On Groups Generated by Involutions of a Semigroup
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The Book of Involutions
The Book of Involutions Max-Albert Knus Alexander Sergejvich Merkurjev Herbert Markus Rost Jean-Pierre Tignol @ @ @ @ @ @ @ @ The Book of Involutions Max-Albert Knus Alexander Merkurjev Markus Rost Jean-Pierre Tignol Author address: Dept. Mathematik, ETH-Zentrum, CH-8092 Zurich,¨ Switzerland E-mail address: [email protected] URL: http://www.math.ethz.ch/~knus/ Dept. of Mathematics, University of California at Los Angeles, Los Angeles, California, 90095-1555, USA E-mail address: [email protected] URL: http://www.math.ucla.edu/~merkurev/ NWF I - Mathematik, Universitat¨ Regensburg, D-93040 Regens- burg, Germany E-mail address: [email protected] URL: http://www.physik.uni-regensburg.de/~rom03516/ Departement´ de mathematique,´ Universite´ catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] URL: http://www.math.ucl.ac.be/tignol/ Contents Pr´eface . ix Introduction . xi Conventions and Notations . xv Chapter I. Involutions and Hermitian Forms . 1 1. Central Simple Algebras . 3 x 1.A. Fundamental theorems . 3 1.B. One-sided ideals in central simple algebras . 5 1.C. Severi-Brauer varieties . 9 2. Involutions . 13 x 2.A. Involutions of the first kind . 13 2.B. Involutions of the second kind . 20 2.C. Examples . 23 2.D. Lie and Jordan structures . 27 3. Existence of Involutions . 31 x 3.A. Existence of involutions of the first kind . 32 3.B. Existence of involutions of the second kind . 36 4. Hermitian Forms . 41 x 4.A. Adjoint involutions . 42 4.B. Extension of involutions and transfer . -
Algebra I (Math 200)
Algebra I (Math 200) UCSC, Fall 2009 Robert Boltje Contents 1 Semigroups and Monoids 1 2 Groups 4 3 Normal Subgroups and Factor Groups 11 4 Normal and Subnormal Series 17 5 Group Actions 22 6 Symmetric and Alternating Groups 29 7 Direct and Semidirect Products 33 8 Free Groups and Presentations 35 9 Rings, Basic Definitions and Properties 40 10 Homomorphisms, Ideals and Factor Rings 45 11 Divisibility in Integral Domains 55 12 Unique Factorization Domains (UFD), Principal Ideal Do- mains (PID) and Euclidean Domains 60 13 Localization 65 14 Polynomial Rings 69 Chapter I: Groups 1 Semigroups and Monoids 1.1 Definition Let S be a set. (a) A binary operation on S is a map b : S × S ! S. Usually, b(x; y) is abbreviated by xy, x · y, x ∗ y, x • y, x ◦ y, x + y, etc. (b) Let (x; y) 7! x ∗ y be a binary operation on S. (i) ∗ is called associative, if (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x; y; z 2 S. (ii) ∗ is called commutative, if x ∗ y = y ∗ x for all x; y 2 S. (iii) An element e 2 S is called a left (resp. right) identity, if e ∗ x = x (resp. x ∗ e = x) for all x 2 S. It is called an identity element if it is a left and right identity. (c) S together with a binary operation ∗ is called a semigroup, if ∗ is as- sociative. A semigroup (S; ∗) is called a monoid if it has an identity element. 1.2 Examples (a) Addition (resp. -
Computational Techniques in Finite Semigroup Theory
COMPUTATIONAL TECHNIQUES IN FINITE SEMIGROUP THEORY Wilf A. Wilson A Thesis Submitted for the Degree of PhD at the University of St Andrews 2018 Full metadata for this item is available in St Andrews Research Repository at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/16521 This item is protected by original copyright Computational techniques in finite semigroup theory WILF A. WILSON This thesis is submitted in partial fulfilment for the degree of Doctor of Philosophy (PhD) at the University of St Andrews November 2018 Declarations Candidate's declarations I, Wilf A. Wilson, do hereby certify that this thesis, submitted for the degree of PhD, which is approximately 64500 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for any degree. I was admitted as a research student at the University of St Andrews in September 2014. I received funding from an organisation or institution and have acknowledged the funders in the full text of my thesis. Date: . Signature of candidate:. Supervisor's declaration I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of PhD in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. Date: . Signature of supervisor: . Permission for publication In submitting this thesis to the University of St Andrews we understand that we are giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being affected thereby. -
Boolean and Abstract Algebra Winter 2019
Queen's University School of Computing CISC 203: Discrete Mathematics for Computing II Lecture 7: Boolean and Abstract Algebra Winter 2019 1 Boolean Algebras Recall from your study of set theory in CISC 102 that a set is a collection of items that are related in some way by a common property or rule. There are a number of operations that can be applied to sets, like [, \, and C. Combining these operations in a certain way allows us to develop a number of identities or laws relating to sets, and this is known as the algebra of sets. In a classical logic course, the first thing you typically learn about is propositional calculus, which is the branch of logic that studies propositions and connectives between propositions. For instance, \all men are mortal" and \Socrates is a man" are propositions, and using propositional calculus, we may conclude that \Socrates is mortal". In a sense, propositional calculus is very closely related to set theory, in that propo- sitional calculus is the study of the set of propositions together with connective operations on propositions. Moreover, we can use combinations of connective operations to develop the laws of propositional calculus as well as a collection of rules of inference, which gives us even more power to manipulate propositions. Before we continue, it is worth noting that the operations mentioned previously|and indeed, most of the operations we have been using throughout these notes|have a special name. Operations like [ and \ apply to pairs of sets in the same way that + and × apply to pairs of numbers. -
