# The Unapologetic Mathematician

## Orders

March 11, 2007 - Posted by | Fundamentals, Orders

1. […] A well-ordering on a set is a special kind of total order: one in which every non-empty subset contains a least […]

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2. […] 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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3. […] A poset which has both least upper bounds and greatest lower bounds is called a lattice. In more detail, […]

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4. […] containment from to those collections of subsets of which are actually topologies, it defines a partial order on the collection of all topologies on […]

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5. […] 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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6. […] let’s consider the collection of all subspaces of . This is a partially-ordered set, where the order is given by containment of the underlying sets. It’s sort of like the power […]

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7. […] Complements and the Lattice of Subspaces We know that the poset of subspaces of a vector space is a lattice. Now we can define complementary subspaces in a way […]

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8. […] we want to introduce a partial order on the collection of partitions called the “dominance order”. Given partitions and , […]

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9. What does the antisymmetry axiom gain / lose you?

Comment by isomorphismes | January 7, 2015 | Reply

10. Antisymmetry makes it so that if two elements satisfy $x\preceq y$ and $y\preceq x$ then we actually have $x=y$. This makes life simpler in some situations.

As a more visual example, imagine the preorder as a graph, with an arrow from $x$ to $y$ if $x\preceq y$ (pointing “up” the order). Then the graph of a preorder can have nontrivial loops, with an arrow from $x$ to $y$ 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 | Reply