Ranks of ideals in inverse semigroups of difunctional binary relations

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 (Publ Math Debrecen 78(2):253–282, 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.