Basic concepts extensions of semirings partially-ordered semirings semirings with infinite sums semialgebras semigroup semirings and semirings of formal power series.

Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic text will provide the reader - from the graduate student to the researcher/practitioner with a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. Key features: (1) Develops q-theory, a new theory that provides a unifying approach to finite semigroup theory via quantization; (2) Contains the only contemporary exposition of the complete theory of the complexity of finite semigroups; (3) Introduces spectral theory into finite semigroup theory; (4) Develops the theory of profinite semigroups from first principles, making connections with spectra of Boolean algebras of regular languages; (5) Presents over 70 research problems, most new, and hundreds of exercises. Additional features: (1) For newcomers, an appendix on elementary finite semigroup theory; (2) Extensive bibliography and index. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, and thereby updates and modernizes the literature of semigroup theory.

background

Semirings: Algebraic Theory and Applications in Computer Science

The q-theory of Finite Semigroups

We survey some important results for weighted tree automata and weighted tree transducers over finite ranked trees and semirings as weight structure. In particular, we address closure properties of the class of recognizable tree series, results on the support of such tree series, the determinization of weighted tree automata, pumping lemmata and decidability results, and finite algebraic characterizations of recognizable tree series. We discuss the equivalence between recognizable tree series and equational, rational, and MSO-definable tree series, and we present a comparison of several other models of recognizability. For weighted tree transducers we show composition and decomposition results, an inclusion diagram of some fundamental classes of tree series transformations, and hierarchies obtained by composing weighted tree transducers. We also discuss other models of weighted tree transducers.

Weighted Tree Automata and Tree Transducers

This chapter presents basic foundations for the theory of weighted automata: semirings and formal power series. A fundamental question is how to extend the star operation (Kleene iteration) from languages to series. For this, we investigate ordered, complete and continuous semirings and the related concepts of star semirings and Conway semirings. We derive natural properties for the Kleene star of cycle-free series and also of matrices often used to analyze the behavior of weighted automata. Finally, we investigate cycle-free linear equations which provide a useful tool for proving identities for formal power series.

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Semirings and Formal Power Series

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.

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Equations of tropical varieties

We give a characterization of different types of (@a,@b)-fuzzy ideals of hemirings, where @a,@b@?{@?,q,@?@?q,@?@?q} and @a @?@?q. Special attention is paid to (@?,@?@?q)-fuzzy prime and semiprime ideals.

Semirings: Algebraic Theory and Applications in Computer Science

Citations

The q-theory of Finite Semigroups

Weighted Tree Automata and Tree Transducers

Semirings and Formal Power Series

Equations of tropical varieties

(α,β)-fuzzy ideals of hemirings

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