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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. 
 
 
 
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enter code hereIs 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',0_2a',1_2) & \hbox{if a || a'} \end{aligned}\right.

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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 L1 $L_1 = (A1,lub1,glb1) A_1,\vee_1,\wedge_1)$ and L2 $L_2 = (A2,lub2,glb2) lexicographically composed into (A1 x A2, lubLA_2,\vee_2,\wedge_2)$, we form the lexicographic product $(A_1\times A_2, glbL) \vee, \wedge)$ where

(a,b) lubL \vee (a',b') = \left\{ \begin{aligned} (a,b) if a' & \hbox{if a' < a; a} \\ (a'b') if & \hbox{if a < a'; a'} \\ (a,b lub2 \vee_2 b') if & \hbox{if a = a'; a'} \\ (a lub1 a',02\vee_1 a',0_2) if & \hbox{if a || a' ; a'} \end{aligned}\right.

(a,b) glbL \wedge (a',b') = \left\{ \begin{aligned} (a,b) if & \hbox{if a < a'; a'} \\ (a'b') if a' a',b') & \hbox{if a' < a; a} \\ (a,b glb2 \wedge_2 b') if & \hbox{if a = a'; a'} \\ (a glb1 a',12\wedge_1 a',0_2) if & \hbox{if a || a'; a'} \end{aligned}\right.

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