An AI, at least in my organization, if a full member with all rights pertaining too. They go through ritual just as any member would--so yes, her children are legacies. We have AIs who have children and we do consider them legacies.
As far as "reverse legacy" I believe your mother could qualify since a legacy is usually a direct relative of a member (a sister, daugther etc.) but I would say that would depend on the organization. I am sure other Alpha Phis can answer that question completely.
Heather
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