On potential properties of a semigroup connected with generation of one-sided congruences

A property $P$ is fulfilled potentially in a semigroup $S$ if $P$ is fulfilled in some supersemigroup $T\supseteq S$. We find necessary and sufficient conditions for a given pair $(a,b)$ of elements of $S$ to belong to the right congruence generated by a given set of pairs $(a_i,b_i)$, $i\in I$, potentially. In connection to this we investigate potential solvability of the equation systems of the form $ax=b$, $ax=bx$, $ax=by$.