We sketch a theory of divisibility and factorization in topological monoids, where finite products are replaced by convergent products. The algebraic case can then be viewed as the special case of discretely topologized topological monoids. We define the topological factorization monoid, a generalization of the factorization monoid for algebraic monoids, and show that it is always topologically...

It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If M is an ordered monoid and F is a field, the ring ^[[iW"]] of all formal power series with well-ordered support is a uf-domain iff M is naturally ordered (i.e...

We consider the class of “well-tempered” integer-valued scoring games, which have the property that the parity of the length of the game is independent of the line of play. We consider disjunctive sums of these games, and develop a theory for them analogous to the standard theory of disjunctive sums of normal-play partizan games. We show that the monoid of well-tempered scoring games modulo ind...

A subsemiring S of $$\mathbb {R}$$ is called a positive semiring provided that consists nonnegative numbers and $$1 \in S$$ . Here we study factorizations in both the additive monoid $$(S,+)$$ multiplicative $$(S\backslash \{0\}, \cdot )$$ In particular, investigate when, for S, have following properties: atomicity, ACCP, bounded factorization property (BFP), finite (FFP), half-factorial (HFP)....

An atomic monoid $M$ is called length-factorial if for every non-invertible element $x \in M$, no two distinct factorizations of $x$ into irreducibles have the same length (i.e., number irreducible factors, counting repetitions). The notion length-factoriality was introduced by J. Coykendall and W. Smith in 2011 under term 'other-half-factoriality': they used to provide a characterization uniqu...

Let M be a commutative cancellative monoid. The set ∆(M), which consists of all positive integers which are distances between consecutive irreducible factorization lengths of elements in M , is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with divisor class group Zn, then it is well-known that ∆(M) ⊆ {1, 2, . . . , n − 2}. Moreover, equality holds fo...

Abstract We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory corresponding monoid T ( ). Results Levy–Wiegand and Levy–Odenthal together with local case yield an explicit description The is typically neither factorial nor cancellative. Nevertheless, we construct transfer homomorphism to graph agglomerat...