Relational depth of transformation semigroups and their ideals

We introduce the concept of relational depth of a finite semigroup $S$ whose $J$-classes form a chain. It captures how far down in the ideal structure one is obliged to go in order to define the semigroup by generators and defining relations. We determine the exact value for the relational depth of an arbitrary ideal in the full transformation monoid, symmetric inverse monoid and in the partial transformation monoid.