On regularity classes of binary relations
The notion of reverse of any binary operation on the unit interval is introduced. The properties of reverses of some binary operations are studied and some applications of reverses are indicated.
We study the deductive strength of the following statements: 𝖱𝖱: every set has a rigid binary relation, 𝖧𝖱𝖱: every set has a hereditarily rigid binary relation, 𝖲𝖱𝖱: every set has a strongly rigid binary relation, in set theory without the Axiom of Choice. 𝖱𝖱 was recently formulated by J. D. Hamkins and J. Palumbo, and 𝖲𝖱𝖱 is a classical (non-trivial) 𝖹𝖥𝖢-result by P. Vopěnka, A. Pultr and Z. Hedrlín.
This article describes a rough subgroup with respect to a normal subgroup of a group, and some properties of the lower and the upper approximations in a group.