Congruence lattices of ideals in categories and (partial) semigroups. (English) Zbl 1515.20009
Memoirs of the American Mathematical Society 1408. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-6269-7/pbk; 978-1-4704-7446-1/ebook). vii, 129 p. (2023).
In this monograph, the congruences of several monoids, categories of transformations, diagrams, matrices and braids together with congruences on their ideals are studied. The authors present an interactive strategy for starting from the smallest ideals stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals developing further the technology from [J. East et al., Adv. Math. 333, 931–1003 (2018; Zbl 1400.20060)]. This technology is used to describe lattices of chains of ideals in several categories: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley-Lieb and Jones partitions; linear and projective linear transformations and partial braids.
Reviewer: Jaak Henno (Tallinn)
MSC:
| 20-02 | Research exposition (monographs, survey articles) pertaining to group theory |
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
| 15A04 | Linear transformations, semilinear transformations |
| 18B40 | Groupoids, semigroupoids, semigroups, groups (viewed as categories) |
| 20M10 | General structure theory for semigroups |
| 20M50 | Connections of semigroups with homological algebra and category theory |
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 categoriesCitations:
Zbl 1400.20060References:
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