Levels of cancellation for monoids and modules

Levels of cancellativity in commutative monoids $M$, determined by stable rank values in $\mathbb{Z}{> 0} \cup {\infty}$ for elements of $M$, are investigated. The behavior of the stable ranks of multiples $ka$, for $k \in \mathbb{Z}{> 0}$ and $a \in M$, is determined. In the case of a refinement monoid $M$, the possible stable rank values in archimedean components of $M$ are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.

💡 Research Summary

The paper introduces and systematically studies the notion of “stable rank” for elements of a commutative (additively written) monoid M, a concept originally motivated by the stable rank of endomorphism rings in algebraic K‑theory. For an element a∈M and a positive integer n, a satisfies the n‑stable‑rank condition if whenever n·a + x = a + y for some x,y∈M there exists an element e∈M such that n·a = a + e and e + x = y. The smallest such n (or ∞ if none exists) is denoted sr_M(a). This invariant measures how “cancellative” a is: the larger the stable rank, the stronger the cancellation property required.

The authors first develop basic properties: Lemma 2.3 shows that if sr_M(a)≤n then (n+1)·a + b = a + c forces n·a + b = c; Theorem 2.4 proves that the stable rank of a sum is bounded by the maximum of the summands’ ranks; Corollary 2.5 shows that adding a unit does not change stable rank.

A central theme is the behavior of stable ranks under multiplication by positive integers. Proposition A (Lemma 4.1, Prop 4.2) proves monotonicity: if k≤ℓ then sr_M(k·a)≥sr_M(ℓ·a). Moreover, if sr_M(a) is finite, then for all k≥sr_M(a)−1 one has sr_M(k·a)≤2. Theorem B (Thms 4.9, 4.12) refines this to an exact formula: \