Retracts and Inheritance 1 Grzegorz Bancerek University of Białystok

The following three propositions are true: (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 substructure 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. (3) Let X be a set, L be a non empty relational structure, S be a full non empty relational substructure 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. Note that there exists a map from S into T which is directed-sups-preserving and monotone. The following proposition is true (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.