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 R -class and each L -class. Here we show that in a certain sense, the converse is true: every finite block-group divides a semigroup of Hall relations on a finite set. © 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.