7.2 Binary Operators Closure
last edited April 19, 2016 7.2 Binary Operators A precise discussion of symmetry benefits from the development of what math- ematicians call a group, which is a special kind of set we have not yet explicitly considered. However, before we define a group and explore its properties, we reconsider several familiar sets and some of their most basic features. Over the last several sections, we have considered many di↵erent kinds of sets. We have considered sets of integers (natural numbers, even numbers, odd numbers), sets of rational numbers, sets of vertices, edges, colors, polyhedra and many others. In many of these examples – though certainly not in all of them – we are familiar with rules that tell us how to combine two elements to form another element. For example, if we are dealing with the natural numbers, we might considered the rules of addition, or the rules of multiplication, both of which tell us how to take two elements of N and combine them to give us a (possibly distinct) third element. This motivates the following definition. Definition 26. Given a set S,abinary operator ? is a rule that takes two elements a, b S and manipulates them to give us a third, not necessarily distinct, element2 a?b. Although the term binary operator might be new to us, we are already familiar with many examples. As hinted to earlier, the rule for adding two numbers to give us a third number is a binary operator on the set of integers, or on the set of rational numbers, or on the set of real numbers. -
An Elementary Approach to Boolean Algebra
Eastern Illinois University The Keep Plan B Papers Student Theses & Publications 6-1-1961 An Elementary Approach to Boolean Algebra Ruth Queary Follow this and additional works at: https://thekeep.eiu.edu/plan_b Recommended Citation Queary, Ruth, "An Elementary Approach to Boolean Algebra" (1961). Plan B Papers. 142. https://thekeep.eiu.edu/plan_b/142 This Dissertation/Thesis is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Plan B Papers by an authorized administrator of The Keep. For more information, please contact [email protected]. r AN ELEr.:ENTARY APPRCACH TC BCCLF.AN ALGEBRA RUTH QUEAHY L _J AN ELE1~1ENTARY APPRCACH TC BC CLEAN ALGEBRA Submitted to the I<:athematics Department of EASTERN ILLINCIS UNIVERSITY as partial fulfillment for the degree of !•:ASTER CF SCIENCE IN EJUCATION. Date :---"'f~~-----/_,_ffo--..i.-/ _ RUTH QUEARY JUNE 1961 PURPOSE AND PLAN The purpose of this paper is to provide an elementary approach to Boolean algebra. It is designed to give an idea of what is meant by a Boclean algebra and to supply the necessary background material. The only prerequisite for this unit is one year of high school algebra and an open mind so that new concepts will be considered reason able even though they nay conflict with preconceived ideas. A mathematical science when put in final form consists of a set of undefined terms and unproved propositions called postulates, in terrrs of which all other concepts are defined, and from which all other propositions are proved. -
Introduction to Linear Logic
Introduction to Linear Logic Beniamino Accattoli INRIA and LIX (École Polytechnique) Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 1 / 49 Outline 1 Informal introduction 2 Classical Sequent Calculus 3 Sequent Calculus Presentations 4 Linear Logic 5 Catching non-linearity 6 Expressivity 7 Cut-Elimination 8 Proof-Nets Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 2 / 49 Outline 1 Informal introduction 2 Classical Sequent Calculus 3 Sequent Calculus Presentations 4 Linear Logic 5 Catching non-linearity 6 Expressivity 7 Cut-Elimination 8 Proof-Nets Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 3 / 49 Quotation From A taste of Linear Logic of Philip Wadler: Some of the best things in life are free; and some are not. Truth is free. You may use a proof of a theorem as many times as you wish. Food, on the other hand, has a cost. Having baked a cake, you may eat it only once. If traditional logic is about truth, then Linear Logic is about food Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 4 / 49 Informally 1 Classical logic deals with stable truths: if A and A B then B but A still holds) Example: 1 A = ’Tomorrow is the 1st october’. 2 B = ’John will go to the beach’. 3 A B = ’If tomorrow is the 1st october then John will go to the beach’. So if tomorrow) is the 1st october, then John will go to the beach, But of course tomorrow will still be the 1st october. Accattoli ( INRIA and LIX (École Polytechnique)) Introduction to Linear Logic 5 / 49 Informally 2 But with money, or food, that implication is wrong: 1 A = ’John has (only) 5 euros’. -
Arxiv:1809.02384V2 [Math.OA] 19 Feb 2019 H Ruet Eaefloigaecmlctd N Oi Sntcerin Clear Not Is T It Saying So Worth and Is Complicated, It Are Following Hand
INVOLUTIVE OPERATOR ALGEBRAS DAVID P. BLECHER AND ZHENHUA WANG Abstract. Examples of operator algebras with involution include the op- erator ∗-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function alge- bras, triangular matrix algebras, (complexifications) of real operator algebras, and an operator algebraic version of the complex symmetric operators stud- ied by Garcia, Putinar, Wogen, Zhu, and others. We investigate the general theory of involutive operator algebras, and give many applications, such as a characterization of the symmetric operator algebras introduced in the early days of operator space theory. 1. Introduction An operator algebra for us is a closed subalgebra of B(H), for a complex Hilbert space H. Here we study operator algebras with involution. Examples include the operator ∗-algebras occurring in noncommutative differential geometry studied re- cently by Mesland, Kaad, Lesch, and others (see e.g. [26, 25, 10] and references therein), (complexifications) of real operator algebras, and an operator algebraic version of the complex symmetric operators studied by Garcia, Putinar, Wogen, Zhu, and many others (see [21] for a survey, or e.g. [22]). By an operator ∗-algebra we mean an operator algebra with an involution † † N making it a ∗-algebra with k[aji]k = k[aij]k for [aij ] ∈ Mn(A) and n ∈ . Here we are using the matrix norms of operator space theory (see e.g. [30]). This notion was first introduced by Mesland in the setting of noncommutative differential geometry [26], who was soon joined by Kaad and Lesch [25]. In several recent papers by these authors and coauthors they exploit operator ∗-algebras and involutive modules in geometric situations. -
Sage 9.4 Reference Manual: Monoids Release 9.4
Sage 9.4 Reference Manual: Monoids Release 9.4 The Sage Development Team Aug 24, 2021 CONTENTS 1 Monoids 3 2 Free Monoids 5 3 Elements of Free Monoids 9 4 Free abelian monoids 11 5 Abelian Monoid Elements 15 6 Indexed Monoids 17 7 Free String Monoids 23 8 String Monoid Elements 29 9 Utility functions on strings 33 10 Hecke Monoids 35 11 Automatic Semigroups 37 12 Module of trace monoids (free partially commutative monoids). 47 13 Indices and Tables 55 Python Module Index 57 Index 59 i ii Sage 9.4 Reference Manual: Monoids, Release 9.4 Sage supports free monoids and free abelian monoids in any finite number of indeterminates, as well as free partially commutative monoids (trace monoids). CONTENTS 1 Sage 9.4 Reference Manual: Monoids, Release 9.4 2 CONTENTS CHAPTER ONE MONOIDS class sage.monoids.monoid.Monoid_class(names) Bases: sage.structure.parent.Parent EXAMPLES: sage: from sage.monoids.monoid import Monoid_class sage: Monoid_class(('a','b')) <sage.monoids.monoid.Monoid_class_with_category object at ...> gens() Returns the generators for self. EXAMPLES: sage: F.<a,b,c,d,e>= FreeMonoid(5) sage: F.gens() (a, b, c, d, e) monoid_generators() Returns the generators for self. EXAMPLES: sage: F.<a,b,c,d,e>= FreeMonoid(5) sage: F.monoid_generators() Family (a, b, c, d, e) sage.monoids.monoid.is_Monoid(x) Returns True if x is of type Monoid_class. EXAMPLES: sage: from sage.monoids.monoid import is_Monoid sage: is_Monoid(0) False sage: is_Monoid(ZZ) # The technical math meaning of monoid has ....: # no bearing whatsoever on the result: it's ....: # a typecheck which is not satisfied by ZZ ....: # since it does not inherit from Monoid_class. -
Interaction Laws of Monads and Comonads
Interaction Laws of Monads and Comonads Shin-ya Katsumata1 , Exequiel Rivas2 , and Tarmo Uustalu3;4 1 National Institute of Informatics, Tokyo, Japan 2 Inria Paris, France 3 Dept. of Computer Science, Reykjavik University, Iceland 4 Dept. of Software Science, Tallinn University of Technology, Estonia [email protected], [email protected], [email protected] Abstract. We introduce and study functor-functor and monad-comonad interaction laws as math- ematical objects to describe interaction of effectful computations with behaviors of effect-performing machines. Monad-comonad interaction laws are monoid objects of the monoidal category of functor- functor interaction laws. We show that, for suitable generalizations of the concepts of dual and Sweedler dual, the greatest functor resp. monad interacting with a given functor or comonad is its dual while the greatest comonad interacting with a given monad is its Sweedler dual. We relate monad-comonad interaction laws to stateful runners. We show that functor-functor interaction laws are Chu spaces over the category of endofunctors taken with the Day convolution monoidal struc- ture. Hasegawa's glueing endows the category of these Chu spaces with a monoidal structure whose monoid objects are monad-comonad interaction laws. 1 Introduction What does it mean to run an effectful program, abstracted into a computation? In this paper, we take the view that an effectful computation does not perform its effects; those are to be provided externally. The computation can only proceed if placed in an environment that can provide its effects, e.g, respond to the computation's requests for input, listen to its output, resolve its nondeterministic choices by tossing a coin, consistently respond to its fetch and store commands. -