historical notes on quasigroups

This entry is about terminological and historical issues (and matters of opinion) on binary operations/algebraic structures, in particular nonassociative ones: magmas, quasigroups, racks and quandles.

Historically, one most often considered everywhere-defined binary algebraic operations; however partial operations were also sometimes considered. Oystein Ore termed everywhere defined operations groupoidal, hence a set with one binary operation that is everywhere defined has been called a groupoid. If the operation is not everywhere defined, then according to the textbook of Bruck, people talked about halfgroupoids. In the nnLab, a groupoid is another structure (a small category where all morphisms are isomorphisms). This however comes from Brandt’s groupoids, which were considered as sets with partially defined operations, equivalent to what we call connected groupoids. As Brandt groupoids assumed primacy, what had been the usual groupoids were now renamed magmas (after Nicolas Bourbaki). In categorical universal algebra, “magma” usually means unital magma. Bourbaki coined some other names of common structures, including semigroup and distributive lattice. In choosing the word “magma”, Bourbaki means to suggest amorphous structure, as a weakening of the common and strong notion of a group. There is actually a tradition in such terminology of weakenings for groups: to use terms suggesting loose matter. For example, heap (synonymous with the earlier and similar flock) for a group with forgotten identity, and wrack (or what is more usual nowadays, rack) for a left distributive left quasigroup. Here wrack was introduced by John Conway (cf. the verb “wreck” and the noun “wreckage”), evidently to conjure up the phrase wrack and ruin, but Conway was also punning on the similar spelling and sound in the name of his coauthor Wraith. (Meanwhile, the term “shelf”, introduced by Alissa Crans, would appear to be based on the homophone “rack” which has a different meaning, thus departing from the tradition, or starting a new one.)

Books on binary structures are quite dominated by the study of nonassociative ones, where quasigroups are the most prominent class (including one sided versions, like left quasigroups and their subclasses, e.g., left racks).


Some consider the concept of quasigroup to be an example of centipede mathematics, and uninteresting due to their lack of deep applications (the latter opinion on quasigroups and loops is however becoming obsolete in view of modern examples and applications). For example, one mathematician has written:

The meeting was dominated by algebraic loop theory. It occurred to me that as a way to use your intellectual resources this was very akin in significance to doing a difficult sudoku, a thought that was made very ironic when one speaker started making loops out of what were essentially sudoku squares.

Nonetheless it can be instructive to ponder these concepts, and there are some nontrivial examples.


  • Hala Orlik Pflugfelder, Historical notes on loop theory, Comment. Math. Univ. Carolin. 41,2 (2000) 359–370 pdf
category: algebra

Last revised on July 9, 2016 at 13:23:22. See the history of this page for a list of all contributions to it.