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

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.

💡 Research Summary

The paper generalizes the well‑known characterisation of kernels in semirings—namely, that they are exactly the so‑called k‑ideals (or subtractive ideals)—to arbitrary universal algebras. The authors fix an algebra A together with a distinguished element ∗ (which plays the role of the additive zero in a semiring) and study subsets I⊆A that are the ∗‑congruence class of ∗, i.e. ∗‑normal sets (also called ∗‑kernels). By examining the congruence generated by I×{∗} they isolate a semicongruence R (a reflexive subalgebra of A×A) and introduce two closure operators:

• ∗‑induction: I∗(I)=RI={x∈A | ∃y∈I, x R y}
• ∗‑deduction: D∗(I)=IR={x∈A | ∃y∈I, y R x}

A set I is called ∗‑inductive if I∗(I)⊆I (equivalently I=I∗(I)), and ∗‑deductive if D∗(I)⊆I (equivalently I=D∗(I)). The paper shows that any ∗‑normal set can be described as a ∗‑clot, i.e. the ∗‑class of a semicongruence generated by I×{∗}. Moreover, a non‑empty subset I containing ∗ is ∗‑normal precisely when it is both ∗‑inductive and ∗‑deductive. The proof proceeds by constructing the full congruence C as the directed union of powers of R (the “zig‑zag” chain) and showing that the two closure conditions guarantee that every element of C’s ∗‑class already lies in I.

Having established this decomposition, the authors define for any variety V (a class of algebras sharing a signature) and a chosen constant ∗ the ∗‑inductive rank and ∗‑deductive rank. The ∗‑inductive rank of V is the smallest n such that for every non‑empty I in any algebra of V, the n‑fold iteration of ∗‑induction stabilises (Iⁿ∗(I)=Iⁿ⁺¹∗(I)). The deductive rank is defined analogously. If no such finite n exists, the rank is ∞.

The paper then computes these ranks for several important varieties:

1. Additive commutative monoids (i.e., commutative monoids under +). The ∗‑inductive rank (with ∗=0) is 1, because one application of ∗‑induction yields the sub‑semigroup generated by I, which is already closed. The ∗‑deductive rank is ∞; the authors exhibit a family of subsets of the natural numbers whose deductive closure requires arbitrarily many steps. Here, ∗‑inductive sets are exactly sub‑semigroups, while ∗‑deductive sets coincide with the classical subtractive sub‑monoids (k‑ideals).

2. 0‑subtractive varieties (varieties possessing a binary term s with s(x,x)=0 and s(x,0)=0). For any such variety the ∗‑inductive and ∗‑deductive ranks are at most 2. Moreover, non‑empty ∗‑inductive, ∗‑deductive, ∗‑normal, and ∗‑clot notions all coincide. This recovers the familiar fact that kernels in a 0‑subtractive variety are precisely the 0‑clots.

3. Modules over a ring R (∗=0). Both ranks are 1, and the ∗‑inductive/∗‑deductive sets are exactly the sub‑modules. The proof uses the linear structure to rewrite any element of the submodule generated by I in the form x+ (−x)+∑rᵢyᵢ, showing that one inductive step already yields the full submodule.

4. Commutative rings with identity (again ∗=0). The situation mirrors modules: both ranks are 1, and the relevant sets are precisely the ideals. The authors give explicit term‑based descriptions of the inductive and deductive closures, showing they coincide with the ideal generated by I.

5. Semirings (no subtraction). Surprisingly, despite lacking a subtraction term, both ranks are still 1. The multiplicative and additive operations together generate a semicongruence that closes in a single step.

The paper also observes that varieties admitting a Mal’cev term (hence having a ternary term p satisfying p(x,x,y)=y and p(x,y,y)=x) automatically have both ranks equal to 1, reflecting the strong internal symmetry that allows immediate closure. Conversely, the infinite deductive rank for commutative monoids illustrates that the absence of any “inverse‑like” operation can make the deductive process arbitrarily long.

In summary, the authors provide a unified framework that splits the notion of a kernel (or normal subalgebra) into two elementary closure properties—∗‑induction and ∗‑deduction—valid in any universal algebra. By introducing rank invariants they quantify how far a given variety is from the ideal situation where one step suffices. This work not only recovers classical results about k‑ideals in semirings but also offers new insights into the structure of kernels across a broad spectrum of algebraic categories, opening avenues for further categorical and computational investigations.