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Loop

A groupoid ↑ is a non-empty set typeset structure together with a binary operation typeset structure.  Quasigroup is a groupoid typeset structure in which every element is invertible with respect to typeset structure. Equivalently,  if we define left translation typeset structure for typeset structure, 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 typeset structure, a subloop typeset structure is called normal, if typeset structure, typeset structure and typeset structure for all typeset structure (p. 60 [1]). These three conditions are to the following couple of conditions typeset structure and typeset structure for all typeset structure.

The left, middle and right nucleus of a loop typeset structure are defined by
typeset structure,
typeset structure,
typeset structure.
The nucleus of a loop typeset structure is
typeset structure.
Each of these nuclei is an associative subloop of typeset structure (Theorem I.3.5 [2]).

The centrum and center of a loop typeset structure are defined by
typeset structure,
typeset structure.

The center of a loop is a normal subloop.

A loop typeset structure is power associative   if for any typeset structure the subloop generated by typeset structure is a group.

References

[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.

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Štefan Porubský: Loop.

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