308 research outputs found

    Semirings which have linearly ordered prime ideals

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    As a generalization of valuation semirings, the main purpose of this paper is to investigate those semirings that their prime ideals are totally ordered by inclusion. First, we prove that the prime ideals of a semiring SS are linearly ordered if and only if for each x,y∈Sx,y \in S, there is a positive integer nn such that either x∣ynx|y^n or y∣xny|x^n. Then we introduce and characterize pseudo-valuation semidomains. It is shown that prime ideals of pseudo-valuation semidomains and also divided ones are linearly ordered.Comment: Some new references added. Some minor typos edite

    Semiring and semimodule issues in MV-algebras

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    In this paper we propose a semiring-theoretic approach to MV-algebras based on the connection between such algebras and idempotent semirings - such an approach naturally imposing the introduction and study of a suitable corresponding class of semimodules, called MV-semimodules. We present several results addressed toward a semiring theory for MV-algebras. In particular we show a representation of MV-algebras as a subsemiring of the endomorphism semiring of a semilattice, the construction of the Grothendieck group of a semiring and its functorial nature, and the effect of Mundici categorical equivalence between MV-algebras and lattice-ordered Abelian groups with a distinguished strong order unit upon the relationship between MV-semimodules and semimodules over idempotent semifields.Comment: This version contains some corrections to some results at the end of Section

    *-Continuous Kleene ω\omega-Algebras for Energy Problems

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    Energy problems are important in the formal analysis of embedded or autonomous systems. Using recent results on star-continuous Kleene omega-algebras, we show here that energy problems can be solved by algebraic manipulations on the transition matrix of energy automata. To this end, we prove general results about certain classes of finitely additive functions on complete lattices which should be of a more general interest.Comment: In Proceedings FICS 2015, arXiv:1509.0282

    Weighted Finite Automata over Strong Bimonoids

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    We investigate weighted finite automata over strings and strong bimonoids. Such algebraic structures satisfy the same laws as semirings except that no distributivity laws need to hold. We define two different behaviors and prove precise characterizations for them if the underlying strong bimonoid satisfies local finiteness conditions. Moreover, we show that in this case the given weighted automata can be determinized

    Weighted automata and multi-valued logics over arbitrary bounded lattices

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    AbstractWe show that L-weighted automata, L-rational series, and L-valued monadic second order logic have the same expressive power, for any bounded lattice L and for finite and infinite words. We also prove that aperiodicity, star-freeness, and L-valued first-order and LTL-definability coincide. This extends classical results of Kleene, Büchi–Elgot–Trakhtenbrot, and others to arbitrary bounded lattices, without any distributivity assumption that is fundamental in the theory of weighted automata over semirings. In fact, we obtain these results for large classes of strong bimonoids which properly contain all bounded lattices
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