general statement, see [4, Corollary to Lemma 2.5 3 .) (c) One may easily characterize those posets P with DCC for which the lattice of M-closed subsets also has DCC : this is the case if and only if every incomparable subset of P is finite»

The attempt to proceed any farther with an analogy to section 2 fails for the following reasons: (d) The pal ideals need not be coupletely Join irreducible elements in the lattice of M-closed subsets, (e) Even in a complete distributive lattice with DCC, there need not be a unique irredunâant representation of an element as a Join of letely Join irreducibles. (f) Non-isomorphic complete tributive lattices with DCC may have the same poset of pletely Join irreducibles.

All of these difficulties are illustrated by the wing 81сф1е example« Let N denote the chain of non- ve integere and S » R и ioo} its completion by adjoining a unit« Let L denote the coaplete distributive lattice with DCC obtained by adjoining a unit to N>^N. L is the lattice of M-closed subsets of the poset P » (N x-ÎO})u

и (lOÎ X N ) of its Join irreducibles. But the element (oo ,0) is not completely join irreducible, and has tely many representations as a Join of completely join ducibles, none of which is irredundant. Furthermore, the completion l' of NX N is a complete distributive ce with DCC having the same set of completely join irredu- eibles as L , but not isomorphic to L •

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