Congruence-simple subsemirings of ℚ⁺

Abstract

Commutative congruence-simple semirings have already been characterized with the exception of the subsemirings of ℝ⁺. Even the class $\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb {Q}^{+})$ of all congruence-simple subsemirings of ℚ⁺ has not been classified yet. We introduce a new large class of the congruence-simple saturated subsemirings of ℚ⁺. We classify all the maximal elements of $\mathit{\mathcal{C}ong\mathcal {S}imp}(\mathbb{Q}^{+})$ and show that every element of $\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb{Q}^{+})\setminus\{\mathbb{Q}^{+}\}$ is contained in at least one of them.