standard definitions when talking about ordered sets What is the standardization if any when it comes to ordered sets?. 
Specifically I'm always confused in the following cases: 
1) When someone say "a partial ordered set":  to me it can mean a strict partial ordered set (The relation is asymmetric and transitive) or a non strict partial ordered set (the relation is antisymmetric, reflexive and transitive), but many books refer to partial ordered sets like they were non strict and excluding the other case, why?. 
2) When someone say "a total ordering": to me it can mean a strict total ordered set or it can mean a non strict total ordered set, but many books refer to total orders like they were non strict and excluding the other case. 
3) When someone say "an ordered set" : to me it can mean any ordered set, partial, complete, strict, or non strict, but many books don't make this distintion and they refer to a "non strict partial orderd". 
Am I missing something? Is it just me the one making these distintions?, please help me clarify this. 
 A: In my experience the term partial order usually — I might even go so far as to say almost always — means a reflexive, transitive, antisymmetric relation; this is the standard definition. A reflexive, transitive, asymmetric relation is usually termed a strict partial order. I have occasionally seen the term partial order used as you suggest, ambiguously to mean either a (non-strict) partial order or a strict partial, so that the reader is forced to rely on context to determine which is meant in any particular instance; in my opinion this usage is better avoided. (The one exception is when one wants to talk about several partial orders at once, some strict and some non-strict, and wants a cover term for the both kinds as well as the specific terms strict partial order and non-strict partial order.)
The same applies to total (or, as I prefer, linear) orders, though not so strongly: I’m pretty sure that I see linear order used ambiguously more often than I do partial order. Still, I’d say that the most common definition is that a linear order is a partial order (in the narrow sense) in which every two elements are comparable, and that a strict linear order is a strict partial order in which every two distinct elements are comparable.
Without a context I would not even attempt to guess what someone means by ordered set, and I would not use the term without defining what class of orders I had in mind. The term can be used very broadly, but I’ve also seen it used very narrowly to mean simply linear order.
A: The definition an ordered set is the same as a partial ordered set, i.e.,

Let $X$ be  a set and $\le$ a relation on $X$. We say that $\le$ orders $X$, or that $\le$ is an order in $X$, if $\le$ has the following properties:
1) If $x\le y$ and $y\le z$, then $x\le z$.
2) For every $x\in X$, $x\le x$.
3) If $x \le y$ and $y\le x$, then $x=y$.

A set $X$ together with an order $\le$ in $X$ is called an ordered set, or partial ordered set. They are same.
Note that two elements $x$ and $y$ of an ordered set $X$ can be incomparable, i.e., it can happen that neither $x\le y$ nor $y\le x$ holds.

Totally ordered set, is also called linearly ordered set or chain. Its definition is this:
Let $X$ be  a set and $<$ a relation on $X$. We say that $X$ is a totally ordered set if $<$ has the following properties:
1) If $x\not= y$ and  then $x<y$ or $y<x$.
2) If $x<y$, then $y<x$ doesn't hold.
3) If $x < y$ and $y<z$, then $x<z$.

Note that two elements $x$ and $y$ of an ordered set $X$ can be comparable.
