Parter, Periodic and Coperiodic Functions on Groups and their Characterization

Abstract

Decomposer functional equations were introduced by the
author and have been completely solved on arbitrary groups. Their solutions
are as decomposer functions and play important role regarding
to decomposition (factorization) of groups by their two subsets. In this
paper, we introduce an important class of strong decomposer functions,
namely parter (or cyclic decomposer) functions. As some important applications
of this topic, we characterize all periodic , coperiodic functions
in arbitrary groups and give general solution of their functional equations:
f(bx) = f(x) , f(xb) = f(x), f(bx) = bf(x) and f(xb) = f(x)b.
Moreover, we characterize all parter functions in arbitrary groups and
completely solve the decomposer equation with the condition which its
-range is a cyclic subgroup of G. Finally, we give some functional characterization
for related projections and b-parts functions and also, we
introduce some uniqueness conditions for b-parts of real numbers.