The Alpha mare only shares her space sparingly and the other horses that do have the honor of getting close with her do so with respect. They are ready in an instant to move out if she gives them the signal. That is why I think so many horses run people over and take control. The ability to invade someones space without having to sit on the edge of respect with the person tells the horse that the person is not a dominant individual. If you correct a horse who is in your space with the tiniest bit of "disrespect" (ex. Pins it's ears, pushes in on you, whacks you with its head, grabs the feedbucket out of your hand, gets overly zealous about a carrot, nips you ect) it shows the horse you are serious about being leader. You can lose that role very easily, with a very tiny infraction being dismissed instead of corrected. A good lead mare doesn't put up with squat...and neither should you. I am all for loving your horse but human displays of affection can and often do blur the line in the horses head and teaches the horse you are not paying attention to it's body language. If they get away with something small it can place them in a position to take over.