Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

## From Loops to Abelian Multiplicative Groups with Zero

Michal Muzalewski
Warsaw University, Bialystok
Wojciech Skaba
Nicolaus Copernicus University, Torun

### Summary.

Elementary axioms and theorems on the theory of algebraic structures, taken from the book [5]. First a loop structure $\langle G, 0, +\rangle$ is defined and six axioms corresponding to it are given. Group is defined by extending the set of axioms with $(a+b)+c = a+(b+c)$. At the same time an alternate approach to the set of axioms is shown and both sets are proved to yield the same algebraic structure. A trivial example of loop is used to ensure the existence of the modes being constructed. A multiplicative group is contemplated, which is quite similar to the previously defined additive group (called simply a group here), but is supposed to be of greater interest in the future considerations of algebraic structures. The final section brings a slightly more sophisticated structure i.e: a multiplicative loop/group with zero: $\langle G, \cdot, 1, 0\rangle$. Here the proofs are a more challenging and the above trivial example is replaced by a more common (and comprehensive) structure built on the foundation of real numbers.

Supported by RPBP.III-24.C6.

#### MML Identifier: ALGSTR_1

The terminology and notation used in this paper have been introduced in the following articles [6] [9] [7] [1] [2] [8] [4] [3]

Contents (PDF format)

#### Bibliography

[1] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[3] Michal Muzalewski. Midpoint algebras. Journal of Formalized Mathematics, 1, 1989.
[4] Michal Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Journal of Formalized Mathematics, 2, 1990.
[5] Wanda Szmielew. \em From Affine to Euclidean Geometry, volume 27. PWN -- D.Reidel Publ. Co., Warszawa -- Dordrecht, 1983.
[6] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[7] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[8] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.
[9] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

Received July 10, 1990