Algebraic and Arithmetic Lattices . Part II 1

The following propositions are true: (1) Let R be a relational structure and S be a full relational substructure of R. Then every full relational substructure of S is a full relational substructure of R. (2) Let X, Y , Z be non empty 1-sorted structures, f be a map from X into Y , and g be a map from Y into Z. If f is onto and g is onto, then g · f is onto. (3) For every non empty 1-sorted structure X and for every subset Y of the carrier of X holds (idX) Y = Y. (4) For every set X and for every element a of 2X⊆ holds ↑a = {Y ;Y ranges over subsets of X: a ⊆ Y }. (5) Let L be an upper-bounded non empty antisymmetric relational structure and a be an element of L. If ⊤L ¬ a, then a = ⊤L. This work has been supported by KBN Grant 8 T11C 018 12.