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.
Categories: math.CO math.GR