Is the Channel shortening similar to Time domain Equalizer (TEQ) I used to read about Time domain equalizer (TEQ) in many lectures, i.e here , But in many other papers,  that's called  Channel shortening i.e here . 

However, in the a two above links, the cost functions is almost similar, but in first paper it's called Time domain equalizer, and in the second, it's channel shortening. 

What's the difference between them (I mean between Time domain equalizer TEQ and Channel Shortening)?  And why do we call it sometimes TEQ and another time channel shortening? 

Thank you.
Most likely people use different words to describe the same concept. This happens quite often. The TEQ is the more general term and also used to equalize other methods that OFDM, while CS is related to OFDM and tries to make the channel shorter so that it fits within the CP duration.

You can check online, you will find that almost all articles related to CS are in OFDM whereas the TEQ is a general term 

Good luck