In a direct democracy, is there a a judicial or executive branch. If so, what is it? It a group of people? Do people elected them?
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There are a lot of good modern examples we could draw from for this. I'm going to look at the early Athenian direct democracy, because it's distinct from modern democracies, and I think it exemplifies direct democratic purism.
Athens had no judges, and court cases were handled by large groups of jurors, selected by lot from the people. On the local level, the people were both judicial and executive.
Athens also had no executive branch (in the modern sense), but made national executive decisions by individual vote in nearly every case where it was possible to carry out a vote. The people didn't have representatives vote for them; the votes were tallied on direct popularity. So, on the national level, the people were also executive.
Very simply, democracy is the process of casting votes to establish the rules of society. So long as a form of democracy is involved, voting exists.
However, voting is not doing, and all three things still need doing: executive (administration or enforcing the law), judicial (adjudicating the law), and legislative (creating the law).
In a direct democracy, you still need police and military, which means you need civil control over the police and military, so the executive branch still exists and would constitute elected offices with appointed cabinet positions.
Similarly, you still need to judge both people and the law, so your judicial branch still exists with judges, lawyers, and other officers of the court — both elected and appointed.
What changes most is the legislative branch. It still exists, but it's only purpose is to create new law and recommend changes to existing law. It no longer does the voting, so it cannot implement law. In a direct democracy, that role — the power to implement law — is given over to the people. Further, in a direct democracy, the now purely administrative legislative role could be given over to the executive branch (it really depends on how large and complex your society is, one hates to give too much power to anyone), which would dissolve the legislative branch entirely.