Length-factoriality in commutative monoids and integral domains

JOURNAL OF ALGEBRA

卷:578

Article

Chapman, Scott T.¹  Coykendall, Jim²  Gotti, Felix³  Smith, William W.⁴ 

[1] Sam Houston State Univ, Dept Math & Stat, Huntsville, TX 77341 USA

[2] Clemson Univ, Dept Math Sci, Clemson, SC 29634 USA

[3] MIT, Dept Math, Cambridge, MA 02139 USA

[4] Univ N Carolina, Dept Math, Chapel Hill, NC 27599 USA

关键词: Length-factoriality;    Factorization;    Unique factorization;    Other-half-factoriality;    Finite-rank monoid;    Dedekind domain;   

DOI  :  10.1016/j.jalgebra.2021.03.010

来源: Elsevier

【 摘 要 】

An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element x is an element of M no two distinct factorizations of x have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved that in the setting of integral domains, length-factoriality can be taken as an alternative definition of a unique factorization domain. However, being a length-factorial monoid is, in general, weaker than being a factorial monoid (i.e., a unique factorization monoid). Here we further investigate length-factoriality. First, we offer two characterizations of a length-factorial monoid M, and we use such characterizations to describe the set of Betti elements and obtain a formula for the catenary degree of M. Then we study the connection between length-factoriality and purely long (resp., purely short) irreducibles, which are irreducible elements that appear in the longer (resp., shorter) part of any unbalanced factorization relation. Finally, we prove that an integral domain cannot contain purely long and purely short irreducibles simultaneously, and we construct a Dedekind domain containing purely long (resp., purely short) irreducibles but not purely short (resp., purely long) irreducibles. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

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