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)....

Irregular (quasi)varieties of groupoids are (quasi)varieties that do not contain semilattices. The regularization of a (strongly) irregular variety V of groupoids is the smallest variety containing V and the variety S of semilattices. Its quasiregularization is the smallest quasivariety containing V and S. In an earlier paper the authors described the lattice of quasivarieties of cancellative c...

By the Quillen-Suslin theorem [Qui76, Sus76], we know that projective modules over a polynomial ring over a field are free. One way of saying this is, that if two projective modules of the same rank are stably isomorphic, then they are isomorphic. That, projective modules of given rank over polynomial rings are stably isomorphic was well known at the time Quillen and Suslin proved their theorem...

State spaces in probabilistic and quantum computation are convex sets, that is, Eilenberg–Moore algebras of the distribution monad. This article studies some computationally relevant properties of convex sets. We introduce the term effectus for a base category with suitable coproducts (so that predicates, as arrows of the shape X → 1 + 1, form effect modules, and states, as arrows of the shape ...

Suppose M is a maximal ideal of a commutative integral domain R and that some power Mn of M is finitely generated. We show that M is finitely generated in each of the following cases: (i) M is of height one, (ii) R is integrally closed and htM = 2, (iii) R = K[X; S̃] is a monoid domain over a field K, where S̃ = S ∪ {0} is a cancellative torsion-free monoid such that ⋂∞ m=1 mS = ∅, and M is the m...

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...

It is shown that if an algebra has more than one element, is freely generated in some variety by one element and has a cancellative endomorphism semigroup then it generates a minimal quasivariety. This is used to construct uncountably many minimal quasivarieties of groupoids that are not varieties. A quasivariety [l] JÍ of algebras will be called implicationally complete or minimal if K has exa...

We present a disambiguation algorithm for weighted automata. The algorithm admits two main stages: a pre-disambiguation stage followed by a transition removal stage. We give a detailed description of the algorithm and the proof of its correctness. The algorithm is not applicable to all weighted automata but we prove sufficient conditions for its applicability in the case of the tropical semirin...