What is the definition of half duplex?Is this system,$A$ give $x$ to $B$,then $B$ give $x$ to $C$,called half duplex?
The receiver/transmitter pair in B's possession is operating in half-duplex mode.  


As you say "$A$ give $x$ to $B$, then $B$ give $x$ to $C$" which specifies that while $B$ ls listening to what $A$ is telling it (the $x$ that is being given to $B$), the transmitter in $B$'s possession must remain silent until $B$ has completely received $x$. It is only then that $B$ can start up his transmitter and send $x$ to $C$. So, $B$ cannot receive and transmit at the same time, that is, $B$ is operating in a half-duplex mode. If $A$ were to send $y$ to $B$ while $B$ is sending $x$ to $C$, $B$ just ignore the transmission from $A$: $B$'s transmitter is active and so its receiver must be in-operational till $B$ is done talking to $C$.