [2,29,37]). Some of our results can be summarized as follows.…”

Section: Introductionmentioning

confidence: 98%

The poset of all logics II: Leibniz classes and hierarchy

Jansana,

Moraschini

2020

Preprint

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A Leibniz class is a class of logics closed under the formation of termequivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of meet-prime logics only.

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Taylor’s modularity conjecture holds for linear idempotent varieties

Cited by 6 publications

References 9 publications

The wonderland of reflections

The wonderland of reflections

Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

The poset of all logics II: Leibniz classes and hierarchy

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