Algebra

Золото-комплексное сечение (Complex Golden Section)

Authors: Sergey Y. Kotkovsky
Comments: 12 Pages. In Russian. Corrections and additions made.

Для широко известной пары чисел золотого сечения и обратной к нему величины обнаружен их комплексный аналог. На основе изучения комплекснозначных бисимметричных матриц второго порядка показана неразрывная связь вещественного и комплексного золотых сечений.

For a well-known pair of numbers of the golden section and the inverse of it, their complex analogue has been found. Based on the study of complex-valued symmetric matrices of the second order, we show the deep inseparable connection between real and complex golden sections.
Category: Algebra

A Boolean Algebra over a Theory

Authors: Shao-Dan Lee
Comments: 9 Pages.

Suppose that L is a first-order language. Let Lu2020 denote the union of L and {t, f} where t(true), f(false) are the nullary operations. We may define a binary relation ‘≤’ such that the sentences set Φ of the language Lu2020 is a preordered set. And we may construct a boolean algebra Φ/∼, denoted Φ ̃ , by an equivalence relation ‘∼’. Then Φ ̃ is a partial ordered set. Let A be a structure of the language L. If Th(A) is a theory of A, then Thu2020(A) is an ultrafilter. If Ψ ⊂ Φ ̃ is a finitely generated filter, then Ψ is principal. We may define a kernel of a homomorphism of the boolean algebra Φ ̃ such that the kernel is a filter. And a filter is a kernel if it is satisfied by some assumptions.
Category: Algebra