Definition of Total Partial Ordering This is a pretty basic question, but my book doesn't give a formal logical definition of total partial ordering.
For a given full partial ordering relation R is anti-symmetric, reflexive, transitive and :
$\forall a,b \in A [ a \neq b \rightarrow ( aRb \wedge \neg bRa ) \vee ( \neg aRb \wedge bRa ) ]$
Is that the correct definition?
Edit
Thank's to Asaf for correcting my mistranslation from Hebrew to English.  The book also calls it a linearly ordered set or a chain.
 A: The original confusion between total and order comes from a mistranslation of the word מלא in Hebrew, hence my answer dealing with both terms.
A total order is a partial ordering in which every two elements are comparable, that is to say what you wrote:
Let $(A,R)$ be a partially ordered set. We say that $R$ is a total order if for all $a,b$ one of the following is true:


*

*$a=b$, or

*$aRb$, or

*$bRa$.


Due to the anti-symmetry of $R$, if both 2 and 3 hold we have that 1 holds, so if $a\neq b$ we must have that only one of the conditions hold.
A total order is also called linear often.

A complete partial order $(A,R)$ is a partial order such that for every nonempty $B\subseteq A$ there exists $y\in A$ such that:


*

*$\forall x\in B, xRy$ (that is $y$ is an upper bound of $B$), and

*$\forall a\in A\left(\forall x\in B\left(xRa\right)\rightarrow bRa\right)$ (every other upper bound of $B$ is an upper bound of $B\cup\lbrace y\rbrace$).


This $y$ is called the least upper bound of $B$.

Example: The real numbers with the usual ordering is a complete and total order. 
However total orders need not be complete; and complete orders need not be total.


*

*The rational numbers with the usual order is a totally ordered, but $\{x\in\mathbb Q\mid x^2<2\}$ does not have a least upper bound (which would have to be $\sqrt 2$. An irrational number).

*The subsets of $\mathbb N$ ordered by inclusion give a complete order which is not total. This is due to the fact $\{0\}$ and $\{1\}$ are incomparable (neither is a subset of the other), but every collection of subsets has a least upper bound - namely its union.

Within a general partially ordered set $(A,R)$ we can talk about subsets of $A$ which are linearly ordered by $R$. Such subset is called a chain. Formally, $B\subseteq A$ is called a chain (in $R$) if:
For every $a,b\in B$ we have either $a=b$, $aRb$ or $bRa$.
An example of a chain in a non-linear order is $\bigg\lbrace A_i = \{n\in\mathbb N\mid n<i\}\bigg\rbrace$. This is a chain in the partial order of $\mathcal P(\mathbb N)$.

Claim: Let $(A,R)$ be a partially ordered set, then $R$ is a total order of $A$ if and only if every nonempty $B\subseteq A$ is a chain in $R$.
Proof: If $R$ is a total order, then every restriction to a subset of $A$ is still a total order; on the other hand if every subset is a chain, then in particular $A$ forms a chain and thus it is linearly ordered.
A: The definition of complete partial order (on $X$) I know is a partial order (on $X$) for which each subset (of $X$) has a supremum.
But then, there is no such thing as a correct definition.
