# Why are are trichotomous strict orders called total orders, when by definition they are not total?

Consider an arbitrary set equipped with a strict order $(X,<)$ such that we have:

$$\forall a,b\in X\left(a\neq b\implies (a<b)\lor (a>b)\right)$$

I frequently see this relation being called a "strict total order" yet the term itself is an oxymoron. For in order for a relation to be total it must be reflexive which would contradict it being a strict order.

Also while on the topic, I would be grateful if someone could explain to me why in order theory there is a focus on partial orders rather then strict orders.

Now I understand any work on partial orders can be transplanted over to give analogous theorems on strict orders, in the same way elementary theorems in linear algebra regarding the rows of a matrix can analogously be translated into theorems on the columns of a matrix. Thus making the distinction between whether we choose strict or partial orders somewhat trivial. However the concept of a strict order seem a bit more intuitive at least when viewing finite posets, for instance the nodes/vertices in a hasse diagram could be viewed as the directed acyclic graphs formed by the covering relations of a strict order.

The point is that a strict order and a non-strict order are essentially the same thing. Given a strict order $<$, you can get a non-strict order by defining $x\leq y$ to mean "$x<y$ or $x=y$". Conversely, given a non-strict order $\leq$, you can get a strict order by defining $x<y$ to mean "$x\leq y$ and $x\neq y$".
So "strict total order" just means "the strict version of a total order", where we're axiomatizing the relation $<$ rather than the relation $\leq$. It really is essentially the same thing, since you can convert between a strict total order and an ordinary (non-strict) total order as described above. Reflexivity really has nothing to do with totality at all: rather, demanding reflexivity just means you are axiomatizing $\leq$ instead of $<$ (regardless of whether your order is total).
• Thanks however I don't understand your statement "Reflexivity really has nothing to do with totality at all". Applying this definition: ncatlab.org/nlab/show/total+relation and the one on wikipedia, it says a relation $R$ is total on $X$ when we have: $$R\cup R^{-1}=X\times X\iff \forall a,b\in X\left(aRb\lor bRa\right)$$ Thus in particular for every $x\in X$ we can take $a=b=x$ so that $aRb\lor bRa\iff xRx$ for every $x\in X$ so $R$ must be reflexive. In this since if $R$ is total on $X$ then $R$ must also be reflexive on $X$. – xijag Aug 30 '17 at 1:48