Simple solutions of the Yang-Baxter equation of cardinality pⁿ

For every prime number p and integer n>1, a simple, involutive, non-degenerate set-theoretic solution (X, r) of the Yang-Baxter equation of cardinality |X| = pⁿ is constructed. Furthermore, for every non- (square-free) positive integer m which is not the square of a prime number, a non-simple, indecomposable, irretractable, involutive, non-degenerate set-theoretic solution (X, r) of the Yang-Baxter equation of cardinality |X| = m is constructed. A recent question of Castelli on the existence of singular solutions of certain type is also answered affirmatively.