  It is stated that Boolean set algebras with unit V, where V is a union of
Cartesian products, are axiomatizable. The axiomatization coincides with that
of cylindric polyadic equality algebras (class CPE). This is an algebraic
representation theorem for the class CPE by relativized polyadic set algebras
in the class Gp. Similar representation theorems are claimed for the classes
strong cylindric polyadic equality algebras (CPES) and cylindric m-quasi
polyadic equality algebras (mCPE). These are polyadic-like equality algebras
with infinite substitution operators and single cylindrifications. They can be
regarded also as infinite transformation systems equipped with diagonals and
cylindrifications. No representation theorem or neat embedding theorem has
proven for this class of algebras yet, except for the locally finite case. The
theorems occuring in the paper answer some unsolved problems.
