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Semigroup Structure of Sets of Solutions to Equation X^s = X^m
 Porubský, Štefan
Using an idempotent semigroup approach we describe the semigroup and group structure of the set of solutions to equation X^m = X^s in successive steps over a periodic commutative semigroup, over multiplicative semigroups of factor rings of residually finite commutative rings and finally over multiplicative semigroups of factor rings of residually finite commutative principal ideal domains. The analysis is done through the use of the maximal subsemigroups and groups corresponding to an idempotent of the corresponding semigroup and in the case of residually finite PID’s employing the available analysis of the Euler-Fermat Theorem as given in [11]. In particular the case when this set of solutions is a union of groups is handled. As a simple application we show a not yet noticed group structure of the set of solutions to x^n = x connected with the message space of RSA cryptosystems and Fermat pseudoprimes.
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