# Loops with Universal and Semi-Universal Flexibility

Riley Britten  Department of Mathematics
University of Denver
Denver, CO 80208 USA<EMAIL_ADDRESS>,  Michael Kinyon  Department of
Mathematics
University of Denver
Denver, CO 80208 USA<EMAIL_ADDRESS>,  Kenneth Kunen  Department of
Mathematics
University of Wisconsin
Madison, WI 57306 USA  and  J. D. Phillips  Department of Mathematics &
Computer Science
Northern Michigan University
Marquette, Michigan 49855 USA<EMAIL_ADDRESS>
###### Abstract.

We study loops which are universal (that is, isotopically invariant) with
respect to the property of flexibility ($xy\cdot x=x\cdot yx$). We also weaken
this to semi-universality, that is, loops in which every left and right
isotope is flexible, but not necessarily every isotope. One of our main
results is that universally flexible, inverse property loops are Moufang
loops. On the other hand, semi-universally flexible, inverse property loops
are diassociative. We also examine the relationship between universally
flexible loops and middle Bol loops. The paper concludes with some open
problems.

###### Key words and phrases:

loop, Moufang loop, flexibility, inverse property, isotopy

###### 2000 Mathematics Subject Classification:

20N05

In memory of Ken Kunen (1943–2020)

##  1\. The story of this paper

This paper was started in 2003 by MK, KK and JDP. Most of the results herein,
especially Theorems 2.2 and 2.3, were proved in that year; only a few were
proved more recently. However, we were unable to settle what we have now
stated as Problem 7.1. This was an unsatisfactory state of affairs because if
the answer turned out to be negative (which we suspect is not the case), it
would imply that in Theorem 2.3, we simply had a more complicated proof of the
diassociativity of Moufang loops than already existing ones. Thus we set the
paper aside in the hope that we could resolve the issue.

Years passed. We occasionally would dig up our notes, think about the problem,
get nowhere, and set it aside once again. And of course, we all got busy with
other projects and other priorities.

In 2022, MK suggested to his PhD student RB that searching for an example to
resolve Problem 7.1 would make a nice topic to fit into their dissertation. As
discussed in §7, that particular search was unsuccessful, but the methodology
led to other useful examples, especially those appearing in §4 and §5.

Ken Kunen passed away in 2020. It is difficult to overstate the influence that
he had on our work. It was thanks to him that both MK and JDP have focused
their research careers on the use of automated deduction tools in quasigroup
theory and other parts of algebra. After mulling it over for a long time, we
decided that the best tribute we could give to him was to finally submit the
paper for publication, leaving Problem 7.1 open with the hope that someone
will be able to resolve it. We also note that Ken deserves full coauthor
status because all of §6, the proof of Theorem 2.3, was entirely his work.

##  2\. Introduction

A _loop_ $(Q,\cdot,1)$ is a set $Q$ with a binary operation $\cdot$ such that
there is an identity element $1$ and for each $a\in Q$, the mappings
$L(a):Q\to Q;x\mapsto a\cdot x$ and $R(a):Q\to Q;x\mapsto x\cdot a$ are
bijections of $Q$. An equivalent and also useful definition is: A loop
$(Q,\cdot,\backslash,/,1)$ is a set $Q$ with three binary operations
$\cdot,\backslash,/:Q\times Q\to Q$ and an element $1\in Q$ satisfying the
equations:

$\displaystyle x\backslash(x\cdot y)=y=(y\cdot x)/x$ $\displaystyle
x\cdot(x\backslash y)=y=(y/x)\cdot x$ $\displaystyle 1\cdot x=x=x\cdot 1\,.$

Basic references for loop theory are [1], [3], [11].

For a loop $Q$ and fixed elements $a,b\in Q$, the binary operations defined by

$x\cdot_{a,b}y:=(x/a)(b\backslash y)\qquad x\backslash_{a,b}\ y:=b\
[(x/a)\backslash y]\qquad x/_{a,b}\ y:=[x/(b\backslash y)]a$

give $Q$ a new loop structure $(Q,\cdot_{a,b},\backslash_{a,b},/_{a,b},ba)$
called a (principal) _loop isotope_ of $Q$. _Right_ loop isotopes are those
with $a=1$ and _left_ loop isotopes are those with $b=1$. Much of loop theory
is devoted to the study of _universal_ or _isotopically invariant_ properties:

######  Definition 2.1.

If $\mathsf{P}$ is a property of loops, then a loop $Q$ is:

  * •

_left semi-universally_ $\mathsf{P}$ iff all left loop isotopes satisfy
$\mathsf{P}$;

  * •

_right semi-universally_ $\mathsf{P}$ iff all right loop isotopes satisfy
$\mathsf{P}$;

  * •

_semi-universally_ $\mathsf{P}$ if it is both left and right semi-universally
$\mathsf{P}$;

  * •

_universally_ $\mathsf{P}$ iff all loop isotopes satisfy $\mathsf{P}$.

“Universally $\mathsf{P}$” may or may not be strictly stronger than “semi-
universally $\mathsf{P}$”, which may or may not be strictly stronger than
“$\mathsf{P}$”. For example, all Moufang loops are universally Moufang.
Moufang loops are alternative and satisfy the inverse property (IP). Every
semi-universally IP loop is Moufang, and every semi-universally alternative
loop is Moufang. Thus, “universally IP” $=$ “semi-universally IP” is strictly
stronger than “IP”, and “universally alternative” $=$ “semi-universally
alternative” is strictly stronger than “alternative”.

We interject here the definitions of the above terms, plus a few others used
in the sequel:

A loop is _Moufang_ iff it satisfies the following four equivalent identities:

$\begin{array}[]{rccrc}M1:&(x\cdot yz)x=xy\cdot zx&&M2:&x(yz\cdot x)=xy\cdot
zx\\\ N1:&(xy\cdot z)y=x(y\cdot zy)&&N2:&x(y\cdot xz)=(xy\cdot x)z\end{array}$

For an element $x$ of a loop, if $1/x=x\backslash 1$, their common value is
denoted by $x^{-1}$. In a loop in which $1/x=x\backslash 1$ holds for all $x$,
the _left inverse property_ (LIP), the _right inverse property_ (RIP), and the
_antiautomorphic inverse property_ (AAIP) are defined by the identities

$\displaystyle x\backslash y$ $\displaystyle=x^{-1}y$ $\displaystyle\text{ or
}L(x)^{-1}=L(x^{-1})$ (LIP) $\displaystyle x/y$ $\displaystyle=xy^{-1}$
$\displaystyle\text{ or }R(y)^{-1}=R(y^{-1})$ (RIP) $\displaystyle(xy)^{-1}$
$\displaystyle=y^{-1}x^{-1}\,.$ (AAIP)

Any two of these identities implies the third, and a loop satisfying all three
is said to have the _inverse property_ (IP). (In fact, all three properties
can be defined in such a way that the property $1/x=x\backslash 1$ is a
theorem, but the equivalent definition above is more convenient for our
purposes.)

Note that every AAIP loop, and in particular every IP loop, is isomorphic to
its opposite loop, so whenever we know that an identity holds in such a loop,
we automatically have the mirror of that identity.

A loop is _diassociative_ iff any two of its elements generate a group, and
_power-associative_ iff any one of its elements generate a group. Three well-
known weakenings of diassociativity are the _left alternative property_
(LAlt), the _right alternative property_ (RAlt), and the _flexible property_
(Flex) defined by

$\displaystyle x\cdot xy$ $\displaystyle=x^{2}y$ $\displaystyle\text{ or
}L(x^{2})=L(x)^{2}$ (LAlt) $\displaystyle xy\cdot y$ $\displaystyle=xy^{2}$
$\displaystyle\text{ or }R(y^{2})=R(y)^{2}$ (RAlt) $\displaystyle x\cdot yx$
$\displaystyle=xy\cdot x$ $\displaystyle\text{ or }[R(x),L(x)]=I$ (Flex)

The term “flexible” has been translated into Russian as “ эластичны” and
translated back into English as “elastic”. In a flexible loop, we have
$1/x=x\backslash 1$ for all $x$. A loop is _alternative_ iff it satisfies both
LAlt and RAlt. As usual in loop theory, we often write such equations as
statements about the left and right multiplications, since proofs are often
simpler when expressed in terms of some permutation group of the loop.

A _left Bol loop_ is one satisfying the identity $x(y\cdot xz)=(x\cdot yx)z$
or equivalently, $L(x\cdot yx)=L(x)L(y)L(x)$. All left Bol loops are power
associative and satisfy the LIP and LAlt. All left Bol loops are universally
left Bol, and any loop which is either (semi-)universally LIP or
(semi-)universally LAlt is left Bol. The opposite loop of a left Bol loop is a
_right Bol loop_ defined by the identity $(xy\cdot z)y=x(yz\cdot y)$ or
$R(yz\cdot y)=R(y)R(z)R(y)$; so loops which are (semi-)universally RIP or
(semi-)universally RAP are right Bol.

A loop which is universally AAIP is called a _middle Bol loop_ ; these are
characterized by the identity $x((yz)\backslash x)=(x/z)(y\backslash x)$ or
equivalently, the mirror of this identity. The operation $x\circ y:=x/y^{-1}$
in a left Bol loop defines a middle Bol loop. A loop is Moufang iff it
satisfies any two, and hence all three, of the Bol properties.

