In this paper a study of properties of the Mitsch order relation µ on a regular semigroup and Nambooripads order <img border=0 width=32 height=32 src="http:/fbpe/img/proy/v27n2/img01.JPG" alt="http:/fbpe/img/proy/v27n2/img01.JPG">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 &#955; to B(S) to be usual order on an orthodox semigroup is also obtained.