The author has previously
introduced a generalized Šilov boundary which seems useful in studying analytic
structure of several dimensions in the spectrum of a uniform algebra A. Related
generalizations of A-convexity, A-polyhedra, etc. are developed here. Several
different but equivalent approaches to these various generalizations are described.
The generalized boundaries discussed here are related to the “q-holomorphic
functions” of the author, and to A-holomorphic convexity.