On the Category of Posets

Let I1 be a relation structure. We say that I1 is discrete if and only if: (Def. 1) The internal relation of I1 = △the carrier of I1. Let us mention that there exists a poset which is strict discrete and non empty and there exists a poset which is strict discrete and empty. Let X be a set. Then △X is an order in X. Observe that 〈∅,△∅〉 is empty. Let P be an empty relation structure. One can check that the internal relation of P is empty. Let us mention that every relation structure which is empty is also discrete. Let P be a relation structure and let I1 be a subset of P . We say that I1 is disconnected if and only if the condition (Def. 2) is satisfied. (Def. 2) There exist subsets A, B of P such that (i) A 6= ∅, (ii) B 6= ∅, (iii) I1 = A ∪ B,