"What are the practical implications of the Trinity for church structure?"
After the first few responders offer blank stares, maybe one will catch something profound (like these folks), and as the Q man (Quentin P. Kinnison) does below in a highly-recommended book.
From a section headed Trinitarian Implications for Church Structures:
...From Trinitarian themes, an ecclesiology forms which expresses serious concern regarding the specialization of ministry. Any specialized ministry in the church occurs within the ministry of all members--the universal priesthood of all believers.
...Viewing the relations of the Trinity as complementary perichoretic subjects, Volf concludes that ecclesial structures must be viewed as complementary and egalitarian. Therefore, he forcefully states:
If one starts from the Trinitarian model I have suggested, then the structure of ecclesial unity cannot be conceived by way of the one, be it the pope, the patriarch or the bishop. Every ecclesial unity held together by a mon-archy, by a "one-[man!]-rule, is monistic and thus also un-trinitarian.In such a church, the Charismata are recognized as universally distributed and are practiced by all in a "polycentric community" where members are participative, fulfilling their calling to serve God and the community in God's mission.
-Quentin P. Kinnison, Transforming Pastoral Leadership: Reimagaining Congregational Leadership for Changing Contexts, pp. 83-84
For more on what polycentrism might mean, see pp. 96-99 of the book.
Yes, buy it now!
I am intrigued by how best to draw/chart out polycentric.
Here are some starting points from Google Images.
Or maybe this could be attempted through set theory. See: