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We show that all subvarieties of the variety of rectangular bands are closed in the variety of all bands. Further, we show that the variety of regular bands is closed. Next, we introduce the varieties of generalized left [right] regular semigroups and show that all such varieties are saturated and, subsequently, epis from generalized left [right] regular semigroups are surjective. [ABSTRACT FROM AUTHOR]

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We show that the posemigroup dominion of a commutative posemigroup is again a commutative posemigroup and all varieties of commutative posemigroups are epimorphically closed. Next, we show that certain subvarieties of the variety of all commutative posemigroups are saturated. Finally, we show that a commutative posemigroup S is saturated if S k is saturated for some positive integer k. [ABSTRACT FROM AUTHOR]