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.

Given 2 complete lattices $L_1 = (A_1,\vee_1,\wedge_1)$ and $L_2 = (A_2,\vee_2,\wedge_2)$, we form the lexicographic product $(A_1\times A_2, \vee, \wedge)$ where

$$ (a,b) \vee (a',b') = \left\{ \begin{aligned} (a,b) & \hbox{ if $a' < a$} \\ (a'b') & \hbox{ if $a < a'$} \\ (a,b \vee_2 b') & \hbox{ if $a = a'$} \\ (a \vee_1 a',0_2) & \hbox{ if $a || a'$} \end{aligned}\right. $$

$$ (a,b) \wedge (a',b') = \left\{ \begin{aligned} (a,b) & \hbox{ if $a < a'$} \\ (a',b') & \hbox{ if $a' < a$} \\ (a,b \wedge_2 b') & \hbox{ if $a = a'$} \\ (a \wedge_1 a',1_2) & \hbox{ if $a || a'$} \end{aligned}\right. $$