nLab Bol loop

A loop $(L,\cdot)$ (in the algebraic sense, of a quasigroup with unit element) is a left Bol loop (resp. right Bol loop) if all triples $a,b,c$ of its elements satisfy:

• the left Bol identity

  $$(a (b a)) c = a (b (a c))$$

• and resp. right Bol identity

  $$((b a) c) a = b ((a c) a)$$

Equivalently, the left multiplication operators $L_a$ and right multiplication operators $R_a$ satisfy the corresponding properties

$$L_{a(b a)} = L_a L_b L_a$$

or

$$R_{(a c) a} = R_a R_c R_a$$

A loop is a Moufang loop iff it is simultaneously a left Bol loop and a right Bol loop.

A core of a right Bol loop $(L,\cdot)$ is the binary algebraic structure $(L,+)$ where

$$a + b = (a\cdot b^{-1})\cdot a$$

(due to nonassociativity of the multiplication, pay attention to the order of brackets!)

Every isotopy of right Bol loops induces an isomorphism between the corresponding cores.

The core of a Moufang loop has the following properties:

$$a + a = a$$

$$a + (a + b) = b$$

$$a+(b+c)=(a+b)+c$$

$$(a+b)^{-1}=a^{-1}+b^{-1}$$

$$(a+b)\cdot c = a\cdot c + b\cdot c$$

$$c\cdot(a+b) = c\cdot a + c\cdot b$$

Related notions: Moufang loop, Bol algebra?, identities of Bol-Moufang type

• Wikipedia, Bol loop
• D. A. Robinson, Bol loops, Trans. Amer. Math. Soc. 123 (1966); Bol quasigroups, Publ. Math. Debrecen 19 (1972), 151–153
• A. Vanžurová, Cores of Bol loops and symmetric groupoids, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2005, no. 3, 153–164 pdf
• V. Volenec, Grupoidi, kvazigrupe i petlje, Zagreb 1982

category: algebra