Another interesting variety properly containing the Moufang loops consists of
loops which are universally flexible (UF); these have been studied in [12, 13,
14, 15, 16, 17, 18]. To express this property as an equation, we expand
$x\cdot_{u,v}(y\cdot_{u,v}x)=(x\cdot_{u,v}y)\cdot_{u,v}x$ in terms of the left
and right division operations to get the following three equivalent equations:

$\displaystyle(x/u)[v\backslash((y/u)(v\backslash x))]$ $\displaystyle=$
$\displaystyle[((x/u)(v\backslash y))/u](v\backslash x)$ (1) $\displaystyle
R(u)^{-1}R(v\backslash x)L(v)^{-1}L(x/u)$ $\displaystyle=$ $\displaystyle
L(v)^{-1}L(x/u)R(u)^{-1}R(v\backslash x)$ (2)
$\displaystyle[R(u)^{-1}R(v\backslash x),\;L(v)^{-1}L(x/u)]$ $\displaystyle=$
$\displaystyle I\ \ .$ (3)

Such loops need not have the IP, but in [17, 18], Syrbu studied UF IP loops,
showing that they share many properties with Moufang loops. This turns out not
to be surprising in view of our first result:

######  Theorem 2.2.

Every universally flexible, IP loop is Moufang.

The equations defining the variety of semi-universally flexible (SUF) loops
are:

$\displaystyle x(v\backslash[y(v\backslash x)])$ $\displaystyle=$
$\displaystyle[x(v\backslash y)](v\backslash x)$ (4) $\displaystyle
R(v\backslash x)L(v)^{-1}L(x)$ $\displaystyle=$ $\displaystyle
L(v)^{-1}L(x)R(v\backslash x)$ and $\displaystyle(x/u)[(y/u)x]$
$\displaystyle=$ $\displaystyle([(x/u)y]/u)x$ (6) $\displaystyle
R(u)^{-1}R(x)L(x/u)$ $\displaystyle=$ $\displaystyle L(x/u)R(u)^{-1}R(x)$ (7)

Moufang’s Theorem implies that Moufang loops are diassociative. This has been
generalized [6] to a wider class of loops called _ARIF_ loops, which are
flexible loops satisfying the (equivalent) identities

$R(xyx)R(y)=R(x)R(yxy)\qquad L(xyx)L(y)=L(x)L(yxy)$

In this paper we offer a (possibly) different generalization:

######  Theorem 2.3.

Every SUF IP loop is diassociative.

In fact, this theorem will turn out to be a corollary of a theorem asserting
that every loop in an even wider class, which we call _stepping up_ , is
diassociative.

##  3\. Semi-universal flexibility

The variety of left SUF loops is defined by the identity (6). Replacing $x$
with $xu$ and $y$ with $yu$ and then rearranging, we obtain

$(x\cdot yu)/u=(x(y\cdot xu))/(xu)\,.$ (8)

This equation implies the following.

######  Lemma 3.1.

For $a,b,c$ in a left SUF loop $Q$, the following are equivalent: (i) $a\cdot
bc=ab\cdot c$; (ii) $ab\cdot ac=a(b\cdot ac)$; (iii) $ab\cdot(a\backslash
c)=a(b(a\backslash c))$.

We will soon show that left SUF loops are power-associative. Until then, we
use left-association to define powers of elements: $x^{n}:=1L(x)^{n}$ for all
$n\in\mathbb{Z}$.

######  Lemma 3.2.

For all $x,y$ in a left SUF loop $Q$ and for all $n\in\mathbb{Z}$, $x\cdot
yx^{n}=xy\cdot x^{n}$.

###### Proof.

From Lemma 3.1, for each $n\in\mathbb{Z}$, $x\cdot yx^{n}=xy\cdot x^{n}$ iff
$x\cdot yx^{n+1}=xy\cdot x^{n+1}$ iff $x\cdot yx^{n-1}=xy\cdot x^{n-1}$. The
case $n=0$ being trivial, an induction argument completes the proof. ∎

######  Theorem 3.3.

Left SUF loops are power-associative.

###### Proof.

Let $Q$ be a left SUF loop. We will prove $x^{m}x^{n}=x^{m+n}$ for all
$m,n\in\mathbb{Z}$. For $m=0$, there is nothing to prove. Taking $y=x^{m}$ in
Lemma 3.2, we have

$x\cdot x^{m}x^{n}=x^{m+1}x^{n}.$ (9)

Assume $x^{m}x^{n}=x^{m+n}$ for some $m,n$. Then $x^{m+n+1}=xx^{m+n}=x\cdot
x^{m}x^{n}=x^{m+1}x^{n}$ by (9). By induction, we have the desired result for
all nonnegative $m$. Replacing $m$ with $-m$, we can then induct again to
complete the argument. ∎

Now we work in SUF loops, that is, loops that are both left and right SUF. We
will use power-associativity without explicit reference. We also have
available both Lemma 3.1 and its dual, so that for $a,b,c$ in an SUF loop $Q$,

$\displaystyle ab\cdot c=a\cdot bc$ $\displaystyle\iff ab\cdot ac=a(b\cdot
ac)$ $\displaystyle\iff(ac\cdot b)c=ac\cdot bc$ (10) $\displaystyle\iff
ab\cdot(a\backslash c)=a(b(a\backslash c))$
$\displaystyle\iff((a/c)b)c=(a/c)\cdot bc$

######  Theorem 3.4.

Let $Q$ be an SUF loop. For all $x,y\in Q$ and for all $m,n\in\mathbb{Z}$,
$x^{m}y\cdot x^{n}=x^{m}\cdot yx^{n}$, that is,
$L(x^{m})R(x^{n})=R(x^{n})L(x^{m})$.

###### Proof.

For $m=0$ or $n=0$, there is nothing to prove. By repeated use of (10), we
have that $x^{m}y\cdot x^{n}=x^{m}\cdot yx^{n}$ iff $x^{m}y\cdot
x^{im+n}=x^{m}\cdot yx^{im+n}$ iff $x^{m+j(im+n)}y\cdot
x^{n}=x^{m+j(im+n)}\cdot yx^{n}$ for all $i,j\in\mathbb{Z}$. For given $m,n$,
we may choose $i$ so that $im+n>0$ and then choose $j$ so that $m+j(im+n)>0$.
Thus it is sufficient to consider the case $m,n>0$. Flexibility is the subcase
$m=n$, so assume that $n>m$. Write $n=qm+r_{1}$ for some
$q,r_{1}\in\mathbb{Z}$ with $0\leq r_{1}<m$. By repeated application of (10),
$x^{m}y\cdot x^{n}=x^{m}\cdot yx^{n}$ iff $x^{m}y\cdot x^{r_{1}}=x^{m}\cdot
yx^{r_{1}}$. Repeating this argument, there exists $0\leq r_{2}<r_{1}$ such
that $x^{m}y\cdot x^{n}=x^{m}\cdot yx^{n}$ iff $x^{r_{2}}y\cdot
x^{r_{1}}=x^{r_{2}}\cdot yx^{r_{1}}$. We can thus eventually reduce the
exponents to the case covered by Lemma 3.2 or its dual. Thus $x^{m}y\cdot
x^{n}=x^{m}\cdot yx^{n}$ holds. The case $m>n$ is proven similarly. ∎

######  Example 3.5.

Table 1 is a left semi-universally flexible loop that is not right semi-
universally flexible. Here $1(3\backslash[2(3\backslash 1)])\neq[1(3\backslash
2)](3\backslash 1)$. This loop is simple and has trivial nuclei. In fact, its
only proper subloops are the two-element subloops. This loop satisfies LIP and
LAlt (which are the same property here). In addition, every mapping
$R(x)L(x)^{-1}$ is an automorphism; this property is stronger than
flexibility. It is easy to check that none of the loops of order 5 have proper
left semi-universal flexibility, and so this example is the smallest such
loop. A computer search using Mace4 shows that this is the only example of
order 6 up to isomorphism.

$\begin{array}[]{c|cccccc|}\cdot&0&1&2&3&4&5\\\ \hline\cr 0&0&1&2&3&4&5\\\
1&1&0&3&2&5&4\\\ 2&2&5&0&4&3&1\\\ 3&3&4&5&0&1&2\\\ 4&4&2&1&5&0&3\\\
5&5&3&4&1&2&0\\\ \hline\cr\end{array}$ Table 1. A left semi-universally
flexible loop that is not right semi-universally flexible

##  4\. UF IP loops are Moufang

In a right SUF LIP loop, we may replace $v$ with $v^{-1}$ in (4) to obtain

$\displaystyle x\cdot v(y\cdot vx)$ $\displaystyle=(x\cdot vy)\cdot vx$ (11)
$\displaystyle R(vx)L(v)L(x)$ $\displaystyle=L(v)L(x)R(vx)$ (12)

Similarly, in a left SUF RIP loop satisfying, replacing $u$ with $u^{-1}$ in
(6) gives

$\displaystyle xu\cdot(yu\cdot x)=(xu\cdot y)u\cdot x$ (13) $\displaystyle
R(u)R(x)L(xu)=L(xu)R(u)R(x)$ (14)

A loop satisfying (11) is clearly flexible (take $v=1$). Taking $y=v^{-1}$ in
(11), we obtain $x\cdot v(v^{-1}\cdot vx)=x\cdot vx$. Cancelling twice, we get
$v^{-1}\cdot vx=x$, which is the LIP. We have proved

######  Lemma 4.1.

  1. (1)

A loop satisfies (11) if and only if it is a left SUF LIP loop.

