Heights of posets associated with Green's relations on semigroups

Given a semigroup S , for each Green's relation K is an element of {G , 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 G- 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) < infinity and 2 <= H-R(S) < infinity, then H-J(S) < H-L(S) + H-R(S) -2. We also show that for each n is an element of 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 G-, R- and H- heights coincide with each other, and are greater or equal to the J- height. Moreover, in a stable, regular semigroup the G-, R-, H- and J- heights are all equal. (c) 2024 The Authors. Published by Elsevier Inc.