Heights of posets associated with Green's relations on semigroups.

Given a semigroup S , for each Green's relation K ∈ { L , R , J , H } on S , the K -height of S , denoted by H K (S) , is the height of the poset of K -classes of S. More precisely, if there is a finite bound on the sizes of chains of K -classes of S , then H K (S) is defined as the maximum size of such a chain; otherwise, we say that S has infinite K -height. We discuss the relationships between these four 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 L -height if and only if it has finite R -height if and only if it has finite J -height. In fact, for a stable semigroup S , if H L (S) = n then H R (S) ≤ 2 n − 1 and H J (S) ≤ 2 n − 1 , and we exhibit a family of examples to prove that these bounds are sharp. Furthermore, we prove that if 2 ≤ H L (S) < ∞ and 2 ≤ H R (S) < ∞ , then H J (S) ≤ H L (S) + H R (S) − 2. We also show that for each n ∈ N there exists a semigroup S such that H L (S) = H R (S) = 2 n + n − 3 and H J (S) = 2 n + 1 − 4. By way of contrast, we prove that for a regular semigroup the L -, R - and H -heights coincide with each other, and are greater or equal to the J -height. Moreover, in a stable, regular semigroup the L -, R -, H - and J -heights are all equal. [ABSTRACT FROM AUTHOR]