  2. (2)

A loop satisfies (13) if and only if it is a right SUF RIP loop.

######  Lemma 4.2.

Every SUF IP loop is alternative.

###### Proof.

Replace $v$ with $yx^{-1}$ in (12) to get

$R(y)L(yx^{-1})L(x)=L(yx^{-1})L(x)R(y)\,.$ (15)

In this, replace $x$ with $xy$ to get

$R(y)L(x^{-1})L(xy)=L(x^{-1})L(xy)R(y)\,.$ (16)

Apply both (15) and (16) to the identity element $1$ to obtain, respectively,

$\displaystyle x\cdot yx^{-1}y$ $\displaystyle=xyx^{-1}\cdot y$ $\displaystyle
xy\cdot x^{-1}y$ $\displaystyle=xyx^{-1}\cdot y\,.$

Thus $x\cdot yx^{-1}y=xy\cdot x^{-1}y$. By Lemma 3.1 and the LIP, $x\cdot
yy=xy\cdot y$, that is RAlt holds. Since $Q$ has the IP, LAlt also holds. ∎

In a UF IP loop, we may replace $u$ with $u^{-1}$ and $v$ with $v^{-1}$ in
(1), to obtain

$\displaystyle xu\cdot v(yu\cdot vx)$ $\displaystyle=(xu\cdot vy)u\cdot vx$
(17) $\displaystyle R(u)R(vx)L(v)L(xu)$ $\displaystyle=L(v)L(xu)R(u)R(vx)$
(18)

Replacing $v$ with $vx^{-1}$, $u$ with $x^{-1}u$, and then $x$ with $x^{-1}$
gives:

$\displaystyle u[vx\cdot(y\cdot xu)v]$ $\displaystyle=[u(vx\cdot y)\cdot xu]v$
(19) $\displaystyle R(xu)R(v)L(vx)L(u)$ $\displaystyle=L(vx)L(u)R(xu)R(v)$
(20)

######  Lemma 4.3.

[17] A loop satisfies (17) or (19) if and only if it is a universally
flexible, IP loop.

###### Proof.

We have established the “if" direction. If, say, (19) holds, then taking $u=1$
gives (13) (after relabelling), and taking $v=1$ gives (11). Then apply Lemma
4.1. ∎

###### Proof of Theorem 2.2.

We use the IP and alternative properties without comment. Take $v=x^{-1}$ in
(20) and rearrange to get

$R(x^{-1})L(u)R(x)=R(u^{-1}x^{-1})L(u)R(xu)$

Multiply on the right by $R(v)$ and apply (20) again to obtain

$\displaystyle R(x^{-1})L(u)R(x)R(v)$ $\displaystyle=$ $\displaystyle
R((xu)^{-1})L((vx)^{-1})[L(vx)L(u)R(xu)R(v)]$ $\displaystyle=$ $\displaystyle
R((xu)^{-1})L((vx)^{-1})R(xu)R(v)L(vx)L(u).$

Apply both sides to $x^{-1}v^{-1}=(vx)^{-1}$:

$\displaystyle[u\cdot x^{-1}v^{-1}x^{-1}]R(x)R(v)$ $\displaystyle=$
$\displaystyle[(vx)^{-1}\cdot(vx)^{-1}(xu)^{-1}]R(xu)R(v)L(vx)L(u)$
$\displaystyle=$ $\displaystyle[(vx)^{-2}]R((xu)^{-1})R(xu)R(v)L(vx)L(u)$
$\displaystyle=$ $\displaystyle[(x^{-1}v^{-1})^{2}v]L(vx)L(u)$
$\displaystyle=$ $\displaystyle x^{-1}L((vx)^{-1})L(vx)L(u)$ $\displaystyle=$
$\displaystyle uR(x^{-1})$

Now replace $x$ with $x^{-1}$, $v$ with $v^{-1}$, and rearrange to get

$uR(xvx)=uR(x)R(v)R(x),$

which is the Moufang identity (N1). ∎

##  5\. Middle Bol loops

Since left Bol loops are universal for both LIP and LAlt, it is natural to
guess that the variety of middle Bol loops (universal for AAIP) is related to
the variety of UF loops, perhaps in the sense that one variety is contained in
the other [15, 16, 17, 18]. It is true that small middle Bol loops are UF and
vice versa. However, it turns out that neither variety is contained in the
other.

######  Example 5.1.

This example shows that a middle Bol loop need not be flexible and a flexible
middle Bol loop need not be SUF. Table 2 is a flexible, middle Bol loop such
that the isotope given by $x\circ y:=x(8\backslash y)$ is not flexible. In
particular, $2\circ(0\circ 2)\neq(2\circ 0)\circ 2$.

$\begin{array}[]{c|cccccccccccccccc|}\cdot&0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15\\\
\hline\cr 0&0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15\\\
1&1&0&3&2&5&4&7&6&9&8&11&10&13&12&15&14\\\
2&2&3&0&1&6&7&4&5&10&11&8&9&14&15&12&13\\\
3&3&2&1&0&7&6&5&4&11&10&9&8&15&14&13&12\\\
4&4&5&6&7&0&1&2&3&12&13&14&15&8&9&10&11\\\
5&5&4&7&6&1&0&3&2&13&12&15&14&9&8&11&10\\\
6&6&7&4&5&2&3&0&1&14&15&12&13&10&11&8&9\\\
7&7&6&5&4&3&2&1&0&15&14&13&12&11&10&9&8\\\
8&8&9&10&11&12&13&14&15&0&1&4&5&2&3&7&6\\\
9&9&8&11&10&13&12&15&14&1&0&5&4&3&2&6&7\\\
10&10&11&8&9&14&15&12&13&4&5&0&1&7&6&2&3\\\
11&11&10&9&8&15&14&13&12&5&4&1&0&6&7&3&2\\\
12&12&13&14&15&8&9&10&11&2&3&7&6&0&1&4&5\\\
13&13&12&15&14&9&8&11&10&3&2&6&7&1&0&5&4\\\
14&14&15&12&13&10&11&8&9&7&6&2&3&4&5&0&1\\\
15&15&14&13&12&11&10&9&8&6&7&3&2&5&4&1&0\\\ \hline\cr\end{array}$
Table 2. A flexible, middle Bol loop with a nonflexible isotope

######  Example 5.2.

The following construction slightly generalizes an example of Shelekhov [14,
16]. Let $V$ be a vector space over a field of characteristic not 2, and let
$\phi:V\to\mathrm{Aut}(V\wedge V)$ be a homomorphism from the additive group
of $V$ into the automorphism group of the wedge product $V\wedge V$. On
$W:=V\oplus(V\wedge V)$, define

$(x,\omega)\cdot(y,\eta):=(x+y,\ \omega+\phi_{-x}(\eta)+\phi_{-2x}(x\wedge
y))$

Then $(W,\cdot)$ is a universally flexible loop, but this loop does not, in
general, have the AAIP. (Shelekhov’s example is the case where
$V=\mathbb{R}^{2}$, $V\wedge V\cong\mathbb{R}$, and $\phi_{x}=e^{x}$.)

######  Example 5.3.

This example was found by using central extensions as described later. Table 3
shows a commutative UF loop which does not have the AAIP. Up to isomorphism,
this loop has only two loop isotopes. The other one, given by $x\circ
y=x(3\backslash y)$ for example, has the AAIP, but is neither middle Bol nor
commutative.

This loop is also minimal in the sense that all of its proper subloops and all
of its factor loops by nontrivial normal subloops are middle Bol loops.

Interestingly, both this loop and its other isotope satisfy the semi-
automorphic inverse property $(xyx)^{-1}=x^{-1}y^{-1}x^{-1}$. Thus this
property is universal in this loop.

