Heights of posets associated with Green's relations on semigroups

Given a semigroup $S$, for each Green's relation $\mathcal{K}\in\{\mathcal{L},\mathcal{R},\mathcal{J},\mathcal{H}\}$ on $S,$ the $\mathcal{K}$-height of $S,$ denoted by $H_{\mathcal{K}}(S),$ is the height of the poset of $\mathcal{K}$-classes of $S.$ More precisely, if there is a finite bound on the sizes of chains of $\mathcal{K}$-classes of $S,$ then $H_{\mathcal{K}}(S)$ is defined as the maximum size of such a chain; otherwise, we say that $S$ has infinite $\mathcal{K}$-height. We discuss the relationships between these four $\mathcal{K}$-heights. The main results concern the class of stable semigroups, which includes all finite semigroups. In particular, we prove that a stable semigroup has finite $\mathcal{L}$-height if and only if it has finite $\mathcal{R}$-height if and only if it has finite $\mathcal{J}$-height. In fact, for a stable semigroup $S,$ if $H_{\mathcal{L}}(S)=n$ then $H_{\mathcal{R}}(S)\leq2^n-1$ and $H_{\mathcal{J}}(S)\leq2^n-1,$ and we exhibit a family of examples to prove that these bounds are sharp. Furthermore, we prove that if $2\leq H_{\mathcal{L}}(S)<\infty$ and $2\leq H_{\mathcal{R}}(S)<\infty,$ then $H_{\mathcal{J}}(S)\leq H_{\mathcal{L}}(S)+H_{\mathcal{R}}(S)-2.$ We also show that for each $n\in\mathbb{N}$ there exists a semigroup $S$ such that $H_{\mathcal{L}}(S)=H_{\mathcal{R}}(S)=2^n+n-3$ and $H_{\mathcal{J}}(S)=2^{n+1}-4.$ By way of contrast, we prove that for a regular semigroup the $\mathcal{L}$-, $\mathcal{R}$- and $\mathcal{H}$-heights coincide with each other, and are greater or equal to the $\mathcal{J}$-height. Moreover, in a stable, regular semigroup the $\mathcal{L}$-, $\mathcal{R}$-, $\mathcal{H}$- and $\mathcal{J}$-heights are all equal.