In this paper a study of properties of the Mitsch order relation ’µ’ on a regular semigroup and Nambooripads order ’ν’ on any arbitrary regular semigroup is made. Mainly a characterization of Lallement order on a regular semigroup is obtained. The necessary and sufficient condition for the restriction of Lallement order ’λ’ to B(S) to be usual order on an orthodox semigroup is also obtained.