On the representation of partially ordered sets
It is well known that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, ⊆) such that the supremum and the infimum of any finite set F of P correspond, respectively to the union and intersection of the images of the elements of F. Here necessary and sufficient conditions are given for similar isomophic representation of a poset where however the supremum and infimum of also infinite subsets I correspond to the union and intersection of images of elements of I.
Kemp, Paula A. "On the representation of partially ordered sets." Rendiconti del Circolo Matematico di Palermo 46, no. 1 (1997): 119-122.
Rendiconti del Circolo Matematico di Palermo