Duality in Relation Structures1

(1) For every relational structure L and for all elements x, y of Lop holds x ≤ y iff xx ≥xy. (2) Let L be a relational structure, x be an element of L, and y be an element of Lop. Then (i) x ≤xy iff x` ≥ y, and (ii) x ≥xy iff x` ≤ y. (3) For every relational structure L holds L is empty iff Lop is empty. (4) For every relational structure L holds L is reflexive iff Lop is reflexive. (5) For every relational structure L holds L is antisymmetric iff Lop is antisymmetric. (6) For every relational structure L holds L is transitive iff Lop is transitive. (7) For every non empty relational structure L holds L is connected iff Lop is connected. Let L be a reflexive relational structure. Observe that Lop is reflexive. Let L be a transitive relational structure. One can verify that Lop is transitive. Let L be an antisymmetric relational structure. One can check that Lop is antisymmetric. Let L be a connected non empty relational structure. One can check that Lop is connected. One can prove the following propositions: