Misc math articles and essays

This page contain my misc math articles and essays not fitting to other categories on my math site.

Logics

Formalistics of generalization: In the framework of ZF formally considered generalizations, such as whole numbers generalizing natural number, rational numbers generalizing whole numbers, real numbers generalizing rational numbers, complex numbers generalizing real numbers, etc. The formal consideration of this may be especially useful for computer proof assistants.

Vertical order of binary relations

Defined a partial order relation between binary relations f and g by the formula g = f|_{dom g}.

Theorem about binary relation limited to a set (WARNING: This theorem appeared to be erroneous.)

Theorem (with proof), expanding and generalizing the following statement:

For two binary relations f and g the formula g = f|_{dom g} is equivalent to conjunction of g being a subset of f and the formula

g g^-1 = f g^-1 (or equivalently g g^-1 = g f^-1).