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Note on Some Order Properties Related to Processes Semantics (I)

Vaida D.

Fundamenta Informaticae

2006

Vol. 73, nr 1-2

307-319

This note begins a study of some elementary properties related to the order structures applied in the algebraic approach to processes semantics. The support examples come from the partially additive semantics developed by Steenstrup (1985) and Manes and Arbib (1986) and from process algebra of Baeten and Weijland (1990). The main sources for the algebraic theory are F.A.Smith (1966) and Golan (1999). We show that different properties can be extended to partially additive distributive algebras more general than sum-ordered partial semirings. One establishes that the support examples constitute multilattices, in the sense of Benado (1955). By the examples, the ordering considered, and the references, this preliminary study is related to Rudeanu et al. (2004) and to the algebraic approach to languages due to Mateescu, e.g., (1996), (1989), (1994).

Semirings in Operations Research and Computer Science: More Algebra

Rudeanu S., Vaida D.

Fundamenta Informaticae

2004

Vol. 61, nr 1

61--85

We undertake an axiomatic study of certain semirings and related structures that occur in operations research and computer science. We focus on the properties A,I,U,G,Z,L that have been used in the algebraic study of path problems in graphs and prove that the only implications linking the above properties are essentially those already known. On the other hand we extend those implications to the framework of left and right variants of A,I,U,G,Z,L, and we also prove two embedding theorems. Further generalizations refer mainly to semiring-like algebras with a partially defined addition, which is needed in semantics. As suggested by idempotency (I) and absorption (A), we also examine in some detail the connections between semirings and distributive lattices, which yield new systems of axioms for the latter.