I'm solving this exercise that's part of my university course on models of computation and I'm trying to understand the definition of the following lattice :
Given the lattice ($2^{(a,b,c,d)}$ ,⊆). We define the function imageR(S) that computes the image of the set S w.r.t. the relation R, whereby R={(a,a),(a,b),(a,c),(b,d),(c,b),(d,a),(d,c)}.For example, imageR({d})={a,c}.
I assumed that the lattice contains a finite set of elements S = {(a);(b);(c);(d);(a,b);(a,c);(a,d);(b,c);(b,d);(c,d);(a,b,c);(a,b,d);(a,c,d);(b,c,d);(a,b,c,d)} which are all the unique combinations of a,b,c,d. Now with the function imageR I can compute the image for each of these elements, for example, imageR({b,d}) = {a,c,d}, but how can I compare/order the elements ? (like how to order {d} and {a,b,c} ?)
this is needed when checking whether imageR is monotone, or for instance function f = imageR(S) U {c} is monotone.
Please correct me if the way I defined S is flawed (I made up this assumption myself). I'm new to this topic and I'm ready to provide any additional details. Thank you.