Core Concepts

This paper presents a novel method for calculating the heights of one- and two-sided congruence lattices of semigroups, utilizing unary algebras and semigroup actions, and applies this method to derive exact formulae for various transformation, matrix, and partition monoids.

Abstract

Brookes, M., East, J., Miller, C., Mitchell, J. D., & Ruškuc, N. (2024). Heights of one- and two-sided congruence lattices of semigroups. *arXiv preprint arXiv:2310.08229v3*.

This paper aims to develop a general method for determining the heights of one- and two-sided congruence lattices of semigroups and apply this method to specific semigroup families.

The authors utilize the framework of unary algebras and semigroup actions to analyze congruence lattices. They leverage the concept of modular elements in lattices for inductive proofs and relate the height of the congruence lattice to the heights of quotients corresponding to strongly connected components. The study focuses on the left S-act associated with an L-class and employs Schützenberger groups for analysis.

- The authors establish a general formula for calculating the height of one- and two-sided congruence lattices of semigroups under specific conditions.
- They derive exact formulae for the heights of congruence lattices for several significant monoids, including the full transformation monoid (Tn), partial transformation monoid (PTn), symmetric inverse monoid (In), monoid of order-preserving transformations (On), full linear monoid (M(n, q)), partition monoid (Pn), Brauer monoid (Bn), and Temperley–Lieb monoid (TLn).
- The study reveals distinct growth patterns in the heights of congruence lattices for different semigroup families. For instance, the height of LCong(Tn) grows linearly with n2^(n−1), while the height of RCong(Tn) grows linearly with the nth Bell number.

The paper provides a powerful framework for analyzing the heights of congruence lattices in semigroups, demonstrating its effectiveness by deriving explicit formulae for important semigroup families. The findings offer valuable insights into the structural complexity of these algebraic structures.

This research significantly contributes to the field of semigroup theory by providing a systematic approach to understanding the heights of congruence lattices. The derived formulae for specific semigroup families offer valuable tools for further investigations into their properties and applications.

The general formula for calculating congruence lattice heights relies on certain assumptions. Future research could explore relaxing these assumptions to broaden the applicability of the method. Additionally, investigating the heights of congruence lattices for other semigroup families and exploring potential connections with other algebraic properties would be worthwhile endeavors.

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Stats

The height of the subgroup lattice (and hence of the left and right congruence lattices) of the symmetric group Sn grows linearly with n.
The height of the subsemigroup lattice of the full transformation monoid Tn grows linearly with n^n.
The height of LCong(Tn) grows linearly with n2^(n−1).
The height of RCong(Tn) grows linearly with the nth Bell number.

Quotes

"The height of a poset P, denoted Ht(P), is the supremum of the cardinalities of chains in P."
"The left congruences of a group G are precisely the equivalence relations whose equivalence classes are the left cosets of a subgroup."

Key Insights Distilled From

by M. Brookes, ... at **arxiv.org** 10-07-2024

Deeper Inquiries

The heights of congruence lattices provide a nuanced perspective on the structural complexity of semigroups, complementing measures like Krohn-Rhodes complexity and word problem complexity. Here's a breakdown of their interrelations:
Krohn-Rhodes Complexity: This measure, rooted in the decomposition of semigroups into wreath products of simpler components, captures the complexity of a semigroup's action. While a high Krohn-Rhodes complexity suggests a rich lattice structure, there's no direct correlation with the height. A semigroup with a simple Krohn-Rhodes decomposition can still have a congruence lattice of substantial height, especially if it possesses many $\mathcal{J}$-classes.
Word Problem Complexity: This measure quantifies the difficulty of determining if two words (sequences of semigroup elements) represent the same element. A more intricate word problem often hints at a more complex congruence lattice, as congruences play a role in word equivalence. However, the relationship isn't straightforward. A semigroup with a decidable but highly complex word problem might have a congruence lattice of relatively small height, and vice-versa.
Interplay and Insights: The heights of congruence lattices offer a distinct lens on complexity, focusing on the length of chains of nested congruences. This can shed light on the fine-grained structure of a semigroup's actions and the interplay between its elements. For instance, a rapidly growing height of the congruence lattice might indicate a rich interplay between the actions of different substructures within the semigroup.
In essence, while these complexity measures are loosely intertwined, they capture different facets of semigroup structure. Analyzing them in conjunction can provide a more comprehensive understanding of a semigroup's complexity and its behavior.

While the unary algebra and semigroup action framework proves effective for many semigroup families, exploring alternative approaches is crucial for broader applicability. Here are some potential avenues:
Combinatorial Methods: Leveraging the combinatorial structure of specific semigroup families could lead to direct height computations. For instance, bijective correspondences between congruences and combinatorial objects like partitions, graphs, or posets could provide alternative routes to determining heights.
Representation Theory: Representing semigroups as transformations on vector spaces could offer insights into their congruence lattices. The structure of invariant subspaces under these representations might be linked to the height, potentially leading to new computational techniques.
Syntactic Methods: Analyzing the defining relations or presentations of semigroups could provide information about their congruences. Techniques from formal language theory, such as analyzing congruence relations on words, might offer alternative ways to study the structure and height of congruence lattices.
Topological and Order-Theoretic Approaches: Viewing congruence lattices as topological spaces or specialized ordered structures might reveal new properties related to their heights. For instance, exploring notions like order dimension or topological invariants could offer fresh perspectives.
Computational Techniques: Developing algorithms tailored to specific semigroup classes or utilizing existing computational algebra systems could facilitate the computation of heights for larger or more complex examples.
Exploring these alternative approaches could lead to a deeper understanding of congruence lattice heights and extend the reach of these techniques to a wider range of semigroups.

The diverse growth patterns of congruence lattice heights across semigroup families have intriguing implications for their applications:
Computational Complexity: The height of a congruence lattice can serve as a proxy for the complexity of certain algorithmic tasks. For instance, in computational group theory, the height of the subgroup lattice influences the complexity of subgroup enumeration algorithms. Similarly, in semigroup applications, a rapidly growing height might indicate increased complexity in tasks like congruence generation or simplification.
Formal Language Theory: Semigroups, particularly syntactic monoids, play a crucial role in formal language theory. The structure of the congruence lattice can impact the complexity of language recognition problems. Different growth patterns might correspond to distinct classes of languages with varying expressive power and computational properties.
Theoretical Physics: Semigroups and their representations appear in areas like quantum mechanics and statistical physics. The structure of the congruence lattice could have implications for the classification of physical states or the analysis of symmetries in physical systems. Distinct growth patterns might reflect different types of physical behavior or phase transitions.
Data Analysis and Machine Learning: Semigroups and their actions are increasingly used in data analysis and machine learning, for instance, in the analysis of sequential data or the study of network dynamics. The height of the congruence lattice could provide insights into the complexity of the underlying data structures or the dynamics of the system being modeled.
Design of Efficient Algorithms: Understanding the growth patterns of congruence lattice heights can guide the design of efficient algorithms for semigroup-related problems. For instance, if a specific semigroup family exhibits slow height growth, it might suggest the existence of efficient algorithms for tasks like congruence testing or finding minimal congruences.
In conclusion, the distinct growth patterns of congruence lattice heights offer valuable information about the inherent complexity and properties of different semigroup families. This knowledge can be leveraged to develop efficient algorithms, analyze computational complexity, and gain deeper insights into applications in computer science, theoretical physics, and other fields.

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