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3 | 3 | 4 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 1 | 2 | 15 | 16 | 5 | 6 | 27 | 28 | 23 | 24 | 31 | 32 | 19 | 20 | 29 | 30 | 17 | 18 | 25 | 26 | 21 | 22
4 | 4 | 3 | 8 | 7 | 10 | 9 | 12 | 11 | 14 | 13 | 2 | 1 | 16 | 15 | 6 | 5 | 28 | 27 | 24 | 23 | 32 | 31 | 20 | 19 | 30 | 29 | 18 | 17 | 26 | 25 | 22 | 21
5 | 5 | 6 | 9 | 10 | 1 | 2 | 14 | 13 | 3 | 4 | 16 | 15 | 8 | 7 | 12 | 11 | 21 | 22 | 25 | 26 | 17 | 18 | 29 | 30 | 19 | 20 | 31 | 32 | 23 | 24 | 27 | 28
6 | 6 | 5 | 10 | 9 | 2 | 1 | 13 | 14 | 4 | 3 | 15 | 16 | 7 | 8 | 11 | 12 | 22 | 21 | 26 | 25 | 18 | 17 | 30 | 29 | 20 | 19 | 32 | 31 | 24 | 23 | 28 | 27
7 | 7 | 8 | 11 | 12 | 14 | 13 | 1 | 2 | 16 | 15 | 3 | 4 | 6 | 5 | 10 | 9 | 23 | 24 | 27 | 28 | 29 | 30 | 17 | 18 | 31 | 32 | 19 | 20 | 21 | 22 | 25 | 26
8 | 8 | 7 | 12 | 11 | 13 | 14 | 2 | 1 | 15 | 16 | 4 | 3 | 5 | 6 | 9 | 10 | 24 | 23 | 28 | 27 | 30 | 29 | 18 | 17 | 32 | 31 | 20 | 19 | 22 | 21 | 26 | 25
9 | 9 | 10 | 13 | 14 | 3 | 4 | 16 | 15 | 7 | 8 | 6 | 5 | 12 | 11 | 2 | 1 | 31 | 32 | 29 | 30 | 27 | 28 | 25 | 26 | 23 | 24 | 21 | 22 | 19 | 20 | 17 | 18
10 | 10 | 9 | 14 | 13 | 4 | 3 | 15 | 16 | 8 | 7 | 5 | 6 | 11 | 12 | 1 | 2 | 32 | 31 | 30 | 29 | 28 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 | 19 | 18 | 17
11 | 11 | 12 | 1 | 2 | 16 | 15 | 3 | 4 | 6 | 5 | 7 | 8 | 10 | 9 | 14 | 13 | 19 | 20 | 17 | 18 | 25 | 26 | 27 | 28 | 21 | 22 | 23 | 24 | 31 | 32 | 29 | 30
12 | 12 | 11 | 2 | 1 | 15 | 16 | 4 | 3 | 5 | 6 | 8 | 7 | 9 | 10 | 13 | 14 | 20 | 19 | 18 | 17 | 26 | 25 | 28 | 27 | 22 | 21 | 24 | 23 | 32 | 31 | 30 | 29
13 | 13 | 14 | 15 | 16 | 8 | 7 | 6 | 5 | 12 | 11 | 10 | 9 | 1 | 2 | 3 | 4 | 30 | 29 | 32 | 31 | 24 | 23 | 22 | 21 | 28 | 27 | 26 | 25 | 18 | 17 | 20 | 19
14 | 14 | 13 | 16 | 15 | 7 | 8 | 5 | 6 | 11 | 12 | 9 | 10 | 2 | 1 | 4 | 3 | 29 | 30 | 31 | 32 | 23 | 24 | 21 | 22 | 27 | 28 | 25 | 26 | 17 | 18 | 19 | 20
15 | 15 | 16 | 5 | 6 | 12 | 11 | 10 | 9 | 2 | 1 | 14 | 13 | 3 | 4 | 7 | 8 | 26 | 25 | 22 | 21 | 20 | 19 | 32 | 31 | 18 | 17 | 30 | 29 | 28 | 27 | 24 | 23
16 | 16 | 15 | 6 | 5 | 11 | 12 | 9 | 10 | 1 | 2 | 13 | 14 | 4 | 3 | 8 | 7 | 25 | 26 | 21 | 22 | 19 | 20 | 31 | 32 | 17 | 18 | 29 | 30 | 27 | 28 | 23 | 24
17 | 17 | 18 | 27 | 28 | 21 | 22 | 23 | 24 | 31 | 32 | 19 | 20 | 30 | 29 | 26 | 25 | 13 | 14 | 11 | 12 | 7 | 8 | 5 | 6 | 16 | 15 | 3 | 4 | 2 | 1 | 9 | 10
18 | 18 | 17 | 28 | 27 | 22 | 21 | 24 | 23 | 32 | 31 | 20 | 19 | 29 | 30 | 25 | 26 | 14 | 13 | 12 | 11 | 8 | 7 | 6 | 5 | 15 | 16 | 4 | 3 | 1 | 2 | 10 | 9
19 | 19 | 20 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 17 | 18 | 32 | 31 | 22 | 21 | 11 | 12 | 5 | 6 | 16 | 15 | 3 | 4 | 1 | 2 | 13 | 14 | 9 | 10 | 8 | 7
20 | 20 | 19 | 24 | 23 | 26 | 25 | 28 | 27 | 30 | 29 | 18 | 17 | 31 | 32 | 21 | 22 | 12 | 11 | 6 | 5 | 15 | 16 | 4 | 3 | 2 | 1 | 14 | 13 | 10 | 9 | 7 | 8
21 | 21 | 22 | 31 | 32 | 17 | 18 | 29 | 30 | 27 | 28 | 25 | 26 | 24 | 23 | 20 | 19 | 7 | 8 | 16 | 15 | 13 | 14 | 2 | 1 | 11 | 12 | 9 | 10 | 5 | 6 | 3 | 4
22 | 22 | 21 | 32 | 31 | 18 | 17 | 30 | 29 | 28 | 27 | 26 | 25 | 23 | 24 | 19 | 20 | 8 | 7 | 15 | 16 | 14 | 13 | 1 | 2 | 12 | 11 | 10 | 9 | 6 | 5 | 4 | 3
23 | 23 | 24 | 19 | 20 | 29 | 30 | 17 | 18 | 25 | 26 | 27 | 28 | 22 | 21 | 32 | 31 | 5 | 6 | 3 | 4 | 2 | 1 | 13 | 14 | 9 | 10 | 11 | 12 | 7 | 8 | 16 | 15
24 | 24 | 23 | 20 | 19 | 30 | 29 | 18 | 17 | 26 | 25 | 28 | 27 | 21 | 22 | 31 | 32 | 6 | 5 | 4 | 3 | 1 | 2 | 14 | 13 | 10 | 9 | 12 | 11 | 8 | 7 | 15 | 16
25 | 25 | 26 | 29 | 30 | 19 | 20 | 31 | 32 | 23 | 24 | 21 | 22 | 28 | 27 | 18 | 17 | 16 | 15 | 1 | 2 | 11 | 12 | 9 | 10 | 5 | 6 | 8 | 7 | 3 | 4 | 13 | 14
26 | 26 | 25 | 30 | 29 | 20 | 19 | 32 | 31 | 24 | 23 | 22 | 21 | 27 | 28 | 17 | 18 | 15 | 16 | 2 | 1 | 12 | 11 | 10 | 9 | 6 | 5 | 7 | 8 | 4 | 3 | 14 | 13
27 | 27 | 28 | 17 | 18 | 31 | 32 | 19 | 20 | 21 | 22 | 23 | 24 | 26 | 25 | 30 | 29 | 3 | 4 | 13 | 14 | 9 | 10 | 11 | 12 | 8 | 7 | 5 | 6 | 16 | 15 | 1 | 2
28 | 28 | 27 | 18 | 17 | 32 | 31 | 20 | 19 | 22 | 21 | 24 | 23 | 25 | 26 | 29 | 30 | 4 | 3 | 14 | 13 | 10 | 9 | 12 | 11 | 7 | 8 | 6 | 5 | 15 | 16 | 2 | 1
29 | 29 | 30 | 25 | 26 | 23 | 24 | 21 | 22 | 19 | 20 | 31 | 32 | 18 | 17 | 28 | 27 | 2 | 1 | 9 | 10 | 5 | 6 | 7 | 8 | 3 | 4 | 16 | 15 | 13 | 14 | 11 | 12
30 | 30 | 29 | 26 | 25 | 24 | 23 | 22 | 21 | 20 | 19 | 32 | 31 | 17 | 18 | 27 | 28 | 1 | 2 | 10 | 9 | 6 | 5 | 8 | 7 | 4 | 3 | 15 | 16 | 14 | 13 | 12 | 11
31 | 31 | 32 | 21 | 22 | 27 | 28 | 25 | 26 | 17 | 18 | 29 | 30 | 20 | 19 | 24 | 23 | 9 | 10 | 8 | 7 | 3 | 4 | 16 | 15 | 13 | 14 | 1 | 2 | 11 | 12 | 5 | 6
32 | 32 | 31 | 22 | 21 | 28 | 27 | 26 | 25 | 18 | 17 | 30 | 29 | 19 | 20 | 23 | 24 | 10 | 9 | 7 | 8 | 4 | 3 | 15 | 16 | 14 | 13 | 2 | 1 | 12 | 11 | 6 | 5

Table 3. A UF loop which does not satisfy AAIP

##  6\. Diassociativity

To prove that a loop is diassociative, it is natural to introduce the
following definition:

######  Definition 6.1.

Given elements $x_{1},\ldots,x_{n}$ in a loop, let $A(x_{1},\ldots,x_{n})$
abbreviate the statement that $x_{1},\ldots,x_{n}$ _associates_ , in the sense
that all associated products of the elements in that order are equal.

