Towards Commutator Theory for Relations. Iii

We derive consequences from the existence of a term which satisfies Mal’cev identities (characterizing permutability) modulo two functions F and G from admissible relations to admissible relations. We also provide characterizations of varieties having a Mal’cev term modulo F and G. Given an algebra A, let Adm(A) denote the set of all reflexive and admissible relations on A (we shall use the words “admissible” and “compatible” interchangeably). If A is an algebra, and F : Adm(A) → Adm(A), G : Adm(A) → Adm(A), we say that a ternary term t of A is Malcev modulo F and G if and only if aF (R)t(a, b, b) and t(a, a, b)G(R)b, whenever a, b ∈ A, R ∈ Adm(A), and aRb. An alternative name for the above notion is a weak difference term modulo F and G, or simply an F -Gdifference term: we used this terminology in [L1, p. 199], in the case when F : Con(A) → Con(A), G : Con(A) → Con(A). For a relation R on some algebra, let R denote the smallest tolerance containing R, and let R denote the converse of R. R is the transitive closure of R, and Cg(R) is the smallest congruence containing R. R denotes the least compatible relation containing R. Notice that if R and S are compatible, then R ◦S is compatible, too. Intersection is denoted by juxtaposition. R◦nS is R◦S ◦R◦S . . . with n−1 occurrences of ◦. R n = R◦nR. By convention, we put R0 = 0, where 0 denotes the identity relation (the smallest reflexive relation). R+ S = ⋃ n R ◦n S. Theorem 1. Suppose that A has a term Malcev modulo F and G. Then for all reflexive admissible relations R,S,R1, R2, · · · ∈ Adm(A), and for arbitrary relations θ, θ1, θ2 ⊆ A 2, the following hold: 2000 Mathematics Subject Classification. Primary 08A30; Secondary 08B05.