On Nov 20, 12:02 pm, Victor Porton <por...@narod.ru> wrote:
> Hi! It's seems a simple problem and I could solve it myself, but I
> want to share it with sci.math.
> Let A is a poset and let Z is its subset (Z is also a poset with
> induced order). Let S is a subset of Z, let t in A.
> Let "inf^Z S" is defined (narrowing our problem we could assume that Z
> is a complete lattice, but I want to consider the more general case).
> Conjecture. (forall X in S: X>=t) => inf^Z S>=t.
> For the case when "inf^A S" is defined (particularly if A is a
> complete lattice) the conjecture is almost obviously true. But what's
> about the general case?
Consider the case where A = {t,u}\cup {0, -1, -2, -3, ...}; give the
negative integers their usual ordering, and define t and u to be
strictly less than any integer, but incomparable between them. Let Z =
A-{t}, and let S = Z-{u}.
Then u = inf^Z(S), and for all X in S={0, -1, -2, -3, ...}, we have
x>=t, but inf^Z(S) = u is not greater than or equal to t.
--
Arturo Magidin
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