The number of such products is the Catalan number
$C_{n-1}=\frac{(2n-2)!}{n!(n-1)!}$. For example, when $n=4$, $C_{4-1}=5$, and
$A(w,x,y,z)$ iff

$w\cdot(x\cdot yz)=w\cdot(xy\cdot z)=wx\cdot yz=(w\cdot xy)\cdot z=(wx\cdot
y)\cdot z\ \ .$

######  Definition 6.2.

In an IP loop, let $H(n)$ abbreviate the statement that for all $a,b$ in the
loop, $A(x_{1},\ldots,x_{n})$ holds for all
$x_{1},\ldots,x_{n}\in\\{1,a,a^{-1},b,b^{-1}\\}$.

######  Lemma 6.3.

In any IP loop $Q$:

  * •

$H(n)\to H(m)$ whenever $m\leq n$.

  * •

$Q$ is diassociative iff $H(n)$ holds for all $n$.

  * •

$H(3)$ holds iff $Q$ is flexible and alternative.

Moufang’s original proof [8, 9] of what we now call Moufang’s Theorem was
inductive. Moufang loops satisfy a number of additional properties which allow
for a proof of Moufang’s Theorem without an explict induction; this is the
approach taken in [3]. More recently, Drápal found a simpler proof in the
spirit of Moufang’s original proof [4].

Our proof of diassociativity will prove $H(n)$ by induction on $n$. Definition
6.5 below introduces a class of loops in which one can step up easily from
$H(n)$ to $H(n+1)$. This class includes all SUF IP loops, and hence all
Moufang loops.

######  Definition 6.4.

If $X$ is a subset of an IP loop $Q$, then $X$ _strongly associates_ iff
$A(x,y,z)$ for all $x,y,z\in X\cup X^{-1}$. Let $\mathit{SA}(x,y,z)$
abbreviate the statement that $\\{x,y,z\\}$ strongly associates.

In Moufang loops, $A(x,y,z)$ is equivalent to $SA(x,y,z)$.

The notation $\mathit{SA}(x,y,z)$ does not imply that $x,y,z$ are distinct. If
$Q$ is also alternative and flexible, then every $\\{a,b\\}$ strongly
associates, and the assertion $\mathit{SA}(a,b,c)$ is naturally written as
$6\cdot 8=48$ equations, since there are $6$ permutatons of $\\{a,b,c\\}$ and
then $2^{3}=8$ choices of which of $a,b,c$ to invert. However, by IP, these
$48$ come in $24$ equivalent pairs, since $A(x,y,z)\leftrightarrow
A(z^{-1},y^{-1},x^{-1})$; for example, $A(a,b^{-1},c^{-1})$ is equivalent to
$A(c,b,a^{-1})$.

######  Definition 6.5.

A _stepping-up loop_ is an alternative, flexible IP loop $Q$ satisfying the
implication $\mathit{SA}(x,y,z)\to\mathit{SA}(x,y,yz)$.

Note that the property $\mathit{SA}(x,y,z)$ is invariant under permutations of
$x,y,z$ and under replacing any of $x,y,z$ by its inverse. Thus in a stepping-
up loop, if $\mathit{SA}(x,y,z)$ holds, then also
$\mathit{SA}(x^{-1},y^{-1},y^{-1}z^{-1})$ , $\mathit{SA}(x,y,zy)$ ,
$\mathit{SA}(x,xy,z)$, $\mathit{SA}(x,x^{-1}y,z)$, etc.

Stepping up loops form a quasivariety because the implication
$\mathit{SA}(x,y,z)\to\mathit{SA}(x,y,yz)$ can be written as a collection of
quasi-identities. Moufang loops are stepping up; this is a consequence of
Moufang’s Theorem or can be shown directly. There exist some Steiner loops
which are stepping up, but not all of them are.

######  Lemma 6.6.

Every stepping-up loop is diassociative.

###### Proof.

We shall prove by induction that $H(n)$ holds for all $n$. We already have
$H(3)$ by the IP, Flex, LAlt and RAlt. Now, we fix $n\geq 3$ and assume
$H(n)$; we shall prove $H(n+1)$, so fix $a,b$, and fix
$x_{0},x_{1},\ldots,x_{n}\in\\{1,a,a^{-1},b,b^{-1}\\}$; we shall prove
$A(x_{0},x_{1},\ldots,x_{n})$. Note that $H(n)$ implies
$\mathit{SA}(x_{1}\cdots x_{j},\;y_{1}\cdots y_{k},\;z_{1}\cdots z_{\ell})$
whenever $j+k+\ell\leq n$ and all
$x_{1},\ldots,x_{j},y_{1},\ldots,y_{k},\allowbreak
z_{1},\ldots,z_{\ell}\in\\{1,a,a^{-1},b,b^{-1}\\}$.

In view of $H(n)$, there are $n$, not $C_{n}$, products that we must prove are
equal, namely the $P_{\ell}:=(x_{0}\cdots x_{\ell})\cdot(x_{\ell+1}\cdots
x_{n})$ for $0\leq\ell<n$.

$P_{0}=P_{1}$ and $P_{n-2}=P_{n-1}$: To prove $P_{0}=P_{1}$, Without loss of
generality, $x_{0}=a$, so we must prove $a\cdot(x_{1}\cdots
x_{n})=(a_{0}x_{1})\cdot(x_{2}\cdots x_{n})$. This is clear from ALT+IP if
$x_{1}\in\\{1,a,a^{-1}\\}$, so Without loss of generality, $x_{1}=b$. Then the
result follows from $\mathit{SA}(a,b,\;x_{2}\cdots x_{n})$, which in turn
follows from $\mathit{SA}(a,b,1)$ and the stepping-up property.

Now, we prove by induction on $\ell$ that $P_{\ell}=P_{0}$, so we fix $\ell$,
assume that $P_{0}=\cdots=P_{\ell-1}$, and we prove $P_{\ell}=P_{0}$. We may
assume that $2\leq\ell\leq n-2$. Without loss of generality, $x_{\ell}=a$. We
first attempt to prove $P_{\ell}=P_{\ell-1}$; that is,

$(x_{0}\cdots x_{\ell-1}a)\cdot(x_{\ell+1}\cdots x_{n})=(x_{0}\cdots
x_{\ell-1})\cdot(ax_{\ell+1}\cdots x_{n})\ \ .$

There are three cases.

Case 1: at least one of $x_{0},x_{\ell-1},x_{\ell+1},x_{n}$ is $1$; then just
apply $H(n)$.

Case 2: at least one of $x_{0},x_{\ell-1},x_{\ell+1},x_{n}$ is $a$ or
$a^{-1}$; then we can verify $\mathit{SA}(x_{0}\cdots
x_{\ell-1},\;a,\;x_{\ell+1}\cdots x_{n})$; for example, if
$x_{\ell+1}=a^{-1}$, we have $\mathit{SA}(x_{0}\cdots
x_{\ell-1},\;a,\;x_{\ell+2}\cdots x_{n})$ from $H(n)$, and then
$\mathit{SA}(x_{0}\cdots x_{\ell-1},\;a,\;x_{\ell+1}\cdots x_{n})$ by the
stepping-up property.

Case 3: assume that $x_{0},x_{\ell-1},x_{\ell+1},x_{n}\in\\{b,b^{-1}\\}$. Now,
we prove instead $P_{\ell}=P_{0}$. Without loss of generality, $x_{0}=b$, so
we must prove

$(bx_{1}\cdots x_{\ell})\cdot(x_{\ell+1}\cdots x_{n})=b\cdot(x_{1}\cdots
x_{\ell}x_{\ell+1}\cdots x_{n})\ \ .$

But now we use $\mathit{SA}(b,\;x_{1}\cdots x_{\ell},\;x_{\ell+1}\cdots
x_{n})$, which follows (using $x_{\ell+1}\in\\{b,b^{-1}\\}$) from
$\mathit{SA}(b,\;x_{1}\cdots x_{\ell},\;x_{\ell+2}\cdots x_{n})$, and this in
turn follows from $H(n)$. ∎

######  Lemma 6.7.

Every SUF IP loop is a stepping-up loop.

###### Proof.

Assume $\mathit{SA}(a,b,c)$. We need to show $\mathit{SA}(a,b,bc)$. We can
write out $\mathit{SA}(a,b,bc)$ explicitly as 48 equations, but, as remarked
above, the IP reduces this to 24, since we can just write the ones for which
$bc$ is not inverted. But also, the hypotheses are symmetric in $a/a^{-1}$
(since $\mathit{SA}(a,b,c)\leftrightarrow\mathit{SA}(a^{-1},b,c)$), so in
verifying $\mathit{SA}(a,b,bc)$, we we need only write the equations in which
$a$ is not inverted. Thus, we need to verify the following twelve:

  1. 1.

$A(a,b,bc):$ $ab\cdot bc=a\cdot bbc.$

  2. 2.

$A(b,a,bc):$ $ba\cdot bc=b\cdot abc.$

  3. 3.

$A(a,bc,b):$ $abc\cdot b=a\cdot bcb.$

  4. 4.

$A(b,bc,a):$ $bbc\cdot a=b\cdot bca.$

  5. 5.

$A(bc,a,b):$ $bca\cdot b=bc\cdot ab.$

  6. 6.

$A(bc,b,a):$ $bcb\cdot a=bc\cdot ba.$

  7. 7.

