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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: 10,390
Registered: 7/15/05
Re: Homomorphism of posets and lattices
Posted: Sep 18, 2013 5:52 AM
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If X,Y are posets, a function f:X -> Y is called an order
homomorphism if x <= y implies f(x) <= f(y).

If X,Y are posets, a bijective function f:X -> Y is called an
order isomorphism if both f and f^(-1) are order homomorphisms.

Posets X,Y are said to be order isomorphic if there exists an
order isomorphism f:X -> Y.

Questions:

Let X,Y be posets and suppose f:X -> Y and g:Y -> X are
order homomorphisms.

(1) If f,g are both injective, must X,Y be order isomorphic?

(2) If f,g are both surjective, must X,Y be order isomorphic?

quasi



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