×
Krause, Ulrich; Maney, Jack; Ponomarenko, Vadim

Diversity in inside factorial monoids. (English) Zbl 1265.20061

Czech. Math. J. 62, No. 3, 811-827 (2012).
Summary: In the recent paper [Czech. Math. J. 62, No. 3, 795-809 (2012; Zbl 1265.20060)], the last two authors introduced and developed the monoid invariant “diversity” and related properties “homogeneity” and “strong homogeneity”. We investigate these properties within the context of inside factorial monoids, in which the diversity of an element counts the number of its different almost primary components. Inside factorial monoids are characterized via diversity and strong homogeneity. A new invariant complementary to diversity, height, is introduced. These two invariants are connected with the well-known invariant of elasticity.

Cited in 2 Documents

MSC:

20M14 Commutative semigroups
20M05 Free semigroups, generators and relations, word problems
11B75 Other combinatorial number theory
13A05 Divisibility and factorizations in commutative rings
11N80 Generalized primes and integers

Keywords:

factorizations into almost primary elements; inside factorial monoids; commutative cancellative monoids; diversity; elasticity; numbers of almost primary components

Citations:

Zbl 1265.20060
Cite Review PDF
Full Text: DOI Link

References:

[1] D.D. Anderson, D. F. Anderson: Elasticity of factorizations in integral domains. J. Pure Appl. Algebra 80 (1992), 217–235. · Zbl 0773.13003 · doi:10.1016/0022-4049(92)90144-5
[2] S.T. Chapman, F. Halter-Koch, U. Krause: Inside factorial monoids and integral domains. J. Algebra 252 (2002), 350–375. · Zbl 1087.13510 · doi:10.1016/S0021-8693(02)00031-5
[3] S.T. Chapman, U. Krause: Cale monoids, Cale domains, and Cale varieties. Arithmetical properties of commutative rings and monoids. Boca Raton, FL: Chapman & Hall/CRC. Lecture Notes in Pure and Applied Mathematics 241 (2005), 142–171. · Zbl 1100.20040
[4] F. Halter-Koch: Ideal Systems. An Introduction to Multiplicative Ideal Theory. Pure and Applied Mathematics, Marcel Dekker. 211. New York, 1998. · Zbl 0953.13001
[5] U. Krause: Eindeutige Faktorisierung ohne ideale Elemente. Abh. Braunschw. Wiss. Ges. 33 (1982), 169–177. (In German.) · Zbl 0518.20062
[6] U. Krause: A characterization of algebraic number fields with cyclic class group of prime power order. Math. Z. 186 (1984), 143–148. · doi:10.1007/BF01161801
[7] U. Krause: Semigroups that are factorial from inside or from outside. Lattices, semigroups, and universal algebra, Proc. Int. Conf., Lisbon/Port. 1988 (1990), 147–161. · Zbl 0736.20039
[8] J. Maney, V. Ponomarenko: Diversity in monoids. Czech. Math. J 62 (2012), 795–809. · Zbl 1265.20060 · doi:10.1007/s10587-012-0046-1
[9] R. J. Valenza: Elasticity of factorization in number fields. J. Number Theory 36 (1990), 212–218. · Zbl 0721.11043 · doi:10.1016/0022-314X(90)90074-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.