I am reading "Lecture Notes on Elementary Topology and Geometry" by I. M. Singer and J. A. Thorpe.
The following definition is the definition for a partial ordering in this book:
Definition. A relation $R\subset A\times A$ is a partial ordering if
(1) $(s_1,s_2)\in R$ and $(s_2,s_3)\in R\implies (s_1,s_3)\in R$ and
(2) $(s_1,s_2)\in R$ and $(s_2,s_1)\in R\implies s_1=s_2.$
Is the above definition in this book ok?
I think the following condition is necessary for a partial ordering $R$:
(0) $(s,s)\in R$ for any $s\in A$.
No, that definition is not "ok", i.e. it's not equivalent to the standard definition which requires property (0) explicitly.
For example, the strict inequality '$<$' on real numbers is a partial order by your definition, but not by the standard definition. If $x<y$ and $y<z$ then $x<z$, so property (1) is satisfied. And property (2) is satisfied vacuously: There are no $x,y$ such that $x<y$ and $y<x$. But property (0) is not satisfied: $x<x$ is false.