Jane is Mary's mother.
John is the Dean of the college.
Can "Mary's mother be an argument and a predicator?
No, predicators are not arguments. The arguments represent the entities involved: in your example "Jane" and "Mary's mother". Syntactically, they are respectively subject and predicative complement.
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No, predicators are not arguments. The arguments represent the entities involved: in your example "Jane" and "Mary's mother". Syntactically, they are respectively subject and predicative complement.
"Predicator" is a syntactic term for the function filled by a verb, which in your examples is "be".