Bifree locally inverse semigroups as expansions

In [Studia Sci. Math. Hungar. 41 (2004) 39–58] we constructed for a completely simple semigroup C an expansion S(C), which is isomorphic to the Birget–Rhodes expansion CPr [J. Algebra 120 (1989) 284–300], if C is a group. Analogous to the fact, proven in [J. Algebra 120 (1989) 284–300], that CPr contains a copy of the free inverse semigroup in case C is the free group on X, we show that S(C) contains a copy of the bifree locally inverse semigroup, if C is the bifree completely simple semigroup on X. As a consequence, among other things, we obtain a new proof of a result due to F. Pastijn [Trans. Amer. Math. Soc. 273 (1982) 631–655] which says that each locally inverse semigroup divides a perfect rectangular band of E-unitary inverse monoids.