The Unapologetic Mathematician

Orders

As a bonus today, I want to define a few more kinds of relations.

A preorder is a relation on a set which is reflexive and transitive. We often write a general preorder as $x\preceq y$ and say that $x$ precedes $y$ or that $y$ succeeds $x$. A set equipped with a preorder is called a preordered set. If we also have that for any two elements $x$ and $y$ there is some element $z$ (possibly the same as $x$ or $y$) that succeeds both of them we call the structure a directed set.

A partial order is a preorder which is also antisymmetric: the only way to have both $x\preceq y$ and $y\preceq x$ is for $x$ and $y$ to be the same element. We call a set with a partial order a partially-ordered set or a “poset”.

Any set gives a partial order on its set of subsets, given by inclusion: if $A$ and $B$ are subsets of a set $X$, then $A$ precedes $B$ if $A$ is contained in $B$. This has the further nice property that it has a top element, $X$ itself, that succeeds every element. It also has a bottom element, the empty subset, that precedes everything. The same sort of construction applies to give the poset of subgroups of any given group. These kinds of partially-ordered sets are very important in logic and set theory, and they’ll come up in more detail later.

Finally, a partial order where for any two elements $x$ and $y$ we either have $x\preceq y$ or $y\preceq x$ is called a total order. Total orders show up over and over, and they’re nice things to have around. I must admit, though, that as far as I’m concerned they’re pretty boring in and of themselves.

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 […]

Pingback by Greatest Lower Bounds and Euclid's Algorithm « The Unapologetic Mathematician | May 4, 2007 | Reply

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

Pingback by Lattices « The Unapologetic Mathematician | May 14, 2007 | Reply

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 […]

Pingback by Topology « The Unapologetic Mathematician | November 5, 2007 | Reply

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 […]

Pingback by Orthogonal Complements and the Lattice of Subspaces « The Unapologetic Mathematician | May 7, 2009 | Reply

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