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The set D_n of all difunctional relations on an n element set is an inversesemigroup under a variation of the usual composition operation. We solve anopen 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 rankof D_n is B(n)+n, where B(n) is the nth Bell number. We also give the rank ofan arbitrary ideal of D_n. Although D_n bears many similarities with familiessuch as the full transformation semigroups and symmetric inverse semigroups(all contain the symmetric group and have a chain of J-classes), we note thatthe fast growth of rank(D_n) as a function of n is a property not shared withthese other families.

Ranks of ideals in inverse semigroups of difunctional binary relations