46 - 5.3 From Partial to Total Orders Given a finite...

Unformatted text preview: 5.3 From Partial to Total Orders Given a finite partial order, we can extend it to a total order. For example, suppose we have a set of tasks T to perform. We wish to decide in what order to perform them. We are not totally free to choose, because some tasks have to be finished before others can be started. We can express this pre-requisite structure by a partial order < on T . We want to find a total order < on T which respects < in the sense that if t < u then t < u . As a more concrete example, consider the partial order ⊆ on P ( { 1 , 2 , 3 } ) given in example 5.5. It is partial because, for example, the sets { 1 } and { 3 } are not contained in each other. A total order ⊆ T which extends this partial order is given by the sequence ∅ , { 3 } , { 2 } , { 1 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 } We are forced to have { 1 } ⊆ T { 1 , 2 } since we wish to respect the partial order, but we have chosen to have { 3 } ⊆ T { 1 } . This process of going from a partial to a total order is called...
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