$A(a,b^{-1},bc):$ $ab^{-1}\cdot bc=a\cdot b^{-1}bc.$

  8. 8.

$A(b^{-1},a,bc):$ $b^{-1}a\cdot bc=b^{-1}\cdot abc.$

  9. 9.

$A(a,bc,b^{-1}):$ $abc\cdot b^{-1}=a\cdot bcb^{-1}.$

  10. 10.

$A(b^{-1},bc,a):$ $b^{-1}bc\cdot a=b^{-1}\cdot bca.$

  11. 11.

$A(bc,a,b^{-1}):$ $bca\cdot b^{-1}=bc\cdot ab^{-1}.$

  12. 12.

$A(bc,b^{-1},a):$ $bcb^{-1}\cdot a=bc\cdot b^{-1}a.$

Now, (10) is immediate from the IP. To prove (2), apply
$L(b)R(c^{-1})R(bc)=R(c^{-1})R(bc)L(b)$ to $ac$. To prove (8), apply
$R(bc)=R(c)L(b)R(c^{-1})R(bc)L(b^{-1})$ to $b^{-1}a$.

Observe that once we have proved an equation, symmetry of the hypotheses
yields a number of other associations. For example, given (8),
$A(b^{-1},a,bc)$, we also have $A(c^{-1},a,cb)$ (interchanging $b,c$), and
then mirror symmetry yields $A(bc,a,c^{-1})$. More formally, the justification
for this symmetry is that (8) really establishes $\forall
x,y,z\,[\mathit{SA}(x,y,z)\to A(y^{-1},x,yz)]$. Thus, we get
$\mathit{SA}(a,b,c)\leftrightarrow\mathit{SA}(a,c,b)\to A(c^{-1},a,cb)$, and
we likewise get
$\mathit{SA}(a,b,c)\leftrightarrow\mathit{SA}(a^{-1},c^{-1},b^{-1})\to
A(c,a^{-1},c^{-1}b^{-1})\leftrightarrow A(bc,a,c^{-1})$.

To prove (5), apply $L(bc)R(c)R(b)=R(c)R(b)L(bc)$ to $ac^{-1}$. The right side
is $bc\cdot ab$. The left side is $((bc\cdot ac^{-1})\cdot c)\cdot b=bca\cdot
b$ by $A(bc,a,c^{-1})$.

We next prove (6). First note that

$bca^{-1}\cdot ba=b\cdot(ca^{-1}\cdot ba)=b\cdot(ca^{-1}b\cdot a)=(bc\cdot
a^{-1}b)\cdot a=(bca^{-1}\cdot b)\cdot a\,.$

For the first ‘=’, apply $[L(b),R(a^{-1})R(ba)]=I$ to $c$. For the second ‘=’,
use $A(ca^{-1},b,a)$, which follows from (8) by symmetry. For the third ‘=’,
apply $[L(b),R(a^{-1}b)R(a)]=I$ to $c$. For the fourth ‘=’, use
$A(bc,a^{-1},b)$, which follows from (5). Now apply
$L(bc)=L(bca^{-1})R(a^{-1})L(ac^{-1}b^{-1})L(bc)R(a)$ to $ba$. The left side
is $bc\cdot ba$. Since $(ba)L(bca^{-1})=(bca^{-1}\cdot b)\cdot a$, the right
side is $bcb\cdot a$.

To prove (3), note that $abc\cdot b=ab\cdot cb=a\cdot bcb$. We are using
$A(ab,c,b)$ and $A(a,b,cb)$, which follow, by symmetry from (2) and (6),
respectively.

To prove (4): Note that $b\cdot a^{-1}cb=ba^{-1}\cdot cb$ (i.e.,
$A(b,a^{-1},cb)$) follows from (5) and symmetry. Thus, applying
$L(b)R(b^{-1}c^{-1})R(bcb)=R(b^{-1}c^{-1})R(bcb)L(b)$ to $a^{-1}cb$, we get
$(ba^{-1})\cdot(bcb)=b\cdot(a^{-1}\cdot bcb)$, and hence,

$(ab^{-1})\cdot(b\cdot(a^{-1}\cdot bcb))=bcb\,.$

Note that $a^{-1}\cdot bcb=a^{-1}bc\cdot b$ (i.e., $A(a^{-1},bc,b)$) by (3)
and symmetry. Apply $L(ab^{-1})R(b^{-1})R(a)=R(b^{-1})R(a)L(ab^{-1})$ to
$b\cdot(a^{-1}\cdot bcb)=b\cdot(a^{-1}bc)\cdot b$. The left side is
$(bcb)R(b^{-1})R(a)=bca$. The right side is
$(ab^{-1})\cdot((b\cdot(a^{-1}bc))\cdot a)=(ab^{-1})\cdot((ba^{-1}b\cdot
c)\cdot a)$ since $A(b,a^{-1}b,c)$ holds by (3) and symmetry. So, we get:

$(ab^{-1})\cdot((ba^{-1}b\cdot c)\cdot a)=bca\,.$

Finally, observe that $ba^{-1}b\cdot c=ba^{-1}\cdot bc$ (i.e.,
$A(ba^{-1},b,c)$) by (6) and symmetry. Apply
$R(a)L(ab^{-1})L(b)=L(ab^{-1})L(b)R(a)$ to $ba^{-1}b\cdot c$. The right side
is $bbc\cdot a$. Since $(ba^{-1}b\cdot c)R(a)L(ab^{-1})=bca$, the left side is
$b\cdot bca$.

We next verify (12): First observe that

$a\cdot(b^{-1}\cdot c^{-1}b^{-1}a)=a\cdot b^{-1}c^{-1}b^{-1}\cdot
a=(ab^{-1}c^{-1}\cdot b^{-1})\cdot a=ab^{-1}\cdot c^{-1}b^{-1}a\,.$

The first two ‘=’s use flexibility plus $A(b^{-1},c^{-1}b^{-1},a)$ and
$A(a,b^{-1}c^{-1},b^{-1})$, which are symmetric variants of (3). The third
applies $[L(ab^{-1}),R(b^{-1})R(a)]=I$ to $c^{-1}$. Next we get

$[bc\cdot(b^{-1}\cdot c^{-1}b^{-1}a)]\cdot(a^{-1}bc)=bcb^{-1}$

by applying $L(a)R(a^{-1}bc)L(a^{-1})L(bc)=L(bc)R(a^{-1}bc)$ to $b^{-1}\cdot
c^{-1}b^{-1}a$. The left side is $(ab^{-1}\cdot
c^{-1}b^{-1}a)R(a^{-1}bc)L(a^{-1})L(bc)=bcb^{-1}$. Finally,

$bc\cdot
b^{-1}a=(b^{-1})R(a)L(bc)=(b^{-1})R(c^{-1}b^{-1}a)L(bc)R(a^{-1}bc)R(a)=\\\
(bcb^{-1})R(a)=(bcb^{-1})\cdot a\,.$

To prove (9), note that $abc\cdot b^{-1}=ab\cdot cb^{-1}=a\cdot bcb^{-1}$,
using $A(ab,c,b^{-1})$ and $A(a,b,cb^{-1})$, which follow, by symmetry from
(8) and (12), respectively.

To prove (11), apply $L(b)R(c)L(b^{-1})L(bc)=L(bc)R(c)$ to $ab^{-1}c^{-1}$.
Using the fact that $A(b,ab^{-1},c)$ (from (9) and symmetry), the left side is
$bc\cdot ab^{-1}$. The right side is $(bc\cdot ab^{-1}c^{-1})\cdot c=bca\cdot
b^{-1}$, since

$bc\cdot
ab^{-1}c^{-1}=aR(b^{-1})R(c^{-1})L(bc)=aL(bc)R(b^{-1})R(c^{-1})=(bca\cdot
b^{-1})\cdot c^{-1}\,.$

To prove (7), we need to show $ab^{-1}\cdot bc=ac$. We begin by showing

$c^{-1}b\cdot cab^{-1}=(c^{-1}\cdot bca)\cdot b^{-1}=c^{-1}\cdot(b\cdot
cab^{-1})\,.$

For the first ‘=’, apply
$L(c^{-1})R(b^{-1})L(c)L(c^{-1}b)=L(c^{-1}b)R(b^{-1})$ to $ca$. The right side
is $(c^{-1}b\cdot ca)\cdot b^{-1}=(c^{-1}\cdot bca)\cdot b^{-1}$, using
$A(c^{-1},b,ca)$, which follows from (8) and symmetry. The left side is
$c^{-1}b\cdot cab^{-1}$. For the second ‘=’, apply
$L(bc)L(c^{-1})=R(b)L(bc)L(c^{-1})R(b^{-1})$ to $ab^{-1}$. The left side is
$c^{-1}\cdot(bc\cdot ab^{-1})=c^{-1}\cdot(b\cdot cab^{-1})$, using
$A(b,c,ab^{-1})$, which follows from (11) and symmetry. The right side is
$(c^{-1}\cdot bca)\cdot b^{-1}$. We shall now get

$(ab^{-1}\cdot bc)\cdot ab^{-1}=ab^{-1}\cdot(b\cdot cab^{-1})=a\cdot
cab^{-1}=ac\cdot ab^{-1}\,,$

and (7) will follow by cancelling the $ab^{-1}$. To prove the second ‘=’,
apply $R(cab^{-1})L(c)L(ab^{-1})=L(c)L(ab^{-1})R(cab^{-1})$ to $c^{-1}b$. The
right side is $a\cdot cab^{-1}$. Since
$(c^{-1}b)R(cab^{-1})=c^{-1}\cdot(b\cdot cab^{-1})$, the left side is
$ab^{-1}\cdot(b\cdot cab^{-1})$. The first ‘=’ follows by FLEX and
$A(b,c,ab^{-1})$, which follows from (11) and symmetry. The third ‘=’ follows
from $A(a,c,ab^{-1})$, which follows from (2) and symmetry.

