On algebraic hierarchies in mathematical repository of Mizar

Abstract

Abstract-Mathematics, especially algebra, uses plenty of structures: groups, rings, integral domains, fields, vector spaces to name a few of the most basic ones. Classes of structures are closely connected -usually by inclusion -naturally leading to hierarchies that has been reproduced in different forms in different mathematical repositories. In this paper we give a brief overview of some existing algebraic hierarchies and report on the latest developments in the Mizar computerized proof assistant system. In particular we present a detailed algebraic hierarchy that has been defined in Mizar and discuss extensions of the hierarchy towards more involved domains. Taking fully formal approach into account we meet new difficulties comparing with its informal mathematical framework.

Ordered Rings and Fields

“…In particular we show that polynomial rings can be ordered in (at least) two different ways [8,5,4,9]. This is the continuation of the development of algebraic hierarchy in Mizar [2,3]. …”

Pythagorean Tuning: Pentatonic and Heptatonic Scale

“…Let H be a heptatonic pythagorean score and fr be an element of H. The heptatonic pythagorean scale of fr in H yielding an element of (the carrier of H) 8 is defined by (Def. 79) it(1) = (the spiral of fifths of (the fourth of fr in H) with fundamental frequency fr in H) (1) and it(2) = (the spiral of fifths of (the fourth of fr in H) with fundamental frequency fr in H) (3) and it(3) = (the spiral of fifths of (the fourth of fr in H) with fundamental frequency fr in H) (5) and it(4) = the fourth of fr in H and it(5) = (the spiral of fifths of (the fourth of fr in H) with fundamental frequency fr in H) (2) and it(6) = (the spiral of fifths of (the fourth of fr in H) with fundamental frequency fr in H) (4) and it (7) = (the spiral of fifths of (the fourth of fr in H) with fundamental frequency fr in H) (6) and it(8) = the octave of (the spiral of fifths of (the fourth of fr in H) with fundamental frequency fr in H) (1) in H.…”

Topological structures as a tool for formal modelling of rough sets

“…In Mizar formalism, axioms are defined as adjectives (called also attributes). The details of the algebraic hierarchy in the Mizar Mathematical Library were presented at FedCSIS conference last year [17].…”