On deterministic and absorbing algebras of binary formulas of polygonometrical theories

D. Yu. Emelyanov ; S. V. Sudoplatov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 20 (2017), pp. 32-44

Résumé

Algebras of distributions of binary isolating and semi-isolating formulas are derived structures for a given theory. These algebras reflect binary links between realizations of $1$-types defined by formulas of the initial theory. Thus these are two sorts of interrelated classification problems: 1) to define, for a given class of theories, what algebras correspond to theories in this class and to classify these algebras; 2) to classify theories in the class in the dependence of algebras of isolating and semi-isolating algebras that defined by these theories. For the finite algebras of binary isolating formulas that description implies the description for the algebra of binary semi-isolating formulas. In the paper, we investigate deterministic, almost deterministic, and absorbing algebras of binary formulas of polygonometrical theories. The properties of determinism and almost determinism for algebras of binary isolating formulas of polygonometrical theories are characterized. As corollary we have that any group generates a deterministic algebra of a polygonometrical theory. The notion of $n$-almost deterministic algebra is introduced, examples and properties of these algebras are stated. A description of these algebras for theories of graphs of regular polyhedrons is given. It is shown that any group is a side-group of a trigonometry with $2$-absorbing algebra of binary isolating formulas.

Détail

Keywords: algebra of binary formulas, deterministic algebra, absorbing algebra, polygonometrical theory.