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Topic: Homomorphism of posets and lattices
Replies: 16   Last Post: Sep 20, 2013 6:22 AM

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quasi

Posts: 9,906
Registered: 7/15/05
Re: Homomorphism of posets and lattices
Posted: Sep 19, 2013 12:55 PM
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William Elliot wrote:
>William Elliot wrote:
>> quasi wrote:
>> >
>> > Questions:
>> >
>> > Let X,Y be posets and suppose f:X -> Y and g:Y -> X are
>> > order homomorphisms [order preserving maps].
>> >
>> > (1) If f,g are both injective, must X,Y be order isomorphic?

>>
>> No.
>>

>> > (2) If f,g are both surjective, must X,Y be order isomorphic?
>>
>> No.
>>
>> What happens if both are bijective?

>
>Clearly, for all x,y, (x <= y --> f(x) <= f(y)).
>If f(x) <= f(y): fgg^-1(x) <= fgg^-1(y)
>
>gf(x) <= gf(y)


Presumably you are trying to answer the question:

If X,Y are posets for which there are bijective order
preserving maps f:X -> Y and g:Y -> X, must X,Y be order
isomorphic?

As far as I can see, what you posted above has no impact on
the question.

quasi



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