Need help with maximal theorem

[Cratylus]

It is from Pinter‘s text.,pg 97 Ex 12

Let <A[MATH]\leq[/MATH]>,<B[MATH]\leq[/MATH]> be posets .Prove the following:
SupposeA×B is ordered lexicographically,:if(a,b) is a maximal element of A×B
then a is the maximal element of A

18 Definition An element m ∈ A is called a maximal element of A if none of the elements of A are strictly greater than m; in symbols, this can be expressed as follows:
∀∀x ∈????≥?∈Aifx≥m then x=m

attempted proof
if (a,b) is a maximal element of A X B and lexicographically ordered then
∀?1∈?∀a1∈A?1≥?a1≥a then a1=a
and
∀?1∈?∀b1∈B?1≥?b1≥b then b1=b
and
∀?2∈?∀a2∈A?2≥?a2≥a then a2=a
and
∀?2∈?∀b2∈B?2≥?b2≥b then b2=b
Then ∀?1∈?,?1≥????∀?2 ???;?2≥?∀a1∈A,a1≥aand∀a2 inA;a2≥a
then a1=A2=a ?
Help

 

[pka]

Let <A[MATH]\leq[/MATH]>,<B[MATH]\leq[/MATH]> be posets .Prove the following:
SupposeA×B is ordered lexicographically,:if(a,b) is a maximal element of A×B
then a is the maximal element of A Help

Suppose that \(a\) is not maximal in \(A\) then \(\exists c\in A\) such that \(a \prec c\).
We are given that \((a,b)\) is maximal in \(A\times B\).
Thus \(a\in A~\&~b\in B\). That means \((c,b)\in A\times B\), moreover the lexicographic ordering means that \((a,b)\prec(c,b)\).
Do you see a contradiction?

 

[Cratylus]

Yah a<c and b <b b<b can’t be less than itself.
How can I improve my proof?
I never thought of using the dual.

 

[Cratylus]

Yah the contradiction is c>a and the fact (a,b) €AX B is maximal
How can I improve my proof?
I never thought of using the dual.

Pinter’s version
4.1 Definition Let A and B be partially ordered classes; by the lexicographic ordering of A × B we mean the following order relation in A × B: If (a1,b1) ∈ A × B and (a2,b2) ∈ A × B, then we let (a1,b1)<=(a2,b2) iffa1<a2

So by this and by your proof we do it for 2 pairs

 

[pka]

Yah a<c and b <b b<b can’t be less than itself.
How can I improve my proof?
I never thought of using the dual.

The contradiction is to the maximality of \((a,b)\).
I simply have no idea what your proof contains. I cannot read it. It appears as gibberish.

 

[Cratylus]

I was trying to do a direct proof starting from the definitions I gave

I copied it from an another site where I posted it.
It double copied ,yikes ?
Since c€A we get (c,b) €. A X B correct?
Here is what it should have said

Suppose (a,b) is a maximal element of AXB
then. a [MATH]\in A [/MATH] and b [MATH]\in B [/MATH] ( from your proof)
But it is lexicographically ordered so
by def (a1,b1) ,(a2,b2) [MATH]\in [/MATH] A X B Let (a1,b1)<=(a2,b2) iff a1<a2
[MATH]\forall[/MATH] a1[MATH]\in A [/MATH]; a1[MATH]\succeq[/MATH]a
[MATH]\forall[/MATH]b1[MATH]\in B [/MATH]; b1. [MATH]\succeq[/MATH] b
[MATH]\forall[/MATH]a2 [MATH]\in A [/MATH]; a2 [MATH]\succeq[/MATH]a
[MATH]\forall[/MATH]b2[MATH]\in B [/MATH]; b2 . [MATH]\succeq[/MATH] b

I hope I was trying to saying for each coordinate in the pair , it has a maximal element?
If not what do I have to do?