Duality in Relation Structures

The articles [15], [18], [19], [21], [20], [7], [8], [10], [1], [2], [6], [14], [11], [16], [12], [17], [3], [4], [23], [9], [5], [22], and [13] provide the terminology and notation for this paper. Let L be a relational structure. We introduce L as a synonym of L`. We now state several propositions: (1) For every relational structure L and for all elements x, y of L 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 L. 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 L is empty. (4) For every relational structure L holds L is reflexive iff L is reflexive. (5) For every relational structure L holds L is antisymmetric iff L is antisymmetric. (6) For every relational structure L holds L is transitive iff L is transitive. (7) For every non empty relational structure L holds L is connected iff L is connected. Let L be a reflexive relational structure. One can check that L is reflexive. Let L be a transitive relational structure. One can check that L is transitive. Let L be an antisymmetric relational structure. Note that L is antisymmetric.