Orders
March 11, 2007  Posted by John Armstrong  Fundamentals, Orders
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[…] A wellordering on a set is a special kind of total order: one in which every nonempty subset contains a least […]
Pingback by WellOrdering « The Unapologetic Mathematician  April 2, 2007 
[…] Lower Bounds and Euclid’s Algorithm One interesting question for any partial order is that of lower or upper bounds. Given a partial order and a subset we say that is a lower […]
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[…] A poset which has both least upper bounds and greatest lower bounds is called a lattice. In more detail, […]
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[…] numbers for sequences is that they’re “directed”. That is, there’s an order on them. It’s a particularly simple order since it’s total — any two elements are […]
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[…] let’s consider the collection of all subspaces of . This is a partiallyordered set, where the order is given by containment of the underlying sets. It’s sort of like the power […]
Pingback by The Sum of Subspaces « The Unapologetic Mathematician  July 21, 2008 
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[…] we want to introduce a partial order on the collection of partitions called the “dominance order”. Given partitions and , […]
Pingback by The Dominance Order on Partitions « The Unapologetic Mathematician  December 17, 2010 
What does the antisymmetry axiom gain / lose you?
Comment by isomorphismes  January 7, 2015 
Antisymmetry makes it so that if two elements satisfy and then we actually have . This makes life simpler in some situations.
As a more visual example, imagine the preorder as a graph, with an arrow from to if (pointing “up” the order). Then the graph of a preorder can have nontrivial loops, with an arrow from to and another one back. The graph of a partial order will be acyclic; partial orders are “simpler” than preorders in the same way acyclic graphs are simpler than general graphs.
Comment by John Armstrong  January 7, 2015 
Thanks!
Comment by isomorphismes  February 19, 2015 