Word problem for semigroups

Let be a semigroup which is not necessarilly finitely presented. Then it is natural to ask if you are given two words then are they equal modulo this presentation. This is the word problem (for semigroups).

Assume the word problem is soluble for . Then my question is,

if we are given two words and and we know they are equal modulo then is it always possible to find the chain of relations connecting these two words?

EDIT: I believe `complete' is the word I am looking for. However, I can't seem to find this anywhere (the WP is soluble for P if and only if P is complete. Apparently).