Equational theories of tropical semirings

✍ Scribed by Luca Aceto; Zoltán Ésik; Anna Ingólfsdóttir

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 446 KB
• Volume: 298
• Category: Article
• ISSN: 0304-3975
• DOI: 10.1016/s0304-3975(02)00864-2

✦ Synopsis

This paper studies the equational theories of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max; +) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a ÿnite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper o ers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.