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A groupoid ↑ is a non-empty set together with a binary operation . Quasigroup is a groupoid in which every element is invertible with respect to . Equivalently, if we define left translation for , and similarly the right translation, then a groupoid with all left and right translations biject is a quasigroup. Loop is a quasigroup with a two-sided identity element .
Given a loop , a subloop is called normal, if , and for all (p. 60 [1]). These three conditions are to the following couple of conditions and for all .
The left, middle and right nucleus of a loop are defined by
,
,
.
The nucleus of a loop is
.
Each of these nuclei is an associative subloop of (Theorem I.3.5 [2]).
The centrum and center of a loop are defined by
,
.
The center of a loop is a normal subloop.
A loop is power associative if for any the subloop generated by is a group.
[1] | Bruck, R. H. (1971). A Survey of Binary Systems. Springer Verlag. |
[2] | Pflugfelder, H. O. (1990). Quasigroups and Loops: Introduction. Berlin: Heldermann Verlag. |
[3] | Belousov, V. D. (1967). Foundations of the Theory of Quasigroups and Loops (Russian). Moscow: Izdat. Nauka. |
Cite this web-page as:
Štefan Porubský: Loop.