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The q-theory of Finite Semigroups

TL;DR: The q-theory of finite semigroups as mentioned in this paper is a theory that provides a unifying approach to finite semigroup theory via quantization, and it is the only contemporary exposition of the complete theory of the complexity of finite semiigroups.
Abstract: 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.
Citations
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Journal Article
TL;DR: In this article, the fundamental isomorphism theorem of π-algebras is proved and some algebraic properties of Hopf π algebbras are studied.
Abstract: This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

1,322 citations

Journal ArticleDOI
TL;DR: The character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of Mobius inversion were generalized in this paper.

109 citations


Cites background or methods or result from "The q-theory of Finite Semigroups"

  • ...This last section will be more demanding of the reader in terms of semigroup theoretic background, but most of the necessary background can be found in [4,11,18, 38 ]....

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  • ...This is an example of one of Green’s relations [11, 15, 18, 38 ]....

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  • ...The book of Almeida [3] contains more modern results, as does the forthcoming book [ 38 ]....

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  • ...If s, t are J -equivalent regular elements, then there are (regular) elements x, y, u, v (J -equivalent to s and t) such that xsy = t and utv = s [11, 18, 38 ]....

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  • ...Alternatively, one can easily verify that each generalized group mapping image of S corresponding to a regular J-class acts by partial permutations on its 0-minimal ideal [18, 38 ]....

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Posted Content
TL;DR: The authors generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of M\"obius inversion.
Abstract: We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of M\"obius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idempotent semigroups and semigroups with basic algebras. Using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers. These include the random walks on chambers of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and Diaconis. Applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups, generalizing and simplifying earlier results of Solomon for the rook monoid.

73 citations

Journal ArticleDOI
01 Nov 2009
TL;DR: In this paper, a modern proof of the Clifford-Munn-Ponizovskii result based on a lemma of J. A. Green is presented, which allows us to circumvent the theory of 0-simple semigroups.
Abstract: Work of Clifford, Munn and Ponizovskii parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovskii result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.

65 citations

References
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Book
01 Jan 1971
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Abstract: I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.

9,254 citations


"The q-theory of Finite Semigroups" refers background in this paper

  • ...This is one of the central notions of category theory; see [177]....

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  • ...For the present text, it is handy to be familiar with a little bit of category theory [177], basic topology [152], and perhaps some combinatorial group theory [176]....

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  • ...We remark that in [177] the left adjoint is drawn on top, while we place it on the bottom....

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  • ...For these reasons we establish in detail the categorical and Galois connection terminology of [177] and also the theory of continuous lattices, as expounded in [92] (see also [142, Chpt....

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  • ...That is, when the right adjoint to ∆ exists, and there is a terminal object, we say that C has finite products; see [177]....

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BookDOI
01 Jan 1967

5,518 citations


"The q-theory of Finite Semigroups" refers background in this paper

  • ...The “monomial map” [108,136,369], going back to the work of Frobenius on induced representations, then placesG inside the iterated wreath product of these simple group divisors....

