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Torsion factors of commutative monoid semirings

Miroslav Korbelář

Abstract

Let $P$ be a finitely generated commutative semiring. It was shown recently that if $P$ is a parasemifield (i.e. the multiplicative reduct of $P$ is a group) then $P$ cannot contain the positive rationals $\mathbb{Q}^+$ as its subsemiring. Equivalently, a commutative parasemifield $P$ finitely generated as a semiring is additively divisible if and only if $P$ is additively idempotent. We generalize this result using weaker forms of these additive properties to a broader class of commutative semirings in the following way. Let $S$ be a semiring that is a factor of a monoid semiring $\mathbb{N}[\mathcal{C}]$ where $\mathcal{C}$ is a submonoid of a free commutative monoid of finite rank. Then the semiring $S$ is additively almost-divisible if and only if $S$ is torsion. In particular, we show that if $S$ is a ring then $S$ cannot contain any non-finitely generated subring of $\mathbb{Q}$.

Torsion factors of commutative monoid semirings

Abstract

Let be a finitely generated commutative semiring. It was shown recently that if is a parasemifield (i.e. the multiplicative reduct of is a group) then cannot contain the positive rationals as its subsemiring. Equivalently, a commutative parasemifield finitely generated as a semiring is additively divisible if and only if is additively idempotent. We generalize this result using weaker forms of these additive properties to a broader class of commutative semirings in the following way. Let be a semiring that is a factor of a monoid semiring where is a submonoid of a free commutative monoid of finite rank. Then the semiring is additively almost-divisible if and only if is torsion. In particular, we show that if is a ring then cannot contain any non-finitely generated subring of .
Paper Structure (5 sections, 15 theorems, 14 equations)

This paper contains 5 sections, 15 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decomposition of saturated submonoids of $\mathbb{N}_{0}^n(+)$
  4. Additively almost-divisible factors of the monoid ring $\mathbb{Z}[\mathcal{C}]$ are torsion
  5. Additively almost-divisible factors of monoid semiring $\mathbb{N}[\mathcal{C}]$ are torsion

Key Result

Proposition 3.1

For every cone $K$ in $\mathbb{R}^n$ there is a decomposition $\mathbf{K}=\{A_i\mid i\in I\}$ of $K$ into disjoint union of relatively open convex subsets $A_i$ of $K$. Moreover, for every $A\in\mathbf{K}$ the set $A\cup\{0\}$ is a cone and for all $x\in A$ and $y\in K\setminus\langle A\rangle$, the

Theorems & Definitions (34)

  • Conjecture 1
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Remark 3.5
  • ...and 24 more