Retracts and Inheritance1

provide the notation and terminology for this paper. 1. POSET RETRACTS One can prove the following propositions: (1) For all binary relations a, b holds a · b = a b. (2) Let X be a set, L be a non empty relational structure, S be a non empty relational substruc-ture of L, f , g be functions from X into the carrier of S, and f , g be functions from X into the carrier of L. If f = f and g = g and f ≤ g, then f ≤ g. Let S be a non empty relational structure and let T be a non empty reflexive antisymmetric relational structure. One can verify that there exists a map from S into T which is directed-sups-preserving and monotone. Next we state the proposition (4) For all functions f , g such that f is idempotent and rng g ⊆ rng f and rng g ⊆ dom f holds f · g = g. Let S be a 1-sorted structure. One can check that there exists a map from S into S which is idempotent. The following four propositions are true: (5) For every up-complete non empty poset L holds every directed-sups-inheriting full non empty relational substructure of L is up-complete. (6) Let L be an up-complete non empty poset and f be a map from L into L. Suppose f is idempotent and directed-sups-preserving. Then Im f is directed-sups-inheriting. (8) 1 Let S, T be non empty relational structures, f be a map from T into S, and g be a map from S into T. If f · g = id S , then rng f = the carrier of S. 1 The proposition (7) has been removed.