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.

The q-theory of Finite Semigroups

. In this paper we prove that the collection of all weakly distributive lattice ordered groups is a radical class and that it fails to be a torsion class.

Lattice ordered groups

We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of H-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and H-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that basically any rational stable model category is Quillen equivalent to modules over a differential graded -algebra (with many objects).

/pdf/hz-algebra-spectra-are-differential-graded-algebras-41rl3l33pv.pdf

HZ -algebra spectra are differential graded algebras

This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of $n$-nerve and Pelissier's thesis. If $M$ is a tractable left proper cartesian model category, we construct a tractable left proper cartesian model structure on the category of $M$-precategories. The procedure can then be iterated, leading to model categories of $(\infty , n)$-categories.

https://hal.archives-ouvertes.fr/hal-00449826/document

Homotopy theory of higher categories

Publisher Summary This chapter introduces the basic notions in theory of quantales. It describes the aspects of algebraic and categorical properties of quantales and quantale modules. This chapter also summarizes the recent developments involving representations and projectivity in quantales. Because of infinitesimal joins, quantales do not form a variety but many of the methods of universal algebra can be still used. Quantales are also applied in linear and other substructural logics and automaton theory. An important moment in the development of the theory of quantales was the realization that quantales give the semantics for propositional linear logic in the same way as Boolean algebras give semantics for classical propositional logic.

Algebraic and Categorical Aspects of Quantales

We show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht-Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.

Classification theory for accessible categories

On homotopy varieties

We show that metric abstract elementary classes are coherent
accessible categories with directed colimits, with concrete
$\aleph_1$-directed colimits and concrete monomorphisms. More
broadly, we define a notion of $\kappa$-concrete Abstract
Elementary Class and develop the theory of such categories,
beginning with a category-theoretic analogue of Shelah's
Presentation Theorem and a proof of the existence of an
Ehrenfeucht-Mostowski functor in case the category is large.

Metric abstract elementary classes as accessible categories

We investigate pure monomorphisms and pure epimorphisms in varieties of universal algebras with applications to equationally compact (= pure injective) algebras, pure projective algebras and perfect varieties.

Purity in algebra

Since its creation by S. Eilenberg and S. MacLane [EM], category theory has brought a number of important concepts. Accessible categories are among them and we going to show how they can help to treat injectivity in algebra, model theory and homotopy theory.

/pdf/injectivity-and-accessible-categories-1u2s5h70w3.pdf

Injectivity and accessible categories

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