This investigation is in Constructive Algebra in the sense of Bishop, Richman, Ruitenburg, Troelstra and van Dalen. The main subject of this paper is the characterization of the left compatible coequality relation $q$ on semigroup with apartness by means of special right consistent subsets $L_{(a)}=\{x\in S:x\in Sa\}$ and by filled product of relations. Also, we give a construction of a quasi-antiorder $c$ such that $q=c\cup c^{-1}$ and some descriptions of classes of relations $c$ and $q$. We also present an assertion on the left class $A(a)$ $(a\in A)$ of the relation $c$ in which we prove that $A(a)$ is a maximal strongly extensional right consistent subset of $S$ such that $a\in A(a)$. Also, we give an assertion on the right class $B(a)$ $(a\in S)$ of the relation $c$ in which we prove that $B(a)$ is a maximal strongly extensional left ideal of $S$ such that $A\in B(a)$. Besides, we provide some descriptions of families $\{A(a):a\in S\}$ and $\{B(a):a\in S\}$.