On some universal criterion for a fixed point

Fixed point criteria may be applied in various fields of mathematics. The problem of finding sufficient conditions for transformations of a certain class to have a fixed point is well known. In the context of element-wise description for the monoid of all endomorphisms of a groupoid, the following were formulated: a bipolar classification of endomorphisms and related mathematical objects. In particular, the concept of a bipolar type of endomorphism of a groupoid (or simply a bipolar type) was stated. Every endomorphism of an arbitrary groupoid has exactly one bipolar type. In this paper, using bipolar types, a fixed point criterion for an arbitrary transformation of a nonempty set (hereinafter, the universal fixed point criterion) is formulated and proved. This criterion is not easy to apply. Further expansion of the range of problems to which this criterion can be applied depends directly on the success in studying the properties of groupoids’ endomorphisms. The paper formulates such general problems (unsolved for today), that the success in their study will expand the possibilities of using the universal fixed point criterion. The connection between the formulated problems and the obtained criterion is discussed. In particular, necessary and sufficient conditions for the Riemann hypothesis on the zeros of the Riemann zeta function to be satisfied are obtained using the universal fixed point criterion.