To verify (1), start with

$ac^{-1}a\cdot b^{-1}=a\cdot c^{-1}ab^{-1}=(ac^{-1}b^{-1}\cdot ba)\cdot
b^{-1}\,.$

The first ‘=’ is by $A(a,c^{-1}a,b^{-1})$ (using (3) and symmetry). For the
second ‘=’, apply $R(ba)R(b^{-1})L(a)=L(a)R(ba)R(b^{-1})$ to $c^{-1}b^{-1}$,
using $c^{-1}b^{-1}\cdot ba=c^{-1}a$ (by (7) and symmetry) to evaluate the
left side. Now cancel the $b^{-1}$ and divide by $ac^{-1}b^{-1}$ to get
$bca^{-1}\cdot ac^{-1}a=ba$, so that $(ac^{-1})R(a)L(bca^{-1})=ba$. Then,
applying $R(bc)L(bca^{-1})=R(a)L(bca^{-1})R(a^{-1})R(bc)$ to $ac^{-1}$ yields:

$bca^{-1}\cdot(ac^{-1}\cdot bc)=bbc\,.$

Now, apply $L(bca^{-1})L(a)R(bc)=R(bc)L(bca^{-1})L(a)$ to $ac^{-1}$, to get

$ab\cdot bc=a\cdot(bca^{-1}\cdot(ac^{-1}\cdot bc))=a\cdot bbc\,.$

∎

Theorem 2.3 now follows:

######  Corollary 6.8.

Every SUF IP loop is diassociative.

##  7\. Open Problems

The main open problem here is the following:

######  Problem 7.1.

Does there exist an SUF IP loop which is not a Moufang loop?

We have spent a lot of effort over a period of many years on both sides of
this question. Using Prover9, we know various cases in which SUF IP loops are
Moufang. For example, every commutative SUF IP loop is Moufang, as is every
nilpotent SUF IP loop of nilpotency class $2$. We are not particularly
motivated to present the proofs of these claims, because they are long
“equation-bashing” proofs and not very enlightening. None of the proofs we
have in special cases seem to give any insight into the general question.

In the negative direction, naive Mace4 searches up to order 20 have been
unsuccessful. As part of their PhD dissertation work [2], one of us (RB)
attacked the problem using central extensions, the same methodology used to
construct Example 5.3. They checked central extensions of the Moufang loops of
orders 16, 32, 64 and 81 by cyclic groups of various orders; see [2] for
details. Unfortunately, none of those cases were successful.

Despite all this, it is still our opinion that proper (i.e. nonMoufang) SUF IP
loops exist.

By Theorem 2.2, an affirmative answer to Problem 7.1 will also be an
affirmative answer to the following.

######  Problem 7.2.

Does there exist an SUF loop which is not a UF loop?

Here we have not searched as extensively as for Problem 7.1.

We have mentioned the property in loops that the inner mappings
$R(x)L(x)^{-1}$ are all automorphisms; this property implies flexibility.

######  Problem 7.3.

Does there exist a loop satisfying the aforementioned property universally,
but which does not satisfy the AAIP?

Example 5.3 also suggests the following problems.

######  Problem 7.4.

Does there exist a UF loop which does not satisfy the semi-automorphic inverse
property $(xyx)^{-1}=x^{-1}y^{-1}x^{-1}$?

######  Problem 7.5.

Does there exist a commutative UF loop with the AAIP which is not a middle Bol
loop?

######  Acknowledgments.

We would like to thank Petr Vojtěchovský for providing us with GAP code for
computing central extensions.

Our investigations were aided by the automated reasoning tool Prover9 and the
finite model builder Mace4 [7], as well as the LOOPS package [10] for GAP [5].

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Index: 1049, GT Entity Name: difficulty initiating sleep

i have trouble falling asleep these days . i would lay in bed for a long time
staring at the ceiling , not feeling sleepy or tired at all and fall asleep .
but after an hour or two , i would wake up like i never slept at all . and
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sleepy . why is this ?

Index: 1049, Constrastive-NER, F-Score: 0.364

i have trouble falling asleep these days . i would lay in bed for a long time
staring at the ceiling , not feeling sleepy or tired at all and fall asleep .
but after an hour or two , i would wake up like i never slept at all . and
then i would be laying in bed for another long time without feeling tired or
sleepy . why is this ?

Index: 1049, Rule-Based-NER, F-Score: 0.667

i have trouble falling asleep these days . i would lay in bed for a long time
staring at the ceiling , not feeling sleepy or tired at all and fall asleep .
but after an hour or two , i would wake up like i never slept at all . and
then i would be laying in bed for another long time without feeling tired or
sleepy . why is this ?

Index: 1051, GT Entity Name: fatigue

i have trouble falling asleep these days . i would lay in bed for a long time
staring at the ceiling , not feeling sleepy or tired at all and fall asleep .
but after an hour or two , i would wake up like i never slept at all . and
then i would be laying in bed for another long time without feeling tired or
sleepy . why is this ?

Index: 1051, Constrastive-NER, F-Score: 0.8

i have trouble falling asleep these days . i would lay in bed for a long time
staring at the ceiling , not feeling sleepy or tired at all and fall asleep .
but after an hour or two , i would wake up like i never slept at all . and
then i would be laying in bed for another long time without feeling tired or
sleepy . why is this ?

Index: 1051, Rule-Based-NER, F-Score: 0.667

i have trouble falling asleep these days . i would lay in bed for a long time
staring at the ceiling , not feeling sleepy or tired at all and fall asleep .
but after an hour or two , i would wake up like i never slept at all . and
then i would be laying in bed for another long time without feeling tired or
sleepy . why is this ?

Index: 1055, GT Entity Name: breast heaviness sensation

i had unprotected sex 19 days ago and i have taken two pregnancy tests since
then , 1 today ( 19 days ) and the other this past tuesday ( 16 days ) . both
test came out negative but i am having pain in my breast and feel as if they
are heavy and swollen

Index: 1055, Constrastive-NER, F-Score: 0.444

i had unprotected sex 19 days ago and i have taken two pregnancy tests since
then , 1 today ( 19 days ) and the other this past tuesday ( 16 days ) . both
test came out negative but i am having pain in my breast and feel as if they
are heavy and swollen

Index: 1055, Rule-Based-NER, F-Score: 0.0

i had unprotected sex 19 days ago and i have taken two pregnancy tests since
then , 1 today ( 19 days ) and the other this past tuesday ( 16 days ) . both
test came out negative but i am having pain in my breast and feel as if they
are heavy and swollen

Index: 1182, GT Entity Name: pelvic pain

i have experienced burning , pains in my pelvis and clit / urethra area as
well as the need to urinate

Index: 1182, Constrastive-NER, F-Score: 0.667

i have experienced burning , pains in my pelvis and clit / urethra area as
well as the need to urinate

Index: 1182, Rule-Based-NER, F-Score: 0.0

i have experienced burning , pains in my pelvis and clit / urethra area as
well as the need to urinate

Index: 1213, GT Entity Name: lower extremity paresthesia

hi my husband has been having tingling sensations in his arms and leg and his
palms and feet sweat with it when we researched we saw cubital tunnel syndrome

Index: 1213, Constrastive-NER, F-Score: 0.667

hi my husband has been having tingling sensations in his arms and leg and his
palms and feet sweat with it when we researched we saw cubital tunnel syndrome

Index: 1213, Rule-Based-NER, F-Score: 0.0

hi my husband has been having tingling sensations in his arms and leg and his
palms and feet sweat with it when we researched we saw cubital tunnel syndrome

Index: 1227, GT Entity Name: myocardial infarction

i believe that i had a heart attack , a while back , but i don ’ t know for
sure . i was at work one day coming in from break and all of a sudden i got
this very uncomfortable feeling in my left arm and jaw , ( i work in a nursing
home ) so they took my blood pressure and it was 196 / 122 and my heart felt
like it was coming out of my chest i sat down and like 7 mins later i had an
odd taste in my mouth but i felt better took some ibuprofen and finished the
shift . i have been diagnosed with a couple heart conditions , but when i
turned 19 my insurance dropped so they didn ’ t really get to really finish
all the testing , i ’ ve felt okay since so i haven ’ t been too worried about
it . if it was a heart attack do people feel somewhat normal afterwards ?
because that was the only thing that really through me off .

