Clical Guitar Repair - one of the ivory pieces broke on the headstock of my guitar?

This item may possibly be repaired by the judicious application of a few drops of SuperGlue. Otherwise, if a matching replacement is not a priority, then the faulty item may be substituted, and, replaced with a cheaper wood, or, plastic version. I have used both methods in repair of violin.(not a Strad !) Your umption that this is not actual ivory, may be correct. If instrument predates 1950's, then this may be the real thing. When putting away an instrument, one should always release the tension on the strings.