By Gottfried Wilhelm Leibniz

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Analogously u is called an upper bound of T if u 2 t for all t E T holds. If here 5 is replaced by < , resp. 2 by >, a is said to be a strict lower (resp. a strict upper) bound of T. Every element of P is a lower bound and at the same time an upper bound of the empty set 0. A subset T C P is said to be bounded from below (resp. from above) if there exists a lower (resp. upper) bound of T in P. If T is bounded from below and from above, it is called bounded. (P,I)is called directed from above (resp.

CUTS. THE DEDEKIND-MACNEILLE COMPLETION < To show this, let hA hB. Then every element of Cs(A) = Ls(Us (A)) is a lower bound of Us (A) and thus infK Us (A) = h~ hB = infK Us(B). And then every lower bound of Us(A) is also a lower bound of Us(B), and so it is in Ls(Us(B)) = Cs(B). Assume now that Cs(A) C Cs(B) holds. By (1) we then have A C Cs(A) = ( S 5 hA) C Cs(B) = ( S h ~ )which , means A C ( S 5 h ~ ) . Therefore hB is an upper bound of A and a lower bound of Us(B). Thus all elements of Us(B) are upper bounds of A, and then Us(B) 2 Us(A) implies hB = infK Us (B) 2 i n f ~Us (A) = hA.

So we assume a < b. Among the chains of P which have a as least and b as greatest element there must be one with a highest number of elements. Then this has a form {a = a l , . . ,an+l = b), where ai < ai+l for i = 1 , . . , n . Here we also must have a1 4 - .. a an+l. ,n), this could be inserted into the previous chain contradicting the fact that this had already a maximal number of elements. In the following we still characterize those relations which are the lower-neighbor-relation of an order relation First we define: <.