Congruence Lattices of Ideals in Categories and (Partial) Semigroups / James East and Nik Ruskuc.

Format:

Book

Author/Creator:

East, James (Professor of mathematics), author.

Ruškuc, Nik, author.

Series:

Memoirs of the American Mathematical Society ; Volume 284.

Memoirs of the American Mathematical Society Series ; Volume 284

Language:

English

Subjects (All):

Congruences (Geometry).

Categories (Mathematics).

Semigroups.

Physical Description:

1 online resource (144 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley-Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories"-- Provided by publisher.

Contents:

Cover

Title page

Acknowledgments

Chapter 1. Introduction

Chapter 2. Categories and partial semigroups

2.1. Preliminaries

2.2. Congruences

2.3. Useful lemmas

Chapter 3. Ideal extensions

3.1. Generation and separation properties

3.2. Liftable congruences

3.3. Congruences on ideal extensions

Chapter 4. Chains of ideals

4.1. Congruences on chains of ideals

4.2. Partial rectangular bands

4.3. Short retractable chains

4.4. Short non-retractable chains

4.5. Visualisation conventions

Chapter 5. Transformation categories

5.1. Definitions and preliminaries on \T

5.2. Green's relations and multiplicative properties in \T

5.3. Congruences on ideals of \T

5.4. Planar and annular reducts of \T

5.5. Other categories of transformations

Chapter 6. Partition categories

6.1. Definitions and preliminaries on \P

6.2. Green's relations and multiplicative properties in \P

6.3. Congruences on ideals of \P

6.4. Planar and annular reducts of \P

6.5. Partial Brauer categories

Chapter 7. Brauer categories

7.1. Definitions and preliminaries on \B

7.2. Green's relations and multiplicative properties in \B

7.3. Congruences on ideals of \B: the odd case

7.4. Congruences on ideals of \B: the even case

7.5. Planar and annular reducts of \B

Chapter 8. Temperley-Lieb and anti-Temperley-Lieb categories

8.1. Multiplicative properties in \TL and \TLpm

8.2. Congruences on ideals of \TL and \TLpm: the odd case

8.3. Congruences on ideals of \TL: the even case

8.4. Congruences on ideals of \TLpm: the even case

Chapter 9. Jones and anti-Jones categories

9.1. Multiplicative properties in \JJ and \Jpm

9.2. Congruences on ideals of \JJ and \Jpm: the odd case

9.3. Congruences on ideals of \JJ and \Jpm: the even case.

Chapter 10. Categories and partial semigroups with ˝-congruences

10.1. ℋ-congruences

10.2. More on chains of ideals

Chapter 11. Linear and projective linear categories

11.1. Definitions and preliminaries on \LL and \PL

11.2. Green's relations and multiplicative properties in \LL and \PL

11.3. Congruences on ideals of \LL

11.4. Visualising the lattices

11.5. Congruences on ideals of \PL

Chapter 12. Partial braid categories

12.1. Definitions and preliminaries on \IB

12.2. Green's relations and multiplicative properties in \IB

12.3. Congruences on ideals of \IB

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

9781470474461

1470474468