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Congruence Lattices of Ideals in Categories and (Partial) Semigroups
About this Title
James East and Nik Ruškuc
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 284, Number 1408
ISBNs: 978-1-4704-6269-7 (print); 978-1-4704-7446-1 (online)
DOI: https://doi.org/10.1090/memo/1408
Published electronically: March 21, 2023
Keywords: Categories,
semigroups,
congruences,
$\mathscr {H}$-congruences,
lattices,
ideals,
chains of ideals,
diagram categories,
partition categories,
Brauer categories,
Temperley-Lieb categories,
Jones categories,
transformation categories,
linear categories,
projective linear categories,
partial braid categories
Table of Contents
Chapters
- Acknowledgments
- 1. Introduction
- 2. Categories and partial semigroups
- 3. Ideal extensions
- 4. Chains of ideals
- 5. Transformation categories
- 6. Partition categories
- 7. Brauer categories
- 8. Temperley–Lieb and anti-Temperley–Lieb categories
- 9. Jones and anti-Jones categories
- 10. Categories and partial semigroups with $\mathscr {H}{}$-congruences
- 11. Linear and projective linear categories
- 12. Partial braid categories
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.- Samson Abramsky, Temperley-Lieb algebra: from knot theory to logic and computation via quantum mechanics, Mathematics of quantum computation and quantum technology, Chapman & Hall/CRC Appl. Math. Nonlinear Sci. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2008, pp. 515–558. MR 2422231
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