The spectrum of prime congruences of a pair

TL;DR

This paper generalizes the theory of prime congruences from idempotent semirings to semiring pairs, including supertropical semirings and hyperrings, establishing an embedding into their prime spectra.

Contribution

It extends the prime congruence spectrum theory to semiring pairs, broadening its applicability beyond idempotent semirings.

Findings

Joo-Mincheva spectrum embeds into semiring pair spectrum

Embedding is surjective under positive e-type restriction

Generalizes prime spectrum theory to new algebraic structures

Abstract

Continuing the study of the structure of semirings, we turn to the spectrum of prime congruences. Joo and Mincheva developed an elegant theory in the special case of idempotent semirings, which is generalized here to ``semiring pairs,'' which include supertropical semirings and various classes of hyperrings. Our main result is that the Joo-Mincheva spectrum can be embedded into the prime spectrum of a semiring pair, and the mapping is onto when the pair satisfies a mild restriction which we call ``positive $e$-type.''