Let S be a commutative monoid,in the category of S-acts,by the injectivity and surjectivity of the mapping of pullback diagrams,a new proof that fractional functors(the category of S-acts to the category of fractional S-acts)preserve flatness properties is given.The relationship between fractional functors and directed colimits is investigated.A sufficient condition for fractional S-acts having a cover is presented.
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