From: aj_dude@my-dejanews.com
Newsgroups: sci.math.research
Subject: [Q] Loops and Quasigroops
Date: Wed, 09 Sep 1998 15:49:46 GMT
I have two questions concerning loops that seem to be easy to people familiar
with non-associative binary systems. Unfortunately, the references to the
questions I found are reviews that are harder available to me. Would someone
give some comments and/or hints if possible (or, if one has a book on his\her
bookshelf, just exgibit in brief some basic ideas of a proof).
Nevertheless, maybe the answers are rather easy, but I am not familiar with
loops...
First, I'll recall some definitions (I believe they are standard, but to
avoid any confusion).
a _groupoid_ (L, *) is a set L equipped by a binary operation *: (x,y)
!--> x*y (or xy simply).
An _isotopy_ of two groupoids (L, *) and (M, . ) is a triple (A,B,C) of
bijections
A: L-->M, B: L--> M, C: L-->M such that
C(x*y) = A(x).B(y) for each x, y from L.
A groupoid is a _quasigroup_ if the equalities ax=b and xa=b have a unique
solution x for any given a,b. A quasigroop is a _loop_ if it also has a
(both-sided) identity element 'e'.
Each (nonempty) quasigroop is isotopic to a loop. Class of loops may be also
regarded as an equationally-presentable algebraic category -- universal
algebras with signature (2,2,2,0) -- (. , \ , /, e) with binary operations:
xy (multiplication); x/y (x divided by y on the left) ; y\x (x divided by
y on the right) with the identities:
xe = ex =x
x(x\y) = x\(xy) = (yx)/x = (y/x)x = y.
A loop is called an _IP-loop_ (loop with inverses) if one has also
(xy)(y\e) = (e/y)(yx) = x.
In IP-loops one has y\e=e/y for each y, so we may put an inversion
operation: y' = e/y.
Then the class of IP-loops may be regarded as equationally presentable with
signature (2,1,0) -- (. , ', e):
xe=ex=x
x'(xy) = (yx)x' = y (then x/y = xy' and y\x = y'x )
A loop is _alternative_ if each its subloop generated by its two elements, is
associative (i.e., generates a subgroup). A loop L is _Moufang loop_ if the
following equivalent conditions hold:
1. Each loop, isotopic to L, is alternative. 2. Each loop, isotopic to L, is
an IP-loop. 3. One of (equivalent) _Moufang identity_ holds: central one:
(xy)(zx) = ( x(yz) ) x; left one: ( (xy)z )y = x( y (zy)) and others.
In particular, each Moufang loop is alternative (and IP-).
So, here are two QUESTIONS:
Q1. Is the class of alternative loops equationally presentable? (i.e., may it
be defined by identities (may be in derived operations), e.g. in alternative
- elastic rings spirit, some kind of (xy)y=x(yy), x(xy)=(xx)y, x(yx)=(xy)x?
Q2. For which IP-loops the following identity holds (recall, y' is the
inverse of y):
(xy)(y'z) = xz? Any? Moufang? If so, what about more general identity
(-ies) in an arbitrary (not necessarily IP- one):
(xy)(y\z) = xz, (x/y)(yz) = xz?
Sorry if the questions are easy (for specialists), but again, I'm not
familiar with that, and my current attempts failed. May be this information
is in original Moufang's works?
Thanks,
--
Alexei Z.
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