NOTES ON CONGRUENCE n-PERMUTABILITY AND SEMIDISTRIBUTIVITY

In [1] T. Dent, K. Kearnes and Á. Szendrei define the derivative, Σ′, of a set of equations Σ and show, for idempotent Σ, that Σ implies congruence modularity if Σ′ is inconsistent (Σ′ |= x ≈ y). In this paper we investigate other types of derivatives that give similar results for congruence n-permutable for some n, and for congruence semidistributivity. In a recent paper [1] T. Dent, K. Kearnes and Á. Szendrei study Maltsev conditions which imply congruence modularity from the point of view of the equations. Given a set of equations Σ they define the derivative Σ′ of Σ. (The definition is given below.) Σ′ ⊇ Σ and if Σ is idempotent then, if Σ′ is inconsistent (that is, Σ′ |= x ≈ y), then any variety V that realizes Σ (each function symbol in Σ can be interpreted as a term of V such that the equations of Σ are satisfied) is congruence modular. While the converse is not true in general, they show that it is true if Σ is a set of linear, idempotent equations. So in particular if Σ consists of Day’s equations (for a fixed n) or Gumm’s equations, then Σ′ is inconsistent. They use these results to prove interesting new results and give easy proofs of several existing theorems. One nice example: one of the equations in Day’s characterization of congruence modularity involves three variables: mi(x, u, u, y) ≈ mi+1(x, u, u, y) for i odd J. B. Nation wondered if there is a two-variable condition implying congruece modularity. In [9] he showed that this is the case. Using their results, the authors of [1] show that the above equation can be replaced by either mi(x, x, x, y) ≈ mi+1(x, x, x, y) for i odd or mi(x, y, y, y) ≈ mi+1(x, y, y, y) for i odd and the resulting set of identities is still implies congruence modularity. The principal results of [1] for congruence modularity (CM) are summarized in the next theorem. Σ idempotent means Σ |= f(x, . . . , x) ≈ x for all operation symbols f occuring in Σ; it is linear if there is no nested composition in the terms occuring in Σ. If Σ |= x ≈ f(w), where w is a sequence of not necessarily distinct variables, then f is weakly independent of its ith place for each i with wi 6= x. (So a Maltsev term is weakly independent of