Congruence lattices of ideals in categories and (partial) semigroups
Abstract
This paper 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 ofa 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.
Citation
East , J & Ruskuc , N 2020 , ' Congruence lattices of ideals in categories and (partial) semigroups ' , Memoirs of the American Mathematical Society , vol. 0 , pp. 2-108 . < https://arxiv.org/abs/2001.01909 >
Publication
Memoirs of the American Mathematical Society
Status
Peer reviewed
ISSN
0065-9266Type
Journal article
Rights
Copyright © 2020 American Mathematical Society. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://www.ams.org/publications/ebooks/memoirs
Description
Funding: UK EPSRC grant EP/S020616/1 (NR).Collections
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