The Polish Academy of Sciences

In this paper D is a non empty set. We now state several propositions: (1) Let D be a non empty set, and let S be a non empty subset of D, and let f be a binary operation on D, and let g be a binary operation on S. Suppose g = f [: S, S :]. Then (i) if f is commutative, then g is commutative, (ii) if f is idempotent, then g is idempotent, and (iii) if f is associative, then g is associative. (2) Let D be a non empty set, and let S be a non empty subset of D, and let f be a binary operation on D, and let g be a binary operation on S, and let d be an element of D, and let d be an element of S. Suppose g = f [:S, S :] and d = d. Then (i) if d is a left unity w.r.t. f , then d is a left unity w.r.t. g, (ii) if d is a right unity w.r.t. f , then d is a right unity w.r.t. g, and (iii) if d is a unity w.r.t. f , then d is a unity w.r.t. g. (3) Let D be a non empty set, and let S be a non empty subset of D, and let f1, f2 be binary operations on D, and let g1, g2 be binary operations on S. Suppose g1 = f1 [: S, S :] and g2 = f2 [:S, S :]. Then