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Topic: Order Embeddings
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William Elliot

Posts: 2,637
Registered: 1/8/12
Order Embeddings
Posted: Jan 19, 2014 8:13 AM
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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.

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