Computable embedding of classes of algebraic structures with congruence relation

It has been shown in the paper that there is an intermediate notion of embedding, which is based on the use of non-injective presentations of algebraic structures, between the computable embedding of classes of algebraic structures based on the enumeration operators and the Turing computable embedding. The problem of equivalence of this notion to the injective computable embedding is related to the problem of effective factorization by enumeration operators.