Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

## Reper Algebras

Michal Muzalewski
Warsaw University, Bialystok

### Summary.

We shall describe $n$-dimensional spaces with the reper operation [7, pages 72-79]. An inspiration to such approach comes from the monograph [9] and so-called Leibniz program. Let us recall that the Leibniz program is a program of algebraization of geometry using purely geometric notions. Leibniz formulated his program in opposition to algebraization method developed by Descartes. The Euclidean geometry in Szmielew's approach [0] {SZMIELEW:1} is a theory of structures $\langle S$; $\parallel, \oplus, O \rangle$, where $\langle S$; $\parallel, \oplus, O \rangle$ is Desarguesian midpoint plane and $O \subseteq S\times S\times S$ is the relation of equi-orthogonal basis. Points $o, p, q$ are in relation $O$ if they form an isosceles triangle with the right angle in vertex $a$. If we fix vertices $a, p$, then there exist exactly two points $q, q'$ such that $O(apq)$, $O(apq')$. Moreover $q \oplus q' = a$. In accordance with the Leibniz program we replace the relation of equi-orthogonal basis by a binary operation $\ast : S\times S \rightarrow S$, called the reper operation. A standard model for the Euclidean geometry in the above sense is the oriented plane over the field of real numbers with the reper operations $\ast$ defined by the condition: $a \ast b = q$ iff the point $q$ is the result of rotating of $p$ about right angle around the center $a$.

#### MML Identifier: MIDSP_3

The terminology and notation used in this paper have been introduced in the following articles [10] [12] [3] [4] [2] [5] [1] [11] [6] [8]

#### Contents (PDF format)

1. Substitutions in tuples
2. Reper Algebra Structure and its Properties
3. Reper Algebra and its Atlas

#### Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
[6] Michal Muzalewski. Midpoint algebras. Journal of Formalized Mathematics, 1, 1989.
[7] Michal Muzalewski. \em Foundations of Metric-Affine Geometry. Dzial Wydawnictw Filii UW w Bialymstoku, Filia UW w Bialymstoku, 1990.
[8] Michal Muzalewski. Atlas of midpoint algebra. Journal of Formalized Mathematics, 3, 1991.
[9] Wanda Szmielew. \em From Affine to Euclidean Geometry, volume 27. PWN -- D.Reidel Publ. Co., Warszawa -- Dordrecht, 1983.
[10] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[11] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[12] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.