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In this paper, homomorphism theorems for ordered semirings are studied. We obtain some characterizations of the smallest compatible quasiorder or order-congruence generated by H. A homomorphism theorem and two isomorphism theorems of ordered semirings based on compatible quasiorders and order-congruences have been given. Finally, we have a characterization for the direct product of ordered semirings by using compatible quasiorders on an ordered semiring.

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In this paper, properties for Hamiltonian semirings are studied. We concluded that subsemirings and all homomorphic images of a Hamiltonian semiring are Hamiltonian semirings. Hamiltonian semirings satisfying the congruence extension property are characterized by the strong congruence extension property. Finally, we concluded that if R × R is a Hamiltonian semiring, then R has strong congruence extension property