Block-groups and Hall relations

TL;DR

This paper explores the structure of Hall relations on finite sets, showing that block-groups are closely related to semigroups of Hall relations, with a focus on their algebraic properties.

Contribution

It demonstrates that every block-group can be embedded into a semigroup of Hall relations, revealing a deep connection between these algebraic structures.

Findings

Hall relations form a semigroup that is a block-group

Every block-group divides a semigroup of Hall relations

The paper characterizes the relationship between block-groups and Hall relations

Abstract

A binary relation on a finite set is called a Hall relation if it contains a permutation of the set. Under the usual relational product, Hall relations form a semigroup which is known to be a block-group, that is, a semigroup with at most one idempotent in each $\mathrsfs{R}$-class and each $\mathrsfs{L}$-class. Here we show that in a certain sense, the converse is true: every block-group divides a semigroup of Hall relations on a finite set.