The problem of reachability of the states of transition systems is considered hereby. The notions of partial unfolding and permutability of two operators (including the notion of statically permutable operators) are presented. New algorithm for reachability problem in terms of insertion modeling is described. An example of application of such algorithm is considered.

If V is a variety of algebras, let L(V) denote the prevariety of all lattices embeddable in congruence lattices of algebras in V . We give some criteria for the first-order axiomatizability or nonaxiomatizability of L(V). One corollary to our results is a nonconstructive proof that every congruence n-permutable variety satisfies a nontrivial congruence identity.