Index: 1227, Constrastive-NER, F-Score: 0.444

i believe that i had a heart attack , a while back , but i don ’ t know for
sure . i was at work one day coming in from break and all of a sudden i got
this very uncomfortable feeling in my left arm and jaw , ( i work in a nursing
home ) so they took my blood pressure and it was 196 / 122 and my heart felt
like it was coming out of my chest i sat down and like 7 mins later i had an
odd taste in my mouth but i felt better took some ibuprofen and finished the
shift . i have been diagnosed with a couple heart conditions , but when i
turned 19 my insurance dropped so they didn ’ t really get to really finish
all the testing , i ’ ve felt okay since so i haven ’ t been too worried about
it . if it was a heart attack do people feel somewhat normal afterwards ?
because that was the only thing that really through me off .

Index: 1227, Rule-Based-NER, F-Score: 0.667

i believe that i had a heart attack , a while back , but i don ’ t know for
sure . i was at work one day coming in from break and all of a sudden i got
this very uncomfortable feeling in my left arm and jaw , ( i work in a nursing
home ) so they took my blood pressure and it was 196 / 122 and my heart felt
like it was coming out of my chest i sat down and like 7 mins later i had an
odd taste in my mouth but i felt better took some ibuprofen and finished the
shift . i have been diagnosed with a couple heart conditions , but when i
turned 19 my insurance dropped so they didn ’ t really get to really finish
all the testing , i ’ ve felt okay since so i haven ’ t been too worried about
it . if it was a heart attack do people feel somewhat normal afterwards ?
because that was the only thing that really through me off .

Index: 1247, GT Entity Name: breast tenderness

i think i am experiencing implantation bleeding but today all of a sudden my
boobs are not sore anymore & i ’ m scared that i ’ m having a miscarriage

Index: 1247, Constrastive-NER, F-Score: 1.0

i think i am experiencing implantation bleeding but today all of a sudden my
boobs are not sore anymore & i ’ m scared that i ’ m having a miscarriage

Index: 1247, Rule-Based-NER, F-Score: 0.0

i think i am experiencing implantation bleeding but today all of a sudden my
boobs are not sore anymore & i ’ m scared that i ’ m having a miscarriage

Index: 1257, GT Entity Name: ear pain

i get nose bleeds every once in a while . right now i feel like the left
nostril may be bleeding soon . it feels like water went into something it wasn
’ t supposed to . it ’ s hard to explain . the left corner of my throat feels
and aches like i ’ m getting an ear ache . i used to get earaches ever so
often , i haven ’ t had one in a while . it settled down now , but there for a
few minutes i thought my nose was going to bleed . is whatever i ’ m having
caused by allergies ? if it continues , i ’ ll see my doctor

Index: 1257, Constrastive-NER, F-Score: 1.0

i get nose bleeds every once in a while . right now i feel like the left
nostril may be bleeding soon . it feels like water went into something it wasn
’ t supposed to . it ’ s hard to explain . the left corner of my throat feels
and aches like i ’ m getting an ear ache . i used to get earaches ever so
often , i haven ’ t had one in a while . it settled down now , but there for a
few minutes i thought my nose was going to bleed . is whatever i ’ m having
caused by allergies ? if it continues , i ’ ll see my doctor

Index: 1257, Rule-Based-NER, F-Score: 0.8

i get nose bleeds every once in a while . right now i feel like the left
nostril may be bleeding soon . it feels like water went into something it wasn
’ t supposed to . it ’ s hard to explain . the left corner of my throat feels
and aches like i ’ m getting an ear ache . i used to get earaches ever so
often , i haven ’ t had one in a while . it settled down now , but there for a
few minutes i thought my nose was going to bleed . is whatever i ’ m having
caused by allergies ? if it continues , i ’ ll see my doctor

Index: 1269, GT Entity Name: contact lens use

i use contacts and last night i put my contacts in a case and thought i soaked
them in contact solution . i wore my glasses to work today and when i was
putting my contacts in at the gym in the evening , my eyes went bloodshot ,
watery , burning , i could not open my eyes , i could not see . i finally was
able to take the contacts out and my eyes still had this feeling - not as
severe was still in pain . i then worked out in my glasses . i tried taking my
glasses off and my eyes would start to burn again . on my drive home from the
gym , i was seeing different . the lights that i saw has a circle around them
that was wider and had a reflecting light in the circle . it ’ s now been a
few hours and i noticed in the bathroom that i soaked my contacts in eye wash
solution and not contacts solution . my eyes are still faintly in pain .
should i be concerned ?

Index: 1269, Constrastive-NER, F-Score: 0.333

i use contacts and last night i put my contacts in a case and thought i soaked
them in contact solution . i wore my glasses to work today and when i was
putting my contacts in at the gym in the evening , my eyes went bloodshot ,
watery , burning , i could not open my eyes , i could not see . i finally was
able to take the contacts out and my eyes still had this feeling - not as
severe was still in pain . i then worked out in my glasses . i tried taking my
glasses off and my eyes would start to burn again . on my drive home from the
gym , i was seeing different . the lights that i saw has a circle around them
that was wider and had a reflecting light in the circle . it ’ s now been a
few hours and i noticed in the bathroom that i soaked my contacts in eye wash
solution and not contacts solution . my eyes are still faintly in pain .
should i be concerned ?

Index: 1269, Rule-Based-NER, F-Score: 0.0

i use contacts and last night i put my contacts in a case and thought i soaked
them in contact solution . i wore my glasses to work today and when i was
putting my contacts in at the gym in the evening , my eyes went bloodshot ,
watery , burning , i could not open my eyes , i could not see . i finally was
able to take the contacts out and my eyes still had this feeling - not as
severe was still in pain . i then worked out in my glasses . i tried taking my
glasses off and my eyes would start to burn again . on my drive home from the
gym , i was seeing different . the lights that i saw has a circle around them
that was wider and had a reflecting light in the circle . it ’ s now been a
few hours and i noticed in the bathroom that i soaked my contacts in eye wash
solution and not contacts solution . my eyes are still faintly in pain .
should i be concerned ?

Index: 1285, GT Entity Name: chest pain

chest / heart pain

Index: 1285, Constrastive-NER, F-Score: 0.8

chest / heart pain

Index: 1285, Rule-Based-NER, F-Score: 0.0

chest / heart pain

Index: 1286, GT Entity Name: vaginal pain

for the past week or so i ’ d say , i ’ ve have unusual vaginal discharge . it
’ s yellowish , and smells bad . i ’ ve had a yeast infection before , but
this doesn ’ t seem like it . no pain , itching , or burning . just the very
strong odor and different discharge . is this a bacterial infection ? are
there over - the - counter medications ?

Index: 1286, Constrastive-NER, F-Score: 0.8

for the past week or so i ’ d say , i ’ ve have unusual vaginal discharge . it
’ s yellowish , and smells bad . i ’ ve had a yeast infection before , but
this doesn ’ t seem like it . no pain , itching , or burning . just the very
strong odor and different discharge . is this a bacterial infection ? are
there over - the - counter medications ?

Index: 1286, Rule-Based-NER, F-Score: 0.0

for the past week or so i ’ d say , i ’ ve have unusual vaginal discharge . it
’ s yellowish , and smells bad . i ’ ve had a yeast infection before , but
this doesn ’ t seem like it . no pain , itching , or burning . just the very
strong odor and different discharge . is this a bacterial infection ? are
there over - the - counter medications ?

Index: 1299, GT Entity Name: muscle rigidity

my arms are twitching when i am lying down or sitting down , i feel nauseous
and feel as though i am ready to vomit , i am really tired and anxious . my
muscles also feel to be tensing up .

Index: 1299, Constrastive-NER, F-Score: 0.8

my arms are twitching when i am lying down or sitting down , i feel nauseous
and feel as though i am ready to vomit , i am really tired and anxious . my
muscles also feel to be tensing up .

Index: 1299, Rule-Based-NER, F-Score: 0.0

my arms are twitching when i am lying down or sitting down , i feel nauseous
and feel as though i am ready to vomit , i am really tired and anxious . my
muscles also feel to be tensing up .

Index: 1301, GT Entity Name: abdominal pain

abdominal and lower back pain

Index: 1301, Constrastive-NER, F-Score: 0.667

abdominal and lower back pain

Index: 1301, Rule-Based-NER, F-Score: 0.0

abdominal and lower back pain

Index: 1346, GT Entity Name: menstrual bleeding

i ’ ve always had irregular periods . i was suppose to get my period on the
1st but i still have not gotten it . i had some very dark red , almost brown
discharge in the middle of my cycle . i am still experiencing symptoms of the
period ( kind of crampy , moody ) like i usually feel , but no bleeding .

Index: 1346, Constrastive-NER, F-Score: 0.0

i ’ ve always had irregular periods . i was suppose to get my period on the
1st but i still have not gotten it . i had some very dark red , almost brown
discharge in the middle of my cycle . i am still experiencing symptoms of the
period ( kind of crampy , moody ) like i usually feel , but no bleeding .

Index: 1346, Rule-Based-NER, F-Score: 0.0

i ’ ve always had irregular periods . i was suppose to get my period on the
1st but i still have not gotten it . i had some very dark red , almost brown
discharge in the middle of my cycle . i am still experiencing symptoms of the
period ( kind of crampy , moody ) like i usually feel , but no bleