6 towards Commutator Theory for Relations

We provide more characterizations of varieties having a term Mal’cev modulo two functions F and G. We characterize varieties neutral in the sense of F , that is varieties satisfying R ⊆ F (R). We present examples of global operators satisfying the homomorphism property, in particular we show that many known commutators satisfy the homomorphism property. See Parts I-III [2] for unexplained notation. We now find more conditions equivalent to the existence of a term Mal’cev modulo F and G throughout a variety, thus complementing Theorem 3 in Part III. Theorem 1. Suppose that V is a variety, F , G are global operators on V for admissible and reflexive relations, F , G are monotone and satisfy the homomorphism property. Then the following are equivalent: (i) V has a term which is Mal’cev modulo FA and GA for every algebra A in V. (ii) In every algebra A ∈ V and for all relations R,S, T ∈ Adm(A), the following holds: T (R ◦ S) ⊆ FA(T ) ◦ S ∪R ◦GA(S). (iii) In the free algebra X in V generated by 2 elements the following holds: T (R ◦ S) ⊆ FX(T ) ◦ S − ◦R ◦GX(S), for all relations R,S, T ∈ Adm(X). (iv) In every algebra A ∈ V and for all relations S, T ∈ Adm(A), the following holds: TS ⊆ FA(T ) ◦ S − ◦GA(T ). (v) In the free algebra X in V generated by 2 elements the following holds: TS ⊆ FX(T ) ◦ S − ◦GX(T ), for all relations S, T ∈ Adm(X). 2000 Mathematics Subject Classification. Primary 08A99; 08B99.