Comultiplications on the localized spheres and moore spaces

Any nilpotent CW-space can be localized at primes in a similar way to the localization of a ring at a prime number. For a collection p of prime numbers which may be empty and a localization X_p of a nilpotent CW-space X at p, we let |C(X)| and |C(X_p)| be the cardinalities of the sets of all homotopy comultiplications on X and X_p, respectively. In this paper, we show that if |C(X)| is finite, then |C(X)| ≥ |C(X_p)|, and if |C(X)| is infinite, then |C(X)| = |C(X_p)|, where X is the k-fold wedge sum V^k_i=1 Sⁿⁱor Moore spaces M(G, n). Moreover, we provide examples to concretely determine the cardinality of homotopy comultiplications on the k-fold wedge sum of spheres, Moore spaces, and their localizations.