A regular semigroup S with a
commutative subsemigroup of idempotents E is called weakly inverse if for any a ∈ S
the set Ea of inverses a′ of a for which a′a ∈ E is nonempty and for all, a,b ∈ S,
Eab⊆ EbEa and Ea= Eb⇒ a = b. In this paper we show that in a weakly
inverse semigroup S with partial identities the ℛ-class R which contains the
partial identities is a right skew semigroup and conversely, every right skew
semigroup R may be so represented. If R satisfies the condition that for every
a,b ∈ R there exists a c ∈ R such that Ra ∩ Rb = Rc, then our considerations
lead to a construction of bisimple weakly inverse semigroup with partial
identities.
