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I am computer scientist, not a mathematician, I've been reading some papers on argumentation in AI that uses the term 'maximal' set without defining it. I think it's left undefined because it's a term used widely in mathematics? The paper at the end of this post. I hope this question isn't too simple for this forum and I hope set theory is ok as a tag?!

Many thanks.

@article{Dung2007642, title = "Computing ideal sceptical argumentation", journal = "Artificial Intelligence", volume = "171", number = "10-15", pages = "642 - 674", year = "2007", note = "Argumentation in Artificial Intelligence", doi = "DOI: 10.1016/j.artint.2007.05.003", author = "P.M. Dung and P. Mancarella and F. Toni" }

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Typically, the term maximal means that a set $A$ has a property but $A \cup \{x \}$ does not have this property for any $x \notin A$. (For example, a maximal matching is a set of edges which is a matching, but if any new edge is added it is no longer a matching.) This is weaker than a maximum set, which is the largest (in cardinality) set with the property.

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  • $\begingroup$ I'd prefer "a set $A$ has a property but no set strictly containing it has this property" (think maximal ideal). $\endgroup$
    – Nik Weaver
    Commented Nov 14, 2015 at 1:56
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I agree with David. A maximal set is defined with some property in mind. In abstract argumentation for instance preferred extensions are maximal admissible sets. I can only guess on the exact context of your question, but the ideal extension of some argumentation framework can be defined as the maximal admissible set that is contained in all preferred extensions. A similar relationship as between grounded and complete semantics.

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