Group theory and generalizations Definition

Bol loop

Definition

A loop is a quasigroup $L$ with an element $e$ such that $ex = xe = x$ for all $x \in L$ . A left Bol loop is a loop $L$ such that $a(b(ac)) = (a(ba))c \quad \text{for all } a, b, c \in L.$ A right Bol loop is a loop $L$ such that $((ca)b)a = c((ab)a) \quad \text{for all } a, b, c \in L.$ A Bol loop is a left Bol loop or a right Bol loop, depending on the convention of the author.

Remarks

Example: every group is a left and a right Bol loop, since both identities follow from associativity. The nonzero octonions under multiplication form a Moufang loop, hence a left and a right Bol loop, and they are not a group.

Non-example: the loop $L = \set{e, a, b, c, d}$ with the multiplication table below, where the row gives the left factor.

$\cdot$	$e$	$a$	$b$	$c$	$d$
$e$	$e$	$a$	$b$	$c$	$d$
$a$	$a$	$e$	$c$	$d$	$b$
$b$	$b$	$c$	$d$	$e$	$a$
$c$	$c$	$d$	$a$	$b$	$e$
$d$	$d$	$b$	$e$	$a$	$c$

Putting $b = e$ in the left Bol identity gives $a(ac) = (aa)c$ . In $L$ , $a(ab) = ac = d$ while $(aa)b = eb = b$ , so $L$ is not left Bol. Putting $b = e$ in the right Bol identity gives $(ca)a = c(aa)$ , and $(ba)a = ca = d$ while $b(aa) = b$ , so $L$ is not right Bol either.

A loop is both left and right Bol exactly when it is a Moufang loop.

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Quasigroup concept
Bol loop definition

Tag	`6BWSNR0PW5`
URI	`m2://6BWSNR0PW5`
Kind	definition
Writing	W0 stub
MSC	20N05
Defines	Bol loop
Uses	Quasigroup
Changed	3 November 2026

Bol loop

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here

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