Nonempty alternative quasigroup equals alternative loop

From Groupprops
Jump to: navigation, search
This article gives a proof/explanation of the equivalence of multiple definitions for the term alternative loop
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a quasigroup:

  1. It is nonempty and is an alternative magma under its multiplication.
  2. It is an alternative loop, i.e., it is an algebra loop and is an alternative magma under its multiplication.

Note that an algebra loop is a quasigroup with a two-sided neutral element, so the above simply says that for a nonempty quasigroup, alternativity guarantees the existence of a neutral element.

Related facts