Day's Theorem is sharp for $n$ even

Lipparini, Paolo

doi:10.48550/arxiv.1902.05995

Cited by 4 publications

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“…Corollary 3.3 is just illustrative. Other examples of Maltsev conditions such that Theorem 3.1 applies to their defining varieties can be found, among many others and with partial overlaps, in Taylor [Ta,Corollary 5.3], Gumm [G,Theorem 7.4(iv)], Tschantz [Ts,Lemmata 3 and 4], Hobby,McKenzie [HoMc,Lemmata 9.4(3), 9.5(3), Theorems 9.8(4), 9.11(4) and 9.15(3)], Siggers [S, Theorem 1.1, Section 3], Kearnes, Kiss [KK,Theorems 3.21(3),4.7,5.23(3),5.28(3),8.13(3) and 8.14(3 [KV,Sections 3.2 and 3.3], Lipparini [DTS,Definitions 2.1,2.7,6.1,7.1,7.6,8.1,Remarks 6.4,8.16,8.19,10.11(c)].…”

Section: The Strong Amalgamation Property For Equilinear Varietiesmentioning

confidence: 99%

Varieties defined by linear equations have the amalgamation property

Lipparini¹

2021

Preprint

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A variety is a class of algebraic structures axiomatized by a set of equations. An equation is linear if there is at most one occurrence of an operation symbol on each side. We show that a variety axiomatized by linear equations has the strong amalgamation property.Suppose further that the language has no constant symbol and, for each equation, either one side is operation-free, or exactly the same variables appear on both sides. Then also the joint embedding property holds.Examples include most varieties defining classical Maltsev conditions. In a few special cases, the above properties are preserved when further unary operations appear in the equations.

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Section: The Strong Amalgamation Property For Equilinear Varietiesmentioning

confidence: 99%

Varieties defined by linear equations have the amalgamation property

Lipparini¹

2021

Preprint

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“…Letting ℓ vary in (2) above and using [7], we get another proof of the result from [21] that a variety V is congruence modular if and only if the congruence identity α(β [3,Theorem 5]. The examples in [17] show that in some cases the bounds given by Corollary 6 are optimal or close to be optimal.…”

mentioning

confidence: 93%

“…Such problems date back at least to [4, p. 173]. See, e. g., [3,6,11,12,14,15,16,17,21] for other problems, comments and results. The reader can find further references in the quoted works.…”

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“…Let us now recall the main definitions together with some needed classical results. See, e. g., [5,8,11,17,19] for further undefined notions and full formal details.…”

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“…We shall not need the explicit definition of Day terms here. See [4,5,11,16,17,21] for full details. From [4] and [7] we get that a variety V has a sequence of Day terms if and only if V has a sequence of Gumm terms.…”

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The Gumm level equals the alvin level in congruence distributive varieties

Lipparini

2020

Preprint

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Congruence modular and congruence distributive varieties are characterized by the existence of sequences of Gumm and Jónsson terms, respectively. Such sequences have variable lengths, in general.It is immediate from the above paragraph that there is a variety with Gumm terms but without Jónsson terms. We prove the quite unexpected result that, on the other hand, if some variety has both kinds of terms, then the minimal lengths of the sequences differ at most by 1.It follows that every r-modular congruence distributive variety is r 2 − r + 2-distributive. Mathematics Subject Classification (2010). Primary 08B10.

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The Jónsson distributivity spectrum

Lipparini

2018

Algebra Univers.

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Suppose throughout that V is a congruence distributive variety. If m ≥ 1, let J V (m) be the smallest natural number k such that the congruence identity α(β •γ •β . . . ) ⊆ αβ •αγ •αβ •. . . holds in V, with m occurrences of • on the left and k occurrences of • on the right. We show that if J V (m) = k, then J V (mℓ) ≤ kℓ, for every natural number ℓ. The key to the proof is an identity which, through a variety, is equivalent to the above congruence identity, but involves also reflexive and admissible relations. If J V (1) = 2, that is, V is 3-distributive, then J V (m) ≤ m, for every m ≥ 3 (actually, a more general result is presented which holds even in nondistributive varieties). If V is m-modular, that is, congruence modularity of V is witnessed by m + 1 Day terms, thenVarious problems are stated at various places. 2010 Mathematics Subject Classification. 08B10. Key words and phrases. Congruence distributive variety; (directed) Jónsson terms; Jónsson distributivity spectrum; congruence identity; identities for reflexive and admissible relations. Work performed under the auspices of G.N.S.A.G.A. Work partially supported by PRIN 2012 "Logica, Modelli e Insiemi". [368][369]. We also need the easy fact that, for every m and k, the condition J V (m) ≤ k is equivalent to a strong Maltsev condition; for example, this is a consequence of the equivalence of (A) and (B) in Theorem 2.1 below.Proposition 1.1. If V and V ′ are congruence distributive varieties, then their non-indexed product V ′′ is such that J V ′′ (m) = max{J V (m), J V ′ (m)}, for every positive natural number m.We do not know whether, for every pair V, V ′ , we always have someIf in Proposition 1.1 we consider the variety of lattices and the mentioned variety from [5, p. 70-71], we get J V ′′ (m) = max{n − 1, m}. By taking the non-indexed product of the variety of n ′ -Boolean algebras and again the variety from [5, p. 70-71], with n ≤ n ′ , we have J V ′′ (m) = n−1, for m ≤ n−1 and J V ′′ (m) = min{m, n ′ }, for m > n−1.The above examples suggest that J V (m) has little influence on the values of J V (m ′ ), for m ′ < m. On the other hand, we are going to show that J V (m) puts some quite restrictive bounds on J V (m ′ ), for m ′ > m, as we already mentioned for the easier case m = 1.

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Day's Theorem is sharp for $n$ even

Cited by 4 publications

References 41 publications

Varieties defined by linear equations have the amalgamation property

Varieties defined by linear equations have the amalgamation property

The Gumm level equals the alvin level in congruence distributive varieties

The Jónsson distributivity spectrum

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