Heights of posets associated with Green's relations on semigroups

Abstract

Given a semigroup S, for each Green's relation K∈{L,R,J,H} on S, the K-height of S, denoted by HK(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 HK(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 HL(S)=n then HR(S)≤2n−1 and HJ(S)≤2n−1, and we exhibit a family of examples to prove that these bounds are sharp. Furthermore, we prove that if 2≤HL(S)<∞ and 2≤HR(S)<∞, then HJ(S)≤HL(S)+HR(S)−2. We also show that for each n∈N there exists a semigroup S such that HL(S)=HR(S)=2n+n−3 and HJ(S)=2n+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.