Bol loop

A loop (L,)(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,ca,b,c of its elements satisfy:

  • the left Bol identity

    (a(ba))c=a(b(ac)) (a (b a)) c = a (b (a c))
  • and resp. right Bol identity

    ((ba)c)a=b((ac)a) ((b a) c) a = b ((a c) a)

Equivalently, the left multiplication operators L aL_a and right multiplication operators R aR_a satisfy the corresponding properties

L a(ba)=L aL bL a L_{a(b a)} = L_a L_b L_a


R (ac)a=R aR cR a 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,)(L,\cdot) is the binary algebraic structure (L,+)(L,+) where

a+b=(ab 1)a 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 = a
a+(a+b)=b a + (a + b) = b
a+(b+c)=(a+b)+c a+(b+c)=(a+b)+c
(a+b) 1=a 1+b 1 (a+b)^{-1}=a^{-1}+b^{-1}
(a+b)c=ac+bc (a+b)\cdot c = a\cdot c + b\cdot c
c(a+b)=ca+cb 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

Last revised on November 11, 2013 at 18:28:02. See the history of this page for a list of all contributions to it.