The Jónsson distributivity spectrum

Abstract

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.

The Tschantz and the alvin higher conditions are equivalent in congruence distributive varieties

“…Classical arguments then give a proof of the following proposition. See, e. g., [7] for full details and for more general notions and arguments.…”

Relation identities in 3-distributive varieties

“…As mentioned in the introduction, the results form Kazda, Kozik, McKenzie and Moore [13] and the observations in [18] show that, for every n, there is some (possibly quite large) k such that α(S •T ) ⊆ αS •αT •αS •. .…”

“…See, e. g., [7,13,18,22] for further details and information. If S is a reflexive binary relation on some algebra, we denote by S the smallest admissible relation containing S. In the following theorem we shall provide bounds for expressions like, say, R(S • T ) • RW , rather than simply R(S • T ).…”

“…Yet it is interesting to check which parts of Theorem 3.1 follow just from the existence of two directed Jónsson terms. The existence of two directed Jónsson terms implies identity (3.2), limited to the case when R is assumed to be a tolerance, by the case ℓ = 2 in equation (3.1) in [18,Proposition 3.1]. Notice that the number of terms is counted in a different way in [18], including also two trivial projections; here we adopt the counting convention from [13].…”

“…If V is congruence distributive, then, by Kazda, Kozik, McKenzie and Moore [13] and by [18,Proposition 3.1], Θ(S • T ) * ⊆ (ΘS • ΘT ) * . If in addition V is congruence n-permutable, we can apply the identities (4.5), (4.6) and (4.9) with ΘS and ΘT in place of, respectively, S and T , obtaining a bounded number of factors on the right.…”