By G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott
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Extra resources for Continuous Lattices and Domains
Sample text
The following also deals with examples, but they play such a very prominent role in what follows that we separate them out. 8. Let L be a poset. (1) The family of all lower sets of L and the family of all upper sets are both complete lattices under ⊆; indeed, both of these families are closed under arbitrary intersections and unions in 2 L . (2) In any poset L, Filt0 L and Filt L are closed under directed unions and hence dcpos. If L is a semilattice, then Filt0 L is a complete lattice; if L is also unital, then Filt L is complete.
In the exercises for this section we comment further on the relation of the concepts we have just introduced. 2. Let L be a poset. (i) For L to be a complete lattice it is sufficient to assume the existence of arbitrary sups (or the existence of arbitrary infs). (ii) For L to be a complete lattice it is sufficient to assume the existence of sups of finite sets and of directed sets (or the existence of finite infs and filtered infs). (iii) If L is a unital semilattice, then for completeness it is sufficient to assume the existence of filtered infs.
Conversely, if Id L is a sup semilattice, then we claim there is a unique element c ∈ ↓a ∨ ↓b with a, b ≤ c. Indeed, there is at least one since ↓a ∨ ↓b is directed; moreover, if c and c1 were two such elements, then ↓c and ↓c1 would be two ideals of L both containing a and b and both contained in ↓a ∨ ↓b. Hence ↓c = ↓c1 = ↓a ∨ ↓b. (vi) Dual statements hold for Filt L, where one assumes L is a semilattice in part (v). 16. Let L be a preordered set, and let L denote the family of all nonempty lower sets of L.
