The $\mathcal{R}$-height of semigroups and their bi-ideals

The $\mathcal{R}$-height of a semigroup $S$ is the height of the poset of $\mathcal{R}$-classes of $S,$ i.e. the supremum of the lengths of chains of $\mathcal{R}$-classes. Given a semigroup $S$ with finite $\mathcal{R}$-height, we establish bounds on the $\mathcal{R}$-height of bi-ideals, one-sided ideals and two-sided ideals; in particular, these substructures inherit the property of having finite $\mathcal{R}$-height. We then investigate whether these bounds can be attained.