arXiv subjects

Ivan Gatica Araus

Publications and source records attributed to Ivan Gatica Araus.

3 recordsLinked to original sources

arXiv2010

$σ$-Set Theory: Introduction to the concepts of $σ$-antielement, $σ$-antiset and Integer Space

In this paper we develop a theory called $σ$-Set Theory, in which we present an axiom system developed from the study of Set Theories of Zermelo-Fraenkel, Neumann-Bernays-Godel and Morse-Kelley. In $σ$-Set Theory, we present the proper existence of objects called $σ$-antielement, $σ$-antiset, natural numbers, antinatural numbers and generated $σ$-set by two $σ$-sets, from which we obtain, among other things, a commutative non-associative algebraic structure called Integer Space $3^{X}$, which corresponds to the algebraic completion of $2^{X}$.

math.LO↗

arXiv2010

$σ$-Relations, $σ$-functions and $σ$-antifunctions

In this article we develop the concepts of $σ$-relation and $σ$-function, following the same steps as in Set Theory. First we define the concept of ordered pair and then we build the Cartesian Product of $σ$-sets so that we can define the concepts of $σ$-relation and $σ$-function. Now, as in $σ$-Set Theory there exist the concepts of $σ$-antielement and $σ$-antiset, we can build the new concepts of $σ$-antifunction, antidentity and antinverse. Finally, in the case that a $σ$-function $f:A\to B$ is bijective and there exist $A^{\st}$ and $B^{\st}$ $σ$-antiset of $A$ and $B$, we get 16 different $σ$-functions which are related in a diagram of $σ$-functions.

math.FA↗

arXiv2009

Characterization and uniqueness of $σ$-Antisets

In this article, we prove that if there exists a $σ$-antiset $X^{*}$, of a $σ$-set $X$, then it is unique and therefore we can present a characterization of the $σ$-antiset.

math.LO↗