The <i>m</i>Θ Quadratic Character in the <i>m</i>Θ Set <img width="40" height="20" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />

Abstract

The modal Θ-valent logic is a logic that contains all the thesis of the classical logical calculus and, besides allows to express notions of possibility, of necessity, and more others. The modal Θ-valent sets are the supports in term of the structure of the Θ-valent rings. A Θ chr (<i>m</i>Θ) is a structure which is rich at the same time of inheritance in the meaning of the romanian academician Gr. C. Moisil, as the algebraic model of a such logic. The set <img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" /> contains the set <img width="10" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image007.png" /> and the elements <img width="25" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image003.png" /> such that the support of <i>x</i> is not congruent to 0 modulo <i>n</i>. In this paper the purpose is to define on <img width="40" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image008.png" />, <i>p</i> prime, a notion of quadratic residues and quadratic character which respects its structure of <i>m</i>Θs. Hoping that this approach will bring something of interest to the notion of quadratic residues. First of all, we construct the modal Θ-valent congruences of (<img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, <i>F_α</i>). We characterize the <i>m</i>Θ set (<img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, <i>F_α</i>) and we then give some arithmetical and intrinsic <i>m</i>Θ parameters of <img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" /> which lead us to the notion of factorial of <i>m</i> without <i>n</i> in <img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, the <i>m</i>Θ quotient of (<img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png" />, <i>F_α</i>) modulo (<img width="30" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image004.png" />) and a complete system of <i>m</i>Θ residues modulo <img width="30" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image004.png" />, <img width="25" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image005.png" />. After that, we define a <i>p</i>-valent modal quadratic residue, <i>p</i> prime. We characterize some properties of <i>p</i>-valent modal quadratic character and <i>p</i>-valent modal quadratic residue of <i>p^k</i> which establish the difference between the <i>m</i>Θ Euler’s theorem and the Euler’s theorem in the classical arithmetic. Later, we establish the theorem for determining the p-valent modal quadratic character of <img width="65" height="10" src="http://article.sciencepublishinggroup.com/journal/247/2471199/image006.png" /> with respect to <i>p^k</i>. This theorem is a non-classical version of Gauss’s lemma. Finally, we establish an example introducing the law of quadratic reciprocity of Gauss.