a finite union of elements on the form for some , the equality in the universal property says , and finally the homomorphism status of implies for all . Any extension of to infinite subsets of (if there even is one) need however not be uniquely determined by these conditions, so there cannot in be any elements corresponding to infinite subsets of .

Lower semilattices

It is similarly possible to define a free functor for lower semilattices, but the combination fails to produce the free

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