Abstract: The bi-semirings whose semigroups (S,+) are semilattices, semigroups (S,·) are rectangular groups and semigroups (S,) are semilattices were studied in this paper. Two partial order relations were constructed on two subsets of a bi-semiring, and a necessary and sufficient condition for that the Green relation on semigroups (S,·) is a bi-semiring congruence was obtained. Furthermore, the equivalent proposition that is a bi-semiring congruence was presented.

Key words: semilattice, rectangular group, congruence, bi-semiring, partial order