The paper deals with varieties having: (i) regular congruences (i.e. congruences uniquely determined by any of their congruence classes), (ii) regular tolerances, (iii) regular compatible reflexive relations, (iv) regular quasiorders. The author shows that the condition (i) \(+\) congruence permutability is equivalent to (ii) and also to (iii) for any variety of algebras. This fact is used for constructing easy identities characterizing varieties with regular and permutable congruences in a way similar to that of the reviewer [Algebra Univers. 11, 159-162 (1980; Zbl 0449.08007)]. As it is noted in the paper, the equivalence of (i) and (iv) was already proved by J. Hagemann. The paper contains analogous results for weak regularity of these relations.