Semigroups that are factorial from inside or from outside. (English) Zbl 0736.20039

Krause, Ulrich

Lattices, semigroups, and universal algebra, Proc. Int. Conf., Lisbon/Port. 1988, 147-161 (1990).

[For the entire collection see Zbl 0724.00010.]
Denote by \((S,.,1)\) a commutative monoid. The notion of factorial monoid is natural. \(S\) is called inside factorial for \((R,\varphi)\) if \(R\) is a factorial monoid and \(\varphi: R\to S\) is a monoid homomorphism such that: (i) \(x|_ Ry\Leftrightarrow\varphi(x)|_ S\varphi(y)\); (ii) For every \(x\in S\), there is some natural \(n\) with \(x^ n\in \varphi (R)\). \(S\) is called outside factorial for \((\psi,T)\) if \(T\) is a factorial monoid and \(\psi: S\to T\) is a monoid homomorphism such that: (i) \(x|_ S y\Leftrightarrow\psi(x)|_ T\psi(y)\); (ii) For every \(x\in T\), there is a natural \(n\) with \(x^ n\in \psi(S)\). \(S\) is called an extraction monoid if the function \(\lambda\) defined on \(S^*\times S^*\) by \(\lambda(x,y)=\sup\{m/n\mid x^ m|_ Sy^ n,\;m\in\mathbb{N}^*,\;n\in N\}\) attains only rational values (not infinite). \(S\) is called of finite type if for any \(x_ i\in S\) the sequence \(\hbox{rad}(x_ 1)\subseteq\hbox{rad}(x_ 2)\subseteq\dots\) becomes stationary, where \(\hbox{rad}(x)=\{y\in S\mid x| y^ n\hbox{ for some }n\in \mathbb{N}\}\). Every factorial monoid is an extraction monoid of finite type.
The main results in the paper are the following ones: 1. If \(S\) is an extraction monoid of finite type and \(A\subseteq S\) is such that any element of \(S^*\) has some component in \(A\), then for each element of \(S\) there is a positive power contained in a factorial monoid generated by finitely many elements from \(A\). 2. The monoid \(S\) is inside factorial if and only if \(S\) is an extraction monoid of finite type such that every element of \(S^*\) has an extractive component. 3. The monoid \(S\) is outside factorial if and only if \(S\) is an integrally closed extraction monoid of finite type such that every element of \(S^*\) has an extractive discrete component. (Here “integrally closed” means that \(x^ n| y^ n\) for some natural \(n\) implies \(x| y\) for \(x,y\in S\), and “discrete” means that, for any \(y\in S^*\) for which \(\lambda(x,y)\in\mathbb{Q}_ +\), \(n\lambda(x,y)\in\mathbb{N}^*\) for some fixed \(n=n(x)\in\mathbb{N}^*\).) Relationships between the Krull monoids [see R. M. Fossum, The divisor class group of a Krull domain (Springer, 1973; Zbl 0256.13001)] and these inside and outside monoids are also established in the last section of the paper.

Reviewer: M.Ştefănescu (Iaşi)

MSC:

20M14	Commutative semigroups
13F15	Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
20M11	Radical theory for semigroups

Keywords:

commutative monoid; factorial monoid; monoid homomorphism; extraction monoid; extraction monoid of finite type; inside factorial; outside factorial; Krull monoids

Citations:

Zbl 0724.00010; Zbl 0256.13001