in this paper, a commutative semigroup will be written as a disjoint :union: of its cancellative subsemigroups. based on this fact we will define the left regular representation of a commutative separative semigroup and show that this representation is faithful. finally concrete examples of commutative separative semigroups, their decompositions and their left regular representations are given.

We prove in a broad combinatorial setting, namely for finitely generated semipositive cancellative reduced binoids, that the Hilbert–Kunz multiplicity is rational number independent of characteristic.

Abstract Partition refinement is a method for minimizing automata and transition systems of various types. Recently, we have developed partition algorithm that generic in the type given system matches run time best known algorithms many concrete types systems, e.g. deterministic as well ordinary, weighted, probabilistic (labelled) systems. Genericity achieved by modelling functors on sets, coal...

Condition $(PWP)$ which was introduced in (Laan, V., {it Pullbacks and flatness properties of acts I}, Commun. Algebra, 29(2) (2001), 829-850), is related to flatness concept of acts over monoids. Golchin and Mohammadzadeh in ({it On Condition $(PWP_E)$}, Southeast Asian Bull. Math., 33 (2009), 245-256) introduced Condition $(PWP_E)$, such that Condition $(PWP)$ implies it, that is, Condition $...