From subtractive ideals of semirings to deductive and inductive sets in general algebras

TL;DR

This work generalizes kernel characterizations from semirings to arbitrary algebras by introducing $\ast$-normal sets and decomposing them into $\ast$-inductive and $\ast$-deductive components, with a nonempty $I$ being $\ast$-normal iff it is both closures. It develops concrete constructions via semicongruences $R$ generated by $I\times\{\ast\}$ and studies the resulting closures $\mathcal{I}_\ast(I)$ and $\mathcal{D}_\ast(I)$, whose iterative unions yield the smallest $\ast$-inductive/deductive sets. A central theme is ranking the requisite closure steps, introducing $\ast$-inductive and $\ast$-deductive ranks for varieties, and deriving sharp bounds in key classes such as $0$-subtractive varieties, Mal'tsev varieties, modules over rings, rings with identity, semirings, and commutative monoids. The results unify several known kernels/ideals phenomena (e.g., subtractive ideals in semirings) and provide a framework connecting logical/closure properties to algebraic structure, with potential implications for understanding congruence generation and normality in general algebraic settings.

Abstract

In this paper we extend the characterisation of kernels in semirings as subtractive ideals to general algebras. We then analyse the counterparts of ``subtractive'' and ``ideal'' in several different algebraic settings.

Theorems & Definitions (34)

• Definition 2.1
• Lemma 2.1
• Definition 2.2
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• proof
• Theorem 2.6
• proof