[Paper Review] Additively irreducible sequences in commutative semigroups

Guoqing Wang|arXiv (Cornell University)|Apr 26, 2015

semigroups and automata theory12 references3 citations

TL;DR

This paper introduces and analyzes additively irreducible sequences in commutative semigroups, defining $\mathrm{D}_a(\mathcal{S})$ as the maximal length of such sequences summing to a given element $a$. It establishes necessary and sufficient conditions for $\mathrm{D}_a(\mathcal{S})$ to be finite when the ascending chain condition holds on principal ideals from $a$, providing sharp bounds and applying the results to finite commutative unitary rings, particularly principal ideal rings.

ABSTRACT

Let $\mathcal{S}$ be a commutative semigroup, and let $T$ be a sequence of terms from the semigroup $\mathcal{S}$. We call $T$ an (additively) {\sl irreducible} sequence provided that no sum of its some terms vanishes. Given any element $a$ of $\mathcal{S}$, let ${ m D}_a(\mathcal{S})$ be the largest length of the irreducible sequence such that the sum of all terms from the sequence is equal to $a$. In case that any ascending chain of principal ideals starting from the ideal $(a)$ terminates in $\mathcal{S}$, we found the sufficient and necessary conditions of ${ m D}_a(\mathcal{S})$ being finite, and in particular, we gave sharp lower and upper bounds of ${ m D}_a(\mathcal{S})$ in case ${ m D}_a(\mathcal{S})$ is finite. We also applied the result to commutative unitary rings. As a special case, the value of ${ m D}_a(\mathcal{S})$ was determined when $\mathcal{S}$ is the multiplicative semigroup of any finite commutative principal ideal unitary ring.

Motivation & Objective

• To define and characterize additively irreducible sequences in commutative semigroups, where no proper subsequence sums to the full sequence sum.
• To investigate the finiteness of $\mathrm{D}_a(\mathcal{S})$, the maximal length of irreducible sequences summing to a fixed element $a$.
• To establish sufficient and necessary conditions for $\mathrm{D}_a(\mathcal{S})$ to be finite under the ascending chain condition on principal ideals starting from $a$.
• To derive sharp upper and lower bounds for $\mathrm{D}_a(\mathcal{S})$ when it is finite.
• To apply the results to finite commutative unitary rings, particularly determining $\mathrm{D}_a(\mathcal{S})$ in multiplicative semigroups of finite commutative principal ideal rings.

Proposed method

• Define $\mathrm{D}_a(\mathcal{S})$ as the maximum length of an irreducible sequence $T$ with $\sigma(T) = a$ in a commutative semigroup $\mathcal{S}$.
• Introduce the group $\Gamma(H_a)$ associated with the structure of the semigroup around $a$, and use the Schützenberger group construction to analyze irreducibility.
• Utilize the concept of $\Psi(a)$, the minimal number of generators of the ideal $\mathfrak{a}$ associated with $a$, to bound $\mathrm{D}_a(\mathcal{S})$.
• Construct explicit irreducible sequences by combining a zero-sum-free sequence in $\Gamma(H_a)$ with a minimal generating sequence for $a$, ensuring no proper subsequence sums to $\sigma(T)$.
• Apply the structure theory of Noetherian semigroups and principal ideal rings to derive finiteness and bounds.
• Use the duality between $\mathrm{D}_a(\mathcal{S})$ and the Davenport constant $\mathrm{D}(G)$ in finite abelian groups to extend results to rings.

Experimental results

Research questions

• RQ1 Under what conditions is $\mathrm{D}_a(\mathcal{S})$ finite for a given element $a$ in a commutative semigroup $\mathcal{S}$?
• RQ2 What are the sharp upper and lower bounds for $\mathrm{D}_a(\mathcal{S})$ when it is finite?
• RQ3 How does the finiteness of $\mathrm{D}_a(\mathcal{S})$ relate to the structure of the semigroup, particularly the finiteness of the Schützenberger group $H_a$?
• RQ4 Can the value of $\mathrm{D}_a(\mathcal{S})$ be explicitly computed for the multiplicative semigroup of a finite commutative principal ideal ring?
• RQ5 What is the relationship between $\mathrm{D}_a(\mathcal{S})$ and the Davenport constant $\mathrm{D}(G)$ in finite abelian groups?

Key findings

• $\mathrm{D}_a(\mathcal{S})$ is finite if and only if the Schützenberger group $H_a$ is finite, under the assumption that every ascending chain of principal ideals from $a$ terminates.
• When $\mathrm{D}_a(\mathcal{S})$ is finite, it satisfies the sharp bounds $\mathrm{D}^*(\Gamma(H_a)) - 1 + \Psi(a) \leq \mathrm{D}_a(\mathcal{S}) \leq \mathrm{D}^*(\Gamma(H_a)) - 1 + \Psi(a)$, where $\mathrm{D}^*(G)$ is the Davenport constant of the group $G$.
• For the multiplicative semigroup of a finite commutative principal ideal unitary ring $R$, the value of $\mathrm{D}_a(\mathcal{S}_R)$ is completely determined and equals $\mathrm{D}^*(\Gamma(H_a)) - 1 + \Psi(a)$.
• The finiteness of $\mathrm{D}_a(\mathcal{S})$ does not require $\Psi(a)$ to be finite, as shown by a counterexample where $\Psi(a)$ is infinite but $\mathrm{D}_a(\mathcal{S}) \leq n$.
• In any commutative semigroup $\mathcal{S}$, $\mathrm{D}(\mathcal{S})$ is finite if and only if $\mathrm{d}(\mathcal{S})$ is finite, and in that case, $\mathrm{D}(\mathcal{S}) = \mathrm{d}(\mathcal{S}) + 1$.
• For a commutative Noetherian semigroup $\mathcal{S}$, $\mathrm{D}(\mathcal{S})$ is finite if and only if the size of $H_a$ is uniformly bounded across all $a \in \mathcal{S}$.