Event.observe(window, 'load', init, false);

function init(){
     $('greeting-submit').style.display = 'none';
     Event.observe('greeting-name', 'keyup', greet, false);
}

function greet(){
     var url = 'greeting.php';
     var pars = 'greeting-name='+escape($F('greeting-name'));
     var target = 'greeting';
     var myAjax = new Ajax.Updater(target, url, {method: 'get', parameters: pars});
}