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Book
01 Jun 1977
TL;DR: In this article, the authors introduce the concept of Free Products with Amalgamation (FPAM) and Small Cancellation Theory over free products with amalgamation and HNN extensions.
Abstract: Chapter I. Free Groups and Their Subgroups 1. Introduction 2. Nielsen's Method 3. Subgroups of Free Groups 4. Automorphisms of Free Groups 5. Stabilizers in Aut(F) 6. Equations over Groups 7. Quadratic sets of Word 8. Equations in Free Groups 9. Abstract Lenght Functions 10. Representations of Free Groups 11. Free Pruducts with Amalgamation Chapter II. Generators and Relations 1. Introduction 2. Finite Presentations 3. Fox Calculus, Relation Matrices, Connections with Cohomology 4. The Reidemeister-Schreier Method 5. Groups with a Single Defining Relator 6. Magnus' Treatment of One-Relator Groups Chapter III. Geometric Methods 1. Introduction 2. Complexes 3. Covering Maps 4. Cayley Complexes 5. Planar Caley Complexes 6. F-Groups Continued 7. Fuchsian Complexes 8. Planar Groups with Reflections 9. Singular Subcomplexes 10. Sherical Diagrams 11. Aspherical Groups 12. Coset Diagrams and Permutation Representations 13. Behr Graphs Chpter IV. Free Products and HNN Extensions 1. Free Products 2. Higman-Neumann-Neumann Extensions and Free Products with Amalgamation 3. Some Embedding Theorems 4. Some Decision Problems 5. One-Relator Groups 6. Bipolar Structures 7. The Higman Embedding Theorem 8. Algebraically Closed Groups Chapter V. Small Cancellation Theory 1. Diagrams 2. The Small Cancellation Hypotheses 3. The Basic Formulas 4. Dehn's Algorithm and Greendlinger's Lemma 5. The Conjugacy Problem 6. The Word Problem 7. The Cunjugacy Problme 8. Applications to Knot Groups 9. The Theory over Free Products 10. Small Cancellation Products 11. Small Cancellation Theory over free Products with Amalgamation and HNN Extensions Bibliography Index of Names Subject Index

3,454 citations


"The q-theory of Finite Semigroups" refers background in this paper

  • ...For the present text, it is handy to be familiar with a little bit of category theory [177], basic topology [152], and perhaps some combinatorial group theory [176]....

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  • ...If we orient Γ and denote by E the set of positively oriented edges, then π1(Γ, v) is freely generated by the homotopy classes of the paths peιepeτ , where e ∈ E \T [176,307,311]....

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  • ...It is well known that π1(Γ, v) is a free group [176, 311]....

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  • ...The reader is referred to [176,311] for more details....

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  • ...Since the free group on A is residually a p-group [176, 209], there is an A-generated group G ∈ Gp such that any two words in A of length n−1 are sent to distinct elements of G....

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Book
01 Jan 1987
TL;DR: In this article, the authors present a largely balanced approach, which combines many elements of the different traditions of the subject, and includes a wide variety of examples, exercises, and applications, in order to illustrate the general concepts and results of the theory.
Abstract: Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.

3,340 citations


"The q-theory of Finite Semigroups" refers background in this paper

  • ...Recall that a compact Hausdorff space is totally disconnected if and only if it has a basis of simultaneously open and closed, that is clopen, subsets for its topology [112,142,152]....

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  • ...In fact it is well known that every compact metric space is a continuous quotient of a Cantor set [152], so the above example is just the tip of the iceberg....

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  • ...For the present text, it is handy to be familiar with a little bit of category theory [177], basic topology [152], and perhaps some combinatorial group theory [176]....

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Book
16 Nov 1981

2,750 citations


"The q-theory of Finite Semigroups" refers background or methods in this paper

  • ...459 7.1 Birkhoff’s Subdirect Representation Theorem . . . . . . . . . . . . . . . 460 7.1.1 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 7.1.2 Elementary examples of smi decompositions . . . . . . . . . . 468 7.2 Locally Dually Algebraic Lattices . . . . . . . . . . . . . . . . . . . . . . . . . ....

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  • ...Imagine what techniques will be needed to address the general case! We then entered into the so-called “abstract spectral theory” of lattices [92], a sweeping generalization of Stone’s duality between Boolean algebras and profinite spaces [59, 92, 112, 142, 330, 331]....

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  • ...This is the classical Stone duality for Boolean algebras [59,112,142,330,331]....

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  • ...Recall that a Boolean algebra [59, 92, 112, 142] is a distributive lattice L with top 1 and bottom 0 such that each element a has a complement ¬a satisfying a ∧ ¬a = 0 and a ∨ ¬a = 1 One can view L as a semiring by taking ∨ as addition and ∧ as multiplication (semirings are the subject of our next chapter)....

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  • ...It is a well-known theorem of Birkhoff [59] that...

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