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.

On cardinalities of Green’s equivalence classes in the semigroup of difunctional binary relations

“…α) = |X \codom(α)|, and call these parameters the rank, domain, codomain, kernel, cokernel, defect and codefect of α respectively. East and Vernitski [1] characterised Green's relations on (a result which is also to be found in [3] in slightly different form). Proof.…”