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: East, James, Ruškuc, Nik
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2023
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Categories (Mathematics)
Congruences (Geometry)
Semigroups
Brauer categories
chains of ideals
ideals
Categories
<inline-formula content-type="math/mathml"> H \mathscr {H} </inline-formula>-congruences
semigroups
projective linear categories
partial braid categories
transformation categories
Jones categories
linear categories
partition categories
diagram categories
congruences
lattices
Temperley-Lieb categories
ISBN:9781470462697, 1470462699
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Abstract 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.
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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.
Author Ruškuc, Nik
East, James
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Copyright Copyright 2023 American Mathematical Society
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1470462699
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Keywords Brauer categories
chains of ideals
ideals
Categories
<inline-formula content-type="math/mathml"> H \mathscr {H} </inline-formula>-congruences
semigroups
projective linear categories
partial braid categories
transformation categories
Jones categories
linear categories
partition categories
diagram categories
congruences
lattices
Temperley-Lieb categories
LCCN 2023014514
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Language English
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Notes Includes bibliographical references (p. 125-129)
April 2023, volume 284, number 1408 (fourth of 6 numbers)
OCLC 1379097150
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PublicationPlace Providence, Rhode Island
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PublicationSeriesTitle Memoirs of the American Mathematical Society
PublicationYear 2023
Publisher American Mathematical Society
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Snippet This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and...
View the abstract.
SourceID askewsholts
proquest
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SubjectTerms Categories (Mathematics)
Congruences (Geometry)
Semigroups
TableOfContents 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
Title Congruence Lattices of Ideals in Categories and (Partial) Semigroups
URI https://www.ams.org/memo/1408/
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Volume 284
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