Band congruences of semigroup were first defined and studied by T. Tamura and N. Kimura in 1955. First general results concerning left zero bands and right zero bands were obtained by P. Dubriel in 1951 and G. Therrin in 1956. This study is concerned with the constructive algebra in the sense of Bishop, van Dalen, Richman and Ruitenburg. The main subject is the characterization of the maximal right zero band coequality relation compatible with the semigroup operation by means of special right consistent left ideal of the semigroup $(S,=,\neq,1,\cdot)$ with apartness. Also, we give a construction of the maximal strongly extensional right consistent left ideals $A(a)$ of $S$ generated by an element $a$ in $S$ such that $a\#A(a)$.