Let $K$ be an anti-ideal of a semigroup $(S,=,\neq,\cdot,\theta)$ with apartness. A construction of the anti-congruence $Q(K)$ and the quasi-antiorder $\theta$, generated by $K$, are presented. Besides, a construction of the anti-order relation $\Theta$ on syntactic semigroup $S/Q(K)$, generated by $\theta$, is given in Bishop's constructive mathematics.