Ranks of ideals in inverse semigroups of difunctional binary relations

TL;DR

This paper determines the minimal size of generating sets for the inverse semigroup of difunctional relations on an n-element set, revealing that the rank grows rapidly with n and differs from related semigroup families.

Contribution

It solves the open problem of finding the rank of D_n and provides the rank of its arbitrary ideals, highlighting unique growth properties.

Findings

Rank of D_n is B(n)+n, where B(n) is the nth Bell number.

Provides the rank of arbitrary ideals in D_n.

Shows that rank(D_n) grows faster than in similar semigroups.

Abstract

The set D_n of all difunctional relations on an n element set is an inverse semigroup under a variation of the usual composition operation. We solve an open problem of Kudryavtseva and Maltcev (2011), which asks: What is the rank (smallest size of a generating set) of D_n? Specifically, we show that the rank of D_n is B(n)+n, where B(n) is the nth Bell number. We also give the rank of an arbitrary ideal of D_n. Although D_n bears many similarities with families such as the full transformation semigroups and symmetric inverse semigroups (all contain the symmetric group and have a chain of J-classes), we note that the fast growth of rank(D_n) as a function of n is a property not shared with these other families.