The office, rank, or authority of an archbishop; archiepiscopal jurisdiction.
From archbishop + -ry (a suffix meaning 'the office or position of,' similar to -ship). While archbishopric refers to the territory, archbishopry emphasizes the role and authority itself. Both terms emerged during the medieval period but with slightly different applications.
English gives you both 'archbishopric' and 'archbishopry'—one is the land they rule, one is the job itself. It's like the difference between 'being king' (kingship) and 'the kingdom' (kingdom). Medieval scribes needed both words to keep their bureaucracy straight!
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