Ranks of ideals in inverse semigroups of difunctional binary relations

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.