Following a general election in Ireland, Dáil Eireann is summoned by the Irish President on the advice of the Taoiseach (the Irish Prime Minister). The President must follow the advice of the Taoiseach.
The President also has the power to dissolve or end the Dáil on the advice of the Taoiseach. If the Taoiseach has the support of the majority of Dáil Eireann, the President cannot refuse to dissolve it.
If the Taoiseach no longer has the support of the majority of Dáil Eireann and he or she advises the President to dissolve the Dáil, the President may refuse.
If the President refuses to dissolve the Dáil, the members of the Dáil must then nominate an alternative Irish Government. Since the foundation of the Irish State, this power of refusal has never been exercised.