Congruence Lattices of Ideals in Categories and (Partial) Semigroups
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...
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| Main Authors: | , |
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| Format: | eBook Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2023
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| Edition: | 1 |
| Series: | Memoirs of the American Mathematical Society |
| Subjects: | |
| ISBN: | 9781470462697, 1470462699 |
| ISSN: | 0065-9266, 1947-6221 |
| Online Access: | Get full text |
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Table of Contents:
- Acknowledgments -- Introduction -- Categories and partial semigroups -- Ideal extensions -- Chains of ideals -- Transformation categories -- Partition categories -- Brauer categories -- Temperley–Lieb and anti-Temperley–Lieb categories -- Jones and anti-Jones categories -- Categories and partial semigroups with <inline-formula content-type="math/mathml"> H \mathscr {H}{} </inline-formula>-congruences -- Linear and projective linear categories -- Partial braid categories
- 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