A closed set $C$ is irreducible if $x$ leads to $y$ for all states $x$ and $y$ in $C$.
Would you consider a set of a singular absorbing state irreducible. Note that this is mostly a question of terminology.
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By "a set", are you saying that the entire state space consists of only one state? – Joe May 10 '22 at 00:32
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Your title and question don't match. Are asking about irreducibility or recurrence? As you're editing to make your question and title match, state the definition you know for that term, whether or not you think it satisfies the definition, and why. – Joe May 10 '22 at 01:31
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If $\ s\ $ is an absorbing state, is $\ {s}\ $ closed? Does $\ x\ $ lead to $\ y\ $ and $\ y\ $ lead to $\ x\ $ for all $\ x,y\in{s}\ $? If the answer to both questions is "yes", then $\ {s}\ $ is an irreducible set, according to the definition. Otherwise it isn't. – lonza leggiera May 17 '22 at 03:22