Is there a book, or a paper where the Lexicographic glb and lub are proven commutative, associative, idempotent and absorbing. I have already proven this, but would like to check proofs, and a short citation in a page limited paper, is better than a list of long proofs. Thank you.

2 complete lattices L1 = (A1,lub1,glb1) and L2 = (A2,lub2,glb2) lexicographically composed into (A1 x A2, lubL, glbL) where

(a,b) lubL (a',b') = (a,b) if a' < a;
                     (a'b') if a < a';
                     (a,b lub2 b') if a = a';
                     (a lub1 a',02) if a || a' ;

(a,b) glbL (a',b') = (a,b) if a < a';
                     (a'b') if a' < a;
                     (a,b glb2 b') if a = a';
                     (a glb1 a',12) if a || a';