I'm the President of my school's Order of Omega; with regard to your question concerning the limitations on membership, only members of social fraternities and sororities may join.
I should clarify this a bit more, this typically means individuals who are members of organizations that are in a campus's IFC, NPC, NPHC, or MGC. At my school, we have an Inter-Greek Council, so when I rewrote our by-laws, I set that as the requirement, although for an organization to be in IGC, they must be a member of one of the four sub-councils that I listed above.
Overall though, what Order of Omega considers to be Greek is what your campus recognizes as Greek through the governing councils for fraternities and sororities.
I think honorary members are not required to be Greek though, but I could be wrong about that.
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Phi Sigma Kappa
Kappa Tetarton Chapter
Southern Illinois University - Carbondale
www.SIUPhiSigs.com
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