Linear orders

Linear orders

Idea

A linear order (also called strict total order or pseudo-order) is the irreflexive version of a total order. A linearly ordered set, or loset, is a set equipped with a linear order.

In classical mathematics, the distinction between linear orders and total orders is merely a terminological technicality, which is not always observed; more precisely, there is a natural bijection between the set of total orders on a given set $S$ and the set of linear orders on $S$, and one distinguishes them by the notation $\lt$ (for the linear order) and $\leq$ (for the total order). In constructive mathematics, however, they are irreducibly different.

Definitions

A linear order on a set $S$ is a (binary) relation $\lt$ with the following properties:

• irreflexivity: $x \nless x$;
• asymmetry: if $x \lt y$, then $y \nless x$;
• transitivity: if $x \lt y \lt z$, then $x \lt z$;
• comparison: if $x \lt z$, then $x \lt y$ or $y \lt z$;
• connectedness: if $x \nless y$ and $y \nless x$, then $x = y$.

In classical mathematics, one may see these versions of asymmetry and connectedness:

• $x \nless y$ or $y \nless x$;
• $x \lt y$ or $y \lt x$ or $x = y$.

Using excluded middle, these are equivalent to asymmetry and connectedness as given above, but they need not hold for all linear orders in constructive mathematics.

Actually, these axioms are overkill; to begin with, irreflexivity is simply a special case of asymmetry and so can be dropped. Additionally, one can either drop transitivity or drop asymmetry (which then requires keeping irreflexivity); they will still follow from the other axioms. Dropping transitivity shows manifestly the duality (see below) between linear orders and total orders (even in constructive mathematics), and dropping asymmetry shows that a linear order is a special kind of connected irreflexive comparison. Keeping transitivity and irreflexivity (thus allowing one to drop asymmetry) shows manifestly that a linear order is a special kind of quasiorder.

In classical mathematics, there are even more options. Now comparison can be dropped, as it follows from transitivity and connectedness. Also, one often combines irreflexivity, asymmetry, and connectedness into a single axiom using exclusive disjunction:

• trichotomy: $x \lt y$ xor $y \lt x$ xor $x = y$.

Thus the most common definition uses only trichotomy and transitivity.

One can also interpret connectedness not as an axiom but as a definition of equality, getting a linear order on a quotient set of $S$. In other words, if $\lt$ is an asymmetric comparison relation on $S$, and we define $x \equiv y$ to mean that neither $x \lt y$ nor $y \lt x$, then $\equiv$ is an equivalence relation and $\lt$ induces a linear order on $S/{\equiv}$.

Examples

Classically, any total order defines an example of a linear order (as explained below), and this also holds constructively in discrete mathematics. Since most linear orders in these cases are usually described in terms of their total orders, the focus here is on constructive analysis. (The first item, however, is an exception.)

• A lexicographic order, even in a classical or discrete context, is more easily described as a linear order than as a total order.
• The big example in analysis is the field of real numbers. Both the Dedekind reals and the Cauchy reals (even if weak countable choice fails so these are not equivalent) have a linear order $\lt$ that extends the (decidable) linear order on the rational numbers. Since the corresponding partial order $\leq$ cannot be proved total (and in some classically invalid versions of constructive mathematics can even be proved not total), $\lt$ is more directly useful than $\leq$ in constructive analysis. In any case, $\lt$ is more fundamental, since $\leq$ can be defined in terms of $\lt$ but not the other way around. (The Mac Neille real numbers have both $\lt$ and $\leq$; however, in this case, neither is necessarily a linear or total order, nor can they necessarily be defined in terms of one another.)
• Baire space and Cantor space, being representable as subspaces of the real line, of course are linearly ordered. It's also interesting to see them as coming from the (decidable) linear orders on $\mathbf{N}$ and $\mathbf{2}$, which they are $\mathbf{N}$-fold products of.

Properties

Relation to total orders

Using excluded middle, one can move between linear orders and total orders using negation; that is, the negation of a linear order is a total order and vice versa. Actually one usually swaps the order too, as follows:

• $x \lt y$ iff $y \nleq x$;
• $x \leq y$ iff $y \nless x$.

To prove this, it's enough to see that the properties of a linear order are dual to the properties of a total order, as follows:

linear ordertotal order
irreflexivityreflexivity
asymmetrytotality
transitivitycomparison
comparisontransitivity
connectednessantisymmetry

Constructively, these are still the default definitions to use; that is, if one is given a linear order or a total order and wants to interpret the other symbol, then one does so using these definitions. However, the result will not necessarily be a total order or a linear order. To be specific, if one starts with a linear order $\lt$ and defines $\leq$ as above, then totality does not follow; and if one starts with a total order $\leq$ and defines $\lt$ as above, then comparison does not follow. Nevertheless, at least $\leq$ will be a partial order, and least $\lt$ will be a quasiorder. Furthermore, the duality between the axioms is still there, even though negation no longer mediates between them; although comparison need not hold for a total order constructively, the duality is preserved if one defines linear orders without using transitivity.

One often sees $x \lt y$ defined as $x \le y$ but $x \ne y$; this is equivalent, but doesn't show the de Morgan duality explicitly. Similarly, one often sees $x \leq y$ defined as $x \lt y$ or $x = y$; this is not even equivalent constructively, although it is classically.

Keep in mind, however, that the only use of excluded middle in the classical theory is the assumption that $x = y$, $x \lt y$, and $x \leq y$ are always either true or false. There is therefore a perfect correspondence between decidable linear orders and decidable total orders on sets with decidable equality.

Classifying topos

There is a classifying topos for inhabited linear orders. It is given by the category of cosimplicial sets, hence by the presheaf topos over the opposite category of the simplicial category.

For more see at classifying topos the section For (inhabited) linear orders.