Affiliations: Faculty of Mathematics and Computer Science, Bucharest
University, Romania, Str.Academiei 14, 010014 Bucharest, Romania
Abstract: 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.
Keywords: semiring, optimal path problem, fuzzy modelling, partial additive semantics, process algebra, complete existential theory
