Transitive Closure and Related Semiring Properties via Eliminants

Closed semirings are algebraic structures that provide a unified approach to a number of seemingly unrelated problems of computer science and operations research. For example, semirings can be used to describe the algebra related to regular expressions, graph-theoretical path problems, and linear equations. We present a new axiomatic formulation of semirings. We introduce the concept of eliminant, which simplifies the treatment of closed semirings considerably and yields very simple proofs of otherwise difficult theorems. We use eliminants to define matrix closure, formulate closure algorithms, and prove their correctness.