  Kamiya, Takemura, and Terao initiated the theory of the characteristic
quasi-polynomial of an integral arrangement, which is a function counting the
elements in the complement of the arrangement modulo positive integers.
  They gave a period of the characteristic quasi-polynomial, called the
LCM-period, and showed that the first constituent of the characteristic
quasi-polynomial coincides with the characteristic polynomial of the
corresponding hyperplane arrangement.
  Recently, Liu, Tran, and Yoshinaga showed that the last constituent of the
characteristic quasi-polynomial coincides with the characteristic polynomial of
the corresponding toric arrangement.
  In addition, by using the theory of toric arrangements, Higashitani, Tran,
and Yoshinaga proved that the LCM-period is the minimum period of the
characteristic quasi-polynomial.
  In this paper, we study an arrangements over a Dedekind domain such that
every residue ring with a nonzero ideal is finite and give algebraic
generalizations of the above results.
