The lattice of z-ideals of every commutative semiring is unconditionally a coherent frame, and under explicit finite-type hypotheses the g-closed ideals form a coherent frame homeomorphic to a prime congruence spectrum.
Banaschewski, Another look at the localic Tychonoff theorem, Comment
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Maximal-Hull $z$-Ideals, Congruence Closures, and Coherent Frames of Commutative Semirings
The lattice of z-ideals of every commutative semiring is unconditionally a coherent frame, and under explicit finite-type hypotheses the g-closed ideals form a coherent frame homeomorphic to a prime congruence spectrum.