In general Nambooripad's order in a regular semigroup is not compatible with multiplication.
The partial order on the set E of idempotents in a semigroup S is defined as follows: For any e and f in E, e ≤ f if and only if e = ef = fe.
This partial order is compatible with multiplication on both sides, that is, if a ≤ b then ac ≤ bc and ca ≤ cb for all c in S. Nambooripad extended these definitions to regular semigroups.
The partial order in a regular semigroup discovered by Nambooripad can be defined in several equivalent ways.
Let a and b be two elements of an arbitrary semigroup S. Then a ≤M b iff there exist t and s in S1 such that tb = ta = a = as = bs.