Algebraic Approaches to Periodic Arithmetical Maps

Let S be a set. For an arithmetical map ψ : Z → S, if for some n ∈ Z = {1, 2, 3, · · · } we have ψ(x+ n) = ψ(x) for all x ∈ Z, then we say that ψ is periodic modulo n and n is a period of ψ. Let P(S) denote the set of all periodic maps ψ : Z → S. If m,n ∈ Z are periods of a map ψ ∈ P(S), then the greatest common divisor (m,n) is also a period of ψ, for we can write (m,n) in the form am + bn with a, b ∈ Z. Thus, any period of ψ ∈ P(S) is a multiple of the smallest (positive) period n(ψ) of ψ. A monoid is a semigroup with identity. Let M be a commutative monoid (considered as an additive one). If ψ1, ψ2 ∈ P(M), then the map ψ1 + ψ2 : x 7→ ψ1(x) +ψ2(x) also lies in P(M) because ψ1 +ψ2 is periodic modulo the least common multiple [n(ψ1), n(ψ2)]. In 1989 the author [S1] introduced triples of the form 〈λ, a, n〉 where λ ∈ M, n ∈ Z and a ∈ R(n) = {0, 1, · · · , n − 1}. We can view 〈λ, a, n〉 as the residue class (or arithmetic sequence)