Orders, Partial Orders, Strict Partial Orders, Total Orders, Strict Total Orders, and Strict Orders

Transitivity is the fundamental property of all relations that we call "something something" order. Of course, an equivalence relation is also transitive, and in fact is also a preorder.

So, maybe, one can start from transitive relations, split them according to whether they are reflexive, irreflexive, or neither. (Obviously, there's nothing new in this taxonomy.) On the irreflexive branch one gets exactly the strict partial orders. On the reflexive branch one gets preorders and their specializations, namely, partial orders and equivalence relations.

On the third branch we find the riff-raff transitive relations, and I'm not sure anybody calls them orders. There are also preorders that are neither partial orders nor equivalence relations, of course. So, maybe one could adopt the definition that an ordering relation is a binary relation that is transitive and either reflexive and antisymmetric or irreflexive.

The only main difference from the definition you consider is that a relation that is transitive and antisymmetric, but neither reflexive not irreflexive, is not considered an order relation.

Totality (linearity) can be specified by saying that for all $a$ and $b$, if $a \neq b$, then either $a R b$ or $b R a$. This works for both reflexive and irreflexive relations. (Thanks to @mlc for reminding me to cover this detail.)

A strict partial order is a relation that's irreflexive and transitive (asymmetric is a consequence). This is the most common definition.

Actually, this notion is completely equivalent to the notion of partial order (a reflexive, antisymmetric and transitive relation).

Indeed, if $X$ is a set and $\Delta_X=\{(x,x):x\in X\}$, we have that

• if $S$ is a strict partial order on $X$, then $S^+=S\cup\Delta_X$ is a partial order;

• if $R$ is a partial order on $X$, then $R^{-}=R\setminus\Delta_X$ is a strict partial order on $X$;

• if $S$ is a strict partial order on $X$, then $S=(S^+)^-$;

• if $R$ is a partial order on $X$, then $R=(R^-)^+$.

You can try your hand in proving the statements.

So any strict partial order determines a unique partial order and conversely. Passing from $S$ to $S^+$ is essentially the same we do when passing from $<$ on numbers to $\le$.

The property of being a linear (or total) order can be expressed by

for all $a,b\in X$, if $a\ne b$, then either $a\mathrel{T}b$ or $b\mathrel{T}a$

where $T$ is a (strict) partial order.

Are strict partial orders useful? Yes. If you compare the two definitions, you see that equality is not necessary in the definition of a strict partial order (not for linear ones, though), which makes them attractive for certain logic frameworks where equality has no particular status with respect to other predicates.