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

Contents (PDF format)

Bibliography

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