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"
}