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ABSTRACT
Let S be a regular semigroup with set of idempotents E(S). We say that S is pure if, for all E(S) and ,
in the natural order implies that
. We study the influence of purity on various constructions which produce semigroups belonging to some familiar classes of regular semigroups as well as examine the relationship of purity and
-unitariness. This condition appears most natural on locally inverse semigroups where it is equivalent to the greatest idempotent pure congruence being a completely simple congruence. We also construct a relatively free completely regular semigroup generated by
where
and apply this result to obtain a criterion for purity for completely regular semigroups.
ACKNOWLEDGMENT
The author is indebted to the referee and David Easdown for their careful reading of the paper.