Semigroups with operation-compatible Green’s quasiorders

We call a semigroup on which the Green’s quasiorder ≤ <inf>J</inf> (≤ <inf>L</inf>, ≤ <inf>R</inf>) is operation-compatible, a ≤ <inf>J</inf>-compatible (≤ <inf>L</inf>-compatible, ≤ <inf>R</inf>-compatible) semigroup. We study the classes of ≤ <inf>J</inf>-compatible, ≤ <inf>L</inf>-compatible and ≤ <inf>R</inf>-compatible semigroups, using the smallest operation-compatible quasiorders containing Green’s quasiorders as a tool. We prove a number of results, including the following. The class of ≤ <inf>L</inf>-compatible (≤ <inf>R</inf>-compatible) semigroups is closed under taking homomorphic images. A regular periodic semigroup is ≤ <inf>J</inf>-compatible if and only if it is a semilattice of simple semigroups. Every negatively orderable semigroup can be embedded into a negatively orderable ≤ <inf>J</inf>-compatible semigroup.