From Monoids to Near-Semirings: the Essence of Monadplus and Alternative
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Semiring Frameworks and Algorithms for Shortest-Distance Problems
Journal of Automata, Languages and Combinatorics u (v) w, x–y c Otto-von-Guericke-Universit¨at Magdeburg Semiring Frameworks and Algorithms for Shortest-Distance Problems Mehryar Mohri AT&T Labs – Research 180 Park Avenue, Rm E135 Florham Park, NJ 07932 e-mail: [email protected] ABSTRACT We define general algebraic frameworks for shortest-distance problems based on the structure of semirings. We give a generic algorithm for finding single-source shortest distances in a weighted directed graph when the weights satisfy the conditions of our general semiring framework. The same algorithm can be used to solve efficiently clas- sical shortest paths problems or to find the k-shortest distances in a directed graph. It can be used to solve single-source shortest-distance problems in weighted directed acyclic graphs over any semiring. We examine several semirings and describe some specific instances of our generic algorithms to illustrate their use and compare them with existing methods and algorithms. The proof of the soundness of all algorithms is given in detail, including their pseudocode and a full analysis of their running time complexity. Keywords: semirings, finite automata, shortest-paths algorithms, rational power series Classical shortest-paths problems in a weighted directed graph arise in various contexts. The problems divide into two related categories: single-source shortest- paths problems and all-pairs shortest-paths problems. The single-source shortest-path problem in a directed graph consists of determining the shortest path from a fixed source vertex s to all other vertices. The all-pairs shortest-distance problem is that of finding the shortest paths between all pairs of vertices of a graph. -
Subset Semirings
University of New Mexico UNM Digital Repository Faculty and Staff Publications Mathematics 2013 Subset Semirings Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy [email protected] Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebraic Geometry Commons, Analysis Commons, and the Other Mathematics Commons Recommended Citation W.B. Vasantha Kandasamy & F. Smarandache. Subset Semirings. Ohio: Educational Publishing, 2013. This Book is brought to you for free and open access by the Mathematics at UNM Digital Repository. It has been accepted for inclusion in Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. Subset Semirings W. B. Vasantha Kandasamy Florentin Smarandache Educational Publisher Inc. Ohio 2013 This book can be ordered from: Education Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: 1-866-880-5373 Copyright 2013 by Educational Publisher Inc. and the Authors Peer reviewers: Marius Coman, researcher, Bucharest, Romania. Dr. Arsham Borumand Saeid, University of Kerman, Iran. Said Broumi, University of Hassan II Mohammedia, Casablanca, Morocco. Dr. Stefan Vladutescu, University of Craiova, Romania. Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/eBooks-otherformats.htm ISBN-13: 978-1-59973-234-3 EAN: 9781599732343 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two SUBSET SEMIRINGS OF TYPE I 9 Chapter Three SUBSET SEMIRINGS OF TYPE II 107 Chapter Four NEW SUBSET SPECIAL TYPE OF TOPOLOGICAL SPACES 189 3 FURTHER READING 255 INDEX 258 ABOUT THE AUTHORS 260 4 PREFACE In this book authors study the new notion of the algebraic structure of the subset semirings using the subsets of rings or semirings. -
On Free Quasigroups and Quasigroup Representations Stefanie Grace Wang Iowa State University
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2017 On free quasigroups and quasigroup representations Stefanie Grace Wang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Wang, Stefanie Grace, "On free quasigroups and quasigroup representations" (2017). Graduate Theses and Dissertations. 16298. https://lib.dr.iastate.edu/etd/16298 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. On free quasigroups and quasigroup representations by Stefanie Grace Wang A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Jonathan D.H. Smith, Major Professor Jonas Hartwig Justin Peters Yiu Tung Poon Paul Sacks The student author and the program of study committee are solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2017 Copyright c Stefanie Grace Wang, 2017. All rights reserved. ii DEDICATION I would like to dedicate this dissertation to the Integral Liberal Arts Program. The Program changed my life, and I am forever grateful. It is as Aristotle said, \All men by nature desire to know." And Montaigne was certainly correct as well when he said, \There is a plague on Man: his opinion that he knows something." iii TABLE OF CONTENTS LIST OF TABLES . -
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. -
Irreducible Representations of Finite Monoids
U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately. -