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 isomorphic 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.
1980 Mathematics Subject Classification: Primary O6A10
Kemp, Paula. "Representation of partially ordered sets." algebra universalis 30, no. 3 (1993): 348-351.