Background
Basic Properties
Structural Results
Examples
Buchsteiner Loops
Michael K. Kinyon
Department of Mathematics
University of Denver
Joint work with P. Csörgő and A. Drápal
See arXiv.org/abs/0708.2358
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Buchsteiner Loops
Michael K. Kinyon
Department of Mathematics
University of Denver
Joint work with P. Csörgő and A. Drápal
See arXiv.org/abs/0708.2358
Other papers are coming soon where
\
{authors} = {Drápal}
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Definition, Motivation & History
Main Questions
Definition
A loop is said to be Buchsteiner loop if it satisfies the following
identity:
x\(xy · z) = (y · zx)/x
or equivalently, the following implication:
xy · z = xu
⇐⇒
Michael K. Kinyon
y · zx = ux
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Definition, Motivation & History
Main Questions
Motivation
Buchsteiner loops are the only “mystery” variety among
those obtained by nuclear identification (cf. preceding talk
by Jedlička)
A Buchsteiner loop is a G-loop, i.e., isomorphic to all of its
loop isotopes (we will discuss this later). Such varieties, if
highly structured (e.g., CC-loops), are always of interest.
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Definition, Motivation & History
Main Questions
History
H. H. Buchsteiner, O nekotorom klasse binarnych lup, Mat.
Issled. 39 (1976), 54–66.
Proved such a loop is isotopically invariant if and only if it
satisfies (xy )\((xy · z)u) = (z(u · yx))/(yx). He called
these “i-loops”.
Did not address if (what we call) Buchsteiner loops are
always i-loops.
Proved an i-loop modulo its nucleus is an abelian group.
Gave one example, which turns out to be CC.
Asked if i-loops are G-loops.
A. S. Basarab, Osborn’s G-loops, QRS 1 (1994), 51-55.
Proved that i-loops are G-loops.
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Definition, Motivation & History
Main Questions
Questions
Are Buchsteiner loops isotopically invariant?
Do there exist examples which are not CC?
If so, do there exist examples for which the factor by the
nucleus is not an elementary abelian 2-group?
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Characterizations
Elementary Properties
Interaction with Other Varieties
Weak inverse properties
Translations
The Buchsteiner identity is equivalent to each of the following:
−1
L−1
x Rz Lx = Rx Rzx for all x, z ∈ Q ,
Rx−1 Ly Rx
=
L−1
x Lxy
for all x, y ∈ Q .
(1)
(2)
Lx
= R(Q) for all x ∈ Q ,
Rx R(Q)
(3)
x
Lx LR
(Q) = L(Q) for all x ∈ Q .
(4)
where R(Q) = {Rx |x ∈ Q}, L(Q) = {Lx |x ∈ Q}.
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Characterizations
Elementary Properties
Interaction with Other Varieties
Weak inverse properties
Autotopisms
A loop is Buchsteiner if and only if each
(Lx , Rx−1 , Lx Rx−1 )
is an autotopism.
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Characterizations
Elementary Properties
Interaction with Other Varieties
Weak inverse properties
The Nucleus
ax · y = a · xy
⇔
xy · a = x · ya
⇔
ya · x = y · ax
And so the three nuclei coincide.
Also: the nucleus of a Buchsteiner loop is a normal subloop.
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Characterizations
Elementary Properties
Interaction with Other Varieties
Weak inverse properties
Left and right inner mappings
Recall the usual generators:
L(x, y) = L−1
xy Lx Ly ,
−1
R(x, y) = Ryx
Rx Ry
In a Buchsteiner loop Q,
R(x, y) = [Lx , Ry ] = L(y, x)−1 .
In particular, the left and right inner mapping groups coincide.
Also: L(a, b) is a pseudoautomorphism with companion
((1/b)/a) · ab
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Characterizations
Elementary Properties
Interaction with Other Varieties
Weak inverse properties
Bol, Moufang, et al
The following are equivalent for a Buchsteiner loop Q:
Q has the left inverse property
Q has the right inverse property
Q is left alternative
Q is right alternative
Q is flexible
Q is left Bol
Q is right Bol
Q is Moufang
Q is extra
Conversely, extra loops are Buchsteiner loops.
Michael K. Kinyon
Buchsteiner Loops
Characterizations
Elementary Properties
Interaction with Other Varieties
Weak inverse properties
Background
Basic Properties
Structural Results
Examples
Osborn and conjugacy closed loops
Osborn loops:
x(yz · x) = ((x · yx)/x) · zx
or
(x · yz)x = xy · (x\(xz · x))
These include Moufang loops and CC-loops as special cases.
Theorem
For a loop Q, any two of the following statements implies the
third.
Q is an Osborn loop,
Q is a Buchsteiner loop,
Each x 2 ∈ N(Q).
Corollary
A CC-loop is Buchsteiner iff each x 2 ∈ N(Q).
Michael K. Kinyon
Buchsteiner Loops
Characterizations
Elementary Properties
Interaction with Other Varieties
Weak inverse properties
Background
Basic Properties
Structural Results
Examples
WIP
Belousov’s notation for inverses:
J(x) · x = 1
x · I(x) = 1
Weak inverse property (WIP):
x · I(yx) = I(y )
or
J(xy ) · x = J(y)
Theorem
A WIP Buchsteiner loop is CC.
(But there are Buchsteiner CC-loops of order 16 which do not
have the WIP.)
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Characterizations
Elementary Properties
Interaction with Other Varieties
Weak inverse properties
WWIP
Doubly weak inverse property (WWIP):
J 2 (x) · J(yx) = J(y)
or
I(xy ) · I 2 (x) = I(y )
Part of a heirarchy of weak inverse properties considered
by various authors, e.g., Karkliňš & Karkliň, Keedwell &
Shcherbacov, etc.
Important because if (α, β, γ) is an autotopism, then so are
(J 2 βI 2 , JγI, JαI) and (IγJ, I 2 αJ 2 , IβJ).
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
Characterizations
Elementary Properties
Interaction with Other Varieties
Weak inverse properties
WWIP
Theorem
Every Buchsteiner loop has the WWIP.
The proof is a series of technical lemmas, mostly involving
the expression η(x) = I(x) · x = x · J(x).
I had automated proofs (using Prover9) last autumn, but I
only recently figured out how to “humanize” a proof.
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
G-loops
Abelian quotients
Refining the quotient
G-loops
Theorem
Every Buchsteiner loop is a G-loop.
−1
Proof: L−1
I(η(x)) ILJ 4 (x) RJ 4 (x) J is an isomorphism from Q to the
isotope at x. QED
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
G-loops
Abelian quotients
Refining the quotient
G-loops
Theorem
Every Buchsteiner loop is a G-loop.
−1
Proof: L−1
I(η(x)) ILJ 4 (x) RJ 4 (x) J is an isomorphism from Q to the
isotope at x. QED
:-)
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
G-loops
Abelian quotients
Refining the quotient
G-loops
Theorem
Every Buchsteiner loop is a G-loop.
−1
Proof: L−1
I(η(x)) ILJ 4 (x) RJ 4 (x) J is an isomorphism from Q to the
isotope at x. QED
:-)
This is really shown by using WWIP to get new autotopisms
from (Lx , Rx−1 , Lx Rx−1 ). Composing those autotopisms gives the
isomorphism above.
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
G-loops
Abelian quotients
Refining the quotient
Abelian quotients
Corollary (Buchsteiner)
If Q is a Buchsteiner loop, then Q/N is an abelian group.
Corollary
A Buchsteiner loop Q is an Al and Ar -loop, i.e., every left and
right inner mapping is an automorphism.
Proof: The companion (J(b)/a) · ab of L(a, b) lives in the
nucleus.
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
G-loops
Abelian quotients
Refining the quotient
Q/N has exponent 4
Theorem
If Q is a Buchsteiner loop, then Q/N is an abelian group of
exponent 4.
The proof uses associator calculus, especially the following
characterization:
Lemma
If Q is a loop with Q/N a group, then Q is Buchsteiner iff
[x, y, z]x = [y, z, x]−1
for all x, y , z ∈ Q.
Here, as usual, [x, y, z] = (x · yz)\(xy · z).
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
CC or not CC? That is the question
Exponent 4?
Nilpotence
CC or not CC?
If Q is a Buchsteiner loop with |Q| < 32, then Q is a
CC-loop.
There are 44 Buchsteiner loops of order 32 which are not
conjugacy closed (Drápal & Kunen)
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
CC or not CC? That is the question
Exponent 4?
Nilpotence
Easy example (Csörgő & Drápal)
Let R be a ring of characteristic 2 satisfying xyz = yzx, but in
which some a, b, c ∈ R satisfy abc 6= acb. (There is a
semigroup ring of order 8 that will do the job.) Then
(x, y , z) · (u, v , w) = (x + u + zv , y + v + zw, z + w)
defines a nonCC Buchsteiner loop on R 3 .
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
CC or not CC? That is the question
Exponent 4?
Nilpotence
Can exponent 4 be achieved?
In the previous examples, Q/N has exponent 2.
Theorem
If |Q| < 64, then Q/N(Q) is an elementary abelian 2-group.
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
CC or not CC? That is the question
Exponent 4?
Nilpotence
Can exponent 4 be achieved?
In the previous examples, Q/N has exponent 2.
Theorem
If |Q| < 64, then Q/N(Q) is an elementary abelian 2-group.
Theorem
There exists a Buchsteiner loop Q of order 64 where Q/N is
not elementary abelian.
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
CC or not CC? That is the question
Exponent 4?
Nilpotence
Subliminal advertisement
There are examples of Buchsteiner loops where the
relationship between the nilpotency class of the loop and the
nilpotency class of the inner mapping group is quite interesting
...
Michael K. Kinyon
Buchsteiner Loops
Background
Basic Properties
Structural Results
Examples
CC or not CC? That is the question
Exponent 4?
Nilpotence
Subliminal advertisement
There are examples of Buchsteiner loops where the
relationship between the nilpotency class of the loop and the
nilpotency class of the inner mapping group is quite interesting
...
But for that you should listen to the next talk!
Michael K. Kinyon
Buchsteiner Loops