# Partial Orders 2021.04.06

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Hello,

I think there are errors in the solution of task 1 of exam 2021.04.06.

My guesses are:

(D1,R1') is not a CPO.

(D6,R6') is not a complete lattice but a lattice.

Am I right?

edited

## 1 Answer

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Yes, you are right. A CPO requires that all directed subsets M have a supremum and that there is a minimal element. In a complete lattice, we must have sup(M) and inf(M) for every subset M. Hence, every complete lattice is also a CPO.

Considering (D1,R1'), it is not a CPO since the entire set does not have a supremum, and the same holds for every infinite subset (since it is a total order, every subset is a directed one). It is a lattice since it is a total order. It is not a complete lattice for the same reasons why it is not a CPO.

(D6,R6') is not a CPO since it does not have a minimal element. However, all directed subsets have a supremum, so if we would add a minimal element, we would obtain a CPO. It is a lattice since it is a total order. It is not a complete lattice for the same reasons why it is not a CPO.

The solutions have been fixed meanwhile.
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