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BASIC DEFINITIONS A Non-empty subset ''I'' of a partially ordered set (''P'',≤) is an ideal, if the following conditions hold: # For every ''x'' in ''I'', ''y'' ≤ ''x'' implies that ''y'' is in ''I''. (''I is a Lower Set '') # For every ''x'', ''y'' in ''I'', there is some element ''z'' in ''I'', such that ''x'' ≤ ''z'' and ''y'' ≤ ''z''. (''I'' is a Directed Set ) While this is the most general way to define an ideal for arbitrary posets, it was originally defined for Lattice s only. In this case, the following equivalent definition can be given: A non-empty subset ''I'' of a lattice (''P'',≤) is an ideal, Iff it is a lower set that is closed under finite joins (suprema), i.e., for all ''x'', ''y'' in ''I'', we find that ''x''''y'' is also in ''I''. The Dual notion of an ideal, i.e. the concept obtained by reversing all ≤ and exchanging with , is a Filter . The terms '''order ideal''' and '''order filter''' are sometimes used for arbitrary lower or upper sets. Wikipedia uses only "ideal/filter (of order theory)" and "lower/upper set" in order to avoid confusion. An ideal or filter is said to be proper if it is not equal to the whole set ''P''. | ||
| ::'''Proof''' Assume The Ideal ''M'' Is Maximal With Respect To Disjointness From The Filter ''F'' Suppose For A Contradiction That ''M'' Is Not Prime, Ie There Exists A Pair Of Elements ''a'' And ''b'' Such That ''a''<math>\wedge</math>''b'' In ''M'' But Neither ''a'' Nor ''b'' Are In ''M'' Consider The Case That For All ''m'' In ''M'', ''m''<math> Ee</math>''a'' Is Not In ''F'' One Can Construct An Ideal ''N'' By Taking The Downward Closure Of The Set Of All Binary Joins Of This Form, Ie ''N'' | { ''x'' ''x''&le ''m''<math> ee</math>''a'' for some ''m'' in ''M''} It is readily checked that ''N'' is indeed an ideal disjoint from ''F'' which is strictly greater than ''M'' But this contradicts the maximality of ''M'' and thus the assumption that ''M'' is not prime |
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