Theory of linear and integer programming
Theory of linear and integer programming
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On the Two-Variable Fragment of the Equational Theory of the Max-Sum Algebra of the Natural Numbers
STACS '00 Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science
Limited subsets of a free monoid
SFCS '78 Proceedings of the 19th Annual Symposium on Foundations of Computer Science
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This paper studies the equational theory 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 finite 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 offers characterizations of the equations that hold in them, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.