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Definitions. A loop is a set L with a binary operation  such that 1) for each a,b in L, there is a unique x in L such that ax=b and there is a unique y in L such that ya=b. 2) there exists a unique element 1 in L such that for every a in L, 1a=a=a1.

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Definitions

Definitions


Definitions

A loop is a set L with a binary operation  such that

1) for each a,b in L, there is a unique x in L such that ax=b and there is a unique y in L such that ya=b.

2) there exists a unique element 1 in L such that for every a in L, 1a=a=a1.

We will simply write ab in place of ab.

A (right) Bol loop is a loop L in which the identity [(xy)z]y = x[(yz)y] holds for all x,y,z in L.


Definitions

The center Z(L) of a loop L is the set of elements z in L such that for all a, b in L,

a(zb)=(az)b=(za)b=z(ab)=(ab)z=a(bz)

The center of a loop is a normal subloop (the kernel of a loop homomorphism).

A loop is centrally nilpotent of class n if the upper central series of L

{1}Z=Z1Z2…Zn…,

where Zi+1 is the full preimage in L of the center of L/Zi, stabilizes with Zn=L,

but Zn-1L.


Definitions

For a, b, c in L,

The commutator (a,b) is defined by

ab=(ba)(a,b)

The associator (a,b,c) is defined by

(ab)c=[a(bc)](a,b,c)

The commutator/associator subloop L is the subloop of L generated by all commutators and all associators.


Definitions

If L is of nilpotence class 2, then Z2=L, so that L/Z(L) is an abelian group.

Therefore Z(L) contains all commutators and all associators.

If zZ(L), then

(za,b)=(a,zb)=(ab)

(za,b,c)=(a,zb,c)=(a,b,zc)=(a,b,c)

Also commutators and associators can be pulled to the right in all equations.

E.g., [x(x,y)][y(x,y,z)]=(xy)(x,y)(x,y,z)




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