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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