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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.

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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).

Since strict orders and non-strict orders are pretty much interchangeable, it usually doesn't matter which one you take as your basic definition or object of study. It's just a matter of convention that non-strict orders are what is usually axiomatized. Each version has a few circumstances where it is more useful (for instance, strict orders tend to be more convenient when talking about well-foundedness, and non-strict orders are more natural if you are also interested in preorders).

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  • $\begingroup$ 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 $X$. $\endgroup$ – xijag Aug 30 '17 at 1:48
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    $\begingroup$ Then treat "strict total" as one unit, and not an adjective modification of "total ordering." $\endgroup$ – Randall Aug 30 '17 at 2:05
  • $\begingroup$ Indeed. Totality entails Reflexivity. Leave it as "strict total order" is the associated "strict version of a total order ... that is not actually total (or reflexive) itself but what the hey." $\endgroup$ – Graham Kemp Aug 30 '17 at 2:14
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    $\begingroup$ I agree that reflexivity has nothing to do with a relation being an order, and the same goes for totality or trichotomy ... Strict total orders are trichotomous, but not reflexive, partial orders are reflexive, but not trichotomous, yet both are orders of some kind. Indeed, both are anti-symmetric and transitive (no cycles!); these two properties seem to be the real essence of an order ... Which raises the question: why don't we have that as a standard definition of 'order'? That is, why don't we define an 'order' as a binary relation that is anti-symmetric and transitive? $\endgroup$ – Bram28 Aug 30 '17 at 2:57

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