An Elegant 3-Basis for Inverse Semigroups

Semigroup Forum, 82(2) (2011), 319-323
http://dx.doi.org/10.1007/s00233-010-9280-8

It is well known that in every inverse semigroup the binary operation and the unary operation of inversion satisfy the following three identities:
x=(xx')x, (xx')(y'y)=(y'y)(xx'), (xy)z=x(yz'').
The goal of this note is to prove the converse, that is, we prove that an algebra of type ?2,1? satisfying these three identities is an inverse semigroup and the unary operation coincides with the usual inversion on such semigroups.