Namespaces
Variants
Actions

Kernel of a loop

From Encyclopedia of Mathematics
Revision as of 20:04, 29 October 2016 by Richard Pinch (talk | contribs) (MSC 20N05)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

2020 Mathematics Subject Classification: Primary: 20N05 [MSN][ZBL]

The set of elements of the loop that are simultaneously left-, right- and middle-associative (or, equivalently, the intersection of the left, right and middle kernels of the loop). An element $a$ of a loop is called left-associative if $a(bc) = (ab)c$ for any $b,c$ in the loop. The set of left-associative elements is called the left kernel of the loop. Right- and middle-associative elements and the corresponding kernels are defined similarly. Left and right kernels can also be defined for quasi-groups, but only loops have a non-empty middle kernel. All the kernels of a loop are subgroups of it. All three kernels of a loop with the invertibility property ($IP$-loop) coincide, and for Moufang loops they form, in addition, a normal subloop (see Loop).

How to Cite This Entry:
Kernel of a loop. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_a_loop&oldid=39526
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article