P-Spaces and the Prime Spectrum of Commutative Semirings

By a semiring we understand a commutative semiring with nonzero identity. The notions of ω-absolutely (semiprime-) irreducible ideals in a semiring R are introduced and we prove that the prime spectrum Spec(R) of R is a P-space if and only if every prime ideal of R is ω-absolutely semiprime-irreducible. Also, Spec(R) is an Artin space totally-ordered by inclusion if and only if every (semiprime) ideal of R is ω-absolutely semiprime-irreducible if and only if the family Spec(R) is well-ordered by inclusion. Further, we characterize when every ideal of R is ω-absolutely irreducible as well as when every (semiprime) prime ideal of R is ω-absolutely irreducible