Countable composition closedness and integer-valued continuous functions in pointfree topology

‎For any archimedeanf-ring A with unit in whichbreakawedge‎ ‎(۱-a)leq ۰ for all ain A‎, ‎the following are shown to be‎ ‎equivalent‎: ‎ ‎۱‎. ‎A is isomorphic to the l-ring {mathfrak Z}L of all‎ ‎integer-valued continuous functions on some frame L‎. ۲‎. ‎A is a homomorphic image of the l-ring C_{Bbb Z}(X)‎ ‎of all integer-valued continuous functions‎, ‎in the usual sense‎, ‎on some topological space X‎. ۳‎. ‎For any family (a_n)_{nin omega} in A there exists an‎ ‎l-ring homomorphism breakvarphi‎ :‎C_{Bbb Z}(Bbb‎ ‎Z^omega)rightarrow A such that varphi(p_n)=a_n for the‎ ‎product projections breakp_n:{Bbb Z^omega}rightarrow Bbb Z‎. ‎This provides an integer-valued counterpart to a familiar result‎ ‎concerning real-valued continuous functions‎.