On the day of the abdication, the Oireachtas, the parliament of the Irish Free State, removed all direct mention of the monarch from the Irish constitution. The next day, it passed the External Relations Act, which gave the monarch limited authority (strictly on the advice of the government) to appoint diplomatic representatives for Ireland and to be involved in the making of foreign treaties. The two acts made the Irish Free State a republic in essence without removing its links to the Commonwealth.
Is there an answer to this question (If it cannot be answered, say "unanswerable"): What is the Oireachtas?
the parliament of the Irish Free State