Semigroups, Categories, and Partial Algebras: ICSAA 2019, Kochi, India, December 9–12 (Springer Proceedings in Mathematics & Statistics, 345)

This book is a collection of selected papers presented at the International Conference on Semigroups and Applications, held at the Cochin University of Science and Technology, India, from December 9{u2013}12, 2019. This book discusses the recent developments in semigroups theory, category theory and the applications of these in various areas of research, including structure theory of semigroups, lattices, rings and partial algebras. This book presents chapters on ordering orders and quotient rings, block groups and Hall{u2019}s relations, quotients of the Booleanization of inverse semigroup, Markov chains through semigroup graph expansions, polycyclic inverse monoids and Thompson group, balanced category and bundle category. This book will be of much value to researchers working in areas of semigroup and operator theory Preface Contents Editors and Contributors Ordering Orders and Quotient Rings 1 Introduction 2 Green's Relations 3 Generalized Inverses 3.1 Inverses Along an Element 3.2 Properties of the Group Inverse 4 Order Relations on General Semigroups and Their Properties 5 Quotient Rings Along Functions 5.1 Properties of Fountain–Gould Quotient Rings 6 Order Relations on a Quotient Ring 6.1 Mitsch Order Relation 6.2 Nambooripad Order Relation 6.3 Sharp Order References A Study on Cayley Graphs of Full Transformation Semigroups 1 Introduction 2 Preliminaries References Block-Groups and Hall Relations 1 Background and Motivation: Straubing's Theorem 2 Block-Groups and Power Semigroups of Groups 3 Hall Relations 4 Representation Theorem References Balanced Categories and the Biorder in Semigroups 1 Categories with Subobjects, Images, and the Biorders in Semigroups 1.1 Morphisms in a Category 1.2 Categories with Subobjects 1.3 Factorization of Morphisms and Categories with Images 2 The Cone Semigroup of a Category and Its Abundance 2.1 The Cone Semigroup calBC and the Idempotent Biorder of It 2.2 The Cone Semigroup calTcalC and Its Abundance 3 The Left Ideal Category of an Abundant Semigroup and Its Cone Semigroup 4 The Isomorphism Between Balanced Category calC and mathbbL(calTcalC) References Category of Principal Left Ideals of Normal Bands 1 Preliminaries 1.1 Normal Categories 1.2 Category of Principal Left Ideals of a Regular Semigroups 2 Normal Category from Normal Bands 2.1 Normal Bands 2.2 Category of Principal Left Ideals of Normal Bands References Quotients of the Booleanization of an Inverse Semigroup 1 Introduction 2 Preliminaries 2.1 First Definitions and Conventions 2.2 Representations and Proper Representations of Semilattices 2.3 The Universal Booleanization of a Semilattice 2.4 The Tight Booleanization of a Semilattice 2.5 Stone Duality for Boolean Algebras 2.6 Boolean Inverse Semigroups 2.7 Stone Groupoids 2.8 The Universal Groupoid of an Inverse Semigroup 2.9 The Tight Groupoid of an Inverse Semigroup 2.10 Stone Duality for Boolean Inverse Semigroups 3 X-to-Join Representations of Semilattices and Inverse Semigroups 3.1 X-to-Join Representations of Semilattices 3.2 Connection with π-Tight Representations 3.3 X-to-Join Representations of Inverse Semigroups 4 Prime and Core Representations 5 Generators and Relations 6 Weakly Meet-Preserving Quotients of B(S) via X-to-Join Representations 7 Intermediate Boundary Quotients of C*-Algebras of Zappa–Szép Product Right LCM Semigroups via X-to-Join Representations 7.1 X-to-Join Representations of Inverse Semigroups in C*-Algebras 7.2 Right LCM Semigroups and Their Left Inverse Hulls 7.3 The C*-Algebra and the Boundary Quotient C*-Algebra of a Right LCM Semigroup 7.4 Zappa–Szép Products: A Construction Due to Brownlowe, Ramagge, Robertson, and Whittaker 7.5 C*A(UbowtieA) and C*U(UbowtieA) via X-to-Join Representations References On the Structure of the Commuting Graph of Brandt Semigroups 1 Introduction 2 Preliminaries 3 Main Results References The Mathematical Work of K. S. S. Nambooripad 1 Introduction 2 Regular Semigroups, Biordered Sets, and Inductive Groupoids 3 Outgrowths of Nambooripad's Inductive Groupoid Approach 4 Regular Semigroups and Cross-Connections 5 Outgrowths of Nambooripad's Cross-Connection Theory 6 Connections Between Regular Semigroups and Other Areas 7 Ph.D. Theses Directed by Nambooripad References Markov Chains Through Semigroup Graph Expansions (A Survey) 1 Introduction 2 Stationary Distribution from Semigroup Expansions 2.1 Markov Chains 2.2 Right Cayley Graphs 2.3 The Karnofsky–Rhodes Expansion 2.4 The McCammond Expansion 2.5 Stationary Distribution 2.6 Loop Graphs 2.7 Kleene Expressions 2.8 From Kleene Expressions to Rational Functions 2.9 Mixing Time 2.10 Another Example References On Various Multiplicity-Free Products of Schur Functions 1 Introduction 2 Terminology and Notation 3 Multiplicity-Free Outer and Kronecker Products of Irreducible Characters 4 Multiplicity-Free Plethysms of Schur Functions References On ω-Identities over Finite Aperiodic Semigroups with Commuting Idempotents 1 Introduction 2 ω-Identities 3 The Automaton of an ω-Term 4 Reversible Aperiodic Quotient of an Automaton 5 The ω-Word Problems over AcapECom and AInv References The Polycyclic Inverse Monoids and the Thompson Groups Revisited 1 Introduction 2 The Thompson Groups via the Polycyclic Inverse Monoids 2.1 The Distributive Inverse Monoid Dn 2.2 The Group Associated with a Polycyclic Inverse Monoid 3 The Structure of Prefix Codes 4 The Tight Completion 4.1 The Lenz Congruence equiv 4.2 The Structure of the Lenz Congruence on Dn 4.3 Handling Right-Infinite Strings 4.4 The Boolean Inverse Monoid Cn 5 The n-Ary Cantor Algebra 5.1 A Restriction Monoid 5.2 n-Ary Cantor Algebras 5.3 The 1-Generated Case References Category of Chain Bundles 1 Preliminaries 2 Category of Chain Bundles 3 Category of Preorder Chains 4 Some Categorical Properties of Chain Bundles References Cancellable Elements in the Lattice of Overcommutative Semigroup Varieties 1 Introduction 2 Preliminaries and Summary 3 G-Sets 4 The Structure of the Lattice mathbbOC 5 Proof of Theorem2.2 References