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 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.

Presentations for tensor categories

“…We begin by defining the partial vine category PV, following [5,28,33,56]. By a string we mean a smooth (tame) embedding s of the unit interval [0, 1] into R 3 , such that:…”

“…Endomorphisms in V and IB form the vine monoids V n [56] and inverse braid monoids IB n [26]. The category IB was studied in [33,Section 12], where it was observed to be an inverse category in the sense of [49] and [16, Section 2.3.2]: for any α ∈ IB, the partial braid obtained by reflecting α in the plane z = 1 2 is the unique element β of IB satisfyng α = αβα and β = βαβ. The categories V and IB are both closed under ⊕, and both contain the braid groups B n (n ∈ N), so they are both PROBs.…”

Classification of congruences of twisted partition monoids

“…Kernels of representations can be equivalently viewed as ideals or as congruences. Understanding congruences is the key motivation for the current article, and indeed for the broader program of which it is a part [23][24][25][26].…”

“…Temperley-Lieb) and Motzkin monoids. The article [23] also developed general machinery for constructing congruences on arbitrary monoids, which has subsequently been applied to infinite partition monoids in [24], and extended to categories and their ideals in [26]. The classification of congruences on P n is stated below in Theorem 2.5, and the lattice Cong(P n ) of all congruences is shown in Figure 2.…”

“…In addition to these factors, a crucial role is also played by certain technical 'separation properties', which were explored in more depth in [26]. Roughly speaking, these properties ensure that pairs of partitions suitably 'separated' by Green's relations [32] generate 'large' principal congruences.…”