Given a group-word w and group G, the set of w-values in G is denoted by Gw verbal subgroup w(G) one generated Gw. The word concise if finite for all groups which finite. We obtain several results supporting conjecture that [u1,…,us] whenever words u1,…,us are non-commutator.

It has been recently observed that fundamental aspects of the classical theory factorization can be greatly generalized by combining languages monoids and preorders. This led to various theorems on existence certain factorizations, herein called ⪯-factorizations, for ⪯-non-units a (multiplicatively written) monoid H endowed with preorder ⪯, where an element u∈H is ⪯-unit if u⪯1H⪯u ⪯-non-unit ot...