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Elasticity in certain block monoids via the Euclidean table. (English) Zbl 1156.20056
The authors compute the irreducibles and elasticities of a basic class of block monoids. Specifically, they consider block monoids over \(\mathbb{Z}_n\) whose support is a two element subset \(\{\overline a,\overline b\}\). Using an earlier result, such monoids are isomorphic to block monoids over \(\mathbb{Z}_n\) restricted to \(\{\overline 1,\overline a\}\), where \(\gcd(a,n)=1\). An algorithm for computing the irreducible blocks is constructed using the concept of a Euclidean table, a variation of the method of continued fractions of {A. Geroldinger} [Colloq. Math. Soc. János Bolyai 51, 723-757 (1987; Zbl 0703.11057)]. This table yields a quick calculation of the elasticity from the Euclidean division algorithm on \(n\) and \(a\), avoiding any consideration of the irreducibles in the monoid. Lastly, the authors work through some computational bounds for the set of all possible elasticities as \(a\) varies and \(n=p\) remains a fixed prime number.

MSC:
20M14 Commutative semigroups
11R27 Units and factorization
11J70 Continued fractions and generalizations
11A55 Continued fractions
13A05 Divisibility and factorizations in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
11B83 Special sequences and polynomials
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