<abstract xmlns="http://www.w3.org/1999/xhtml">In this article, using the Mizar system [3], [4], we define a structure [1], [6] in order to build a Pythagorean pentatonic scale and a Pythagorean heptatonic scale<xref ref-type="fn" rid="j_forma-2018-0022_fn_001_w2aab3b7ab1b6b1ab1aaAa">1</xref> [5], [7].</abstract>

Pythagorean Tuning: Pentatonic and Heptatonic Scale

<abstract xmlns="http://www.w3.org/1999/xhtml">In the article we continue in the Mizar system [8], [2] the formalization of fuzzy implications according to the monograph of Baczyński and Jayaram <italic>“Fuzzy Implications”</italic> [1]. We develop a framework of Mizar attributes allowing us for a smooth proving of basic properties of these fuzzy connectives [9]. We also give a set of theorems about the ordering of nine fundamental implications: Łukasiewicz (<italic>I</italic>LK), Gödel (<italic>I</italic>GD), Reichenbach (<italic>I</italic>RC), Kleene-Dienes (<italic>I</italic>KD), Goguen (<italic>I</italic>GG), Rescher (<italic>I</italic>RS), Yager (<italic>I</italic>YG), Weber (<italic>I</italic>WB), and Fodor (<italic>I</italic>FD).This work is a continuation of the development of fuzzy sets in Mizar [6]; it could be used to give a variety of more general operations on fuzzy sets [13]. The formalization follows [10], [5], and [4].</abstract>

Fundamental Properties of Fuzzy Implications

<abstract xmlns="http://www.w3.org/1999/xhtml">We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1].The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring <italic>A</italic> is called the prime spectrum of <italic>A</italic> denoted by Spectrum <italic>A</italic>. A new functor Spec generates Zariski topology to make Spectrum <italic>A</italic> a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism <italic>h</italic> : <italic>A</italic> → <italic>B</italic>, we defined (Spec <italic>h</italic>) : Spec <italic>B</italic> → Spec <italic>A</italic> by (Spec <italic>h</italic>)(𝔭) = <italic>h</italic>−1(𝔭) where 𝔭 <italic>2</italic> Spec <italic>B</italic>.</abstract>

Zariski Topology

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Formalized Mathematics

Volume 26 (2018): Issue 4 (December 2018)

Pythagorean Tuning: Pentatonic and Heptatonic Scale

Fundamental Properties of Fuzzy Implications

Zariski Topology

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