From Loops to Abelian Multiplicative Groups with Zero 1
Micha l Muzalewski, Wojciech Skaba · 1990
Summary. Elementary axioms and theorems on the theory of algebraic structures, taken from the book [5]. First a loop structure 〈G,0,+ 〉 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: 〈G,·,1,0〉. 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.