I saw this video, where Kevin-Prince Boateng was sent off for picking up the ball and kicking it at the stands after racist chants from the fans. I wondered if this is possible: that a player can be sent off, while not receiving a red card which will affect his permanent record and likely result in being banned from several matches. Is this possible? And has it ever happened? (I don't think Boateng's case really counted, because of the circumstances.)
The yellow card communicates a caution and the red card communicates a sending-off.
The sending-off is the sanction; the showing of a red card is simply the official signal for this sanction.
Focusing on the concrete example you mentioned, it should be stated that leaving the field of play without being substituted is not a sending-off offence. Here is the complete list of sending-off offences from aforementioned Law 12:
A player, substitute or substituted player who commits any of the following offences is sent off:
- denying the opposing team a goal or an obvious goal-scoring opportunity by deliberately handling the ball (except a goalkeeper within their penalty area)
- denying a goal or an obvious goal-scoring opportunity to an opponent whose overall movement is towards the offender’s goal by an offence punishable by a free kick (unless as outlined below)
- serious foul play
- spitting at an opponent or any other person
- violent conduct
- using offensive, insulting or abusive language and/or gestures
- receiving a second caution in the same match
However, kicking the ball deliberately at the spectators in the manner that Boateng did is an unsportsmanlike behavior, for what he had to be at least cautioned if not sent off considering the force he used during the kick.