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Order Embeddings
Posted:
Jan 19, 2014 8:13 AM


Let (S,<=) be an (partially) ordered set and A a subset of S with the inherited order, namely, <= /\ AxA
Thus, A is order embedded in S.
When S isn't a lattice and A is a lattice would would you call these cases?
A has the inherited order.
For all a,b in A, a inf_S b and a sup_S b exist and additionally a inf_S b = a inf_A b and a sup_S b = a sup_A b. Note that because of those requirements A has the inherited order.
For all a,b in A, if a inf_S b exists, then a inf_S b = a inf_A b and if a sup_S b exist, a sup_S b = a sup_A b.
I'd call them respectively: an order embedding of an ordered subset that happens to be a lattice; an embedding of lattice; a pseudo lattice embedding.
How would you describe these three distinctions? Do some already have terms that describe them? What use are they? Have you an example or two?
I'd think the 2nd would be the useful one, the others of little or no significance.



