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

Kernels of semiring homomorphisms are precisely the subtractive ideals. We extend this decomposition of normality to general algebras by introducing inductive and deductive sets, which turn out to correspond to the two directions of the biconditional in Mal'tsev's criterion for a congruence class. In semirings, inductivity recovers idealhood for subsets containing zero, and deductivity of an ideal recovers subtractivity. Their generation processes, described by polynomials or equivalently by reflexive compatible relations (semicongruences), give two ranks measuring the numbers of steps required uniformly in a variety. We show that apart from the trivial cases, any pair of positive integers or infinity is the inductive-deductive rank pair of some variety. For the variety of semirings, the rank pair is $(1,\infty)$, while for any non-trivial Mal'tsev variety, it is $(1,1)$. For varieties of finite-group actions, inductive rank is expressed exactly in terms of directed Cayley-graph diameters, while deductive rank is related to undirected diameters after symmetrization. We compute both ranks of these action varieties for every finite abelian group in terms of its invariant factors, and obtain their complete spectrum. We also establish special spectrum theorems for subtractive varieties and several classes of ordered algebras. Finally, we characterise structural properties of algebras and varieties by conditions on induction and deduction.