I am using selectMenu as:
<a4j:region selfRendered="true" id="text">
<h:selectOneListbox id="symbol" value="#{user.name}">
<f:selectItems value="#{user.temp_list}" />
</h:selectOneListbox>
<a4j:commandButton action="#{user.getData}" reRender="symbol" value="Get"/>
</a4j:region>
temp_list.add(new SelectItem("Sunday"));
temp_list.add(new SelectItem("Monday"));
temp_list.add(new SelectItem("Tuesday"));
temp_list.remove(0); this.setTemp_list(temp_list);