Additively irreducible sequences in commutative semigroups

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 ${\rm D}_a(\mathcal{S})$ 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 ${\rm D}_a(\mathcal{S})$ being finite, and in particular, we gave sharp lower and upper bounds of ${\rm D}_a(\mathcal{S})$ in case ${\rm D}_a(\mathcal{S})$ is finite. We also applied the result to commutative unitary rings. As a special case, the value of ${\rm D}_a(\mathcal{S})$ was determined when $\mathcal{S}$ is the multiplicative semigroup of any finite commutative principal ideal unitary ring.

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