Minimal generating set of the semigroup of partitioned binary relations

The problem of determining the size of a minimal generating set (the rank) of a semigroup has both a combinatorial and algebraic nature. For example, the rank of full transformation semigroup and the partition monoid of degree n eventually does not depend on n, however, the rank of the semigroup of binary relations \({\mathscr {B}}_n\) ${\mathcal{B}}_n$ increases exponentially with n. In this paper, we show that the rank of the semigroup of partitioned binary relations \(\mathscr{P}\mathscr{B}_n\) $\mathcal{P}{\mathcal{B}}_n$ also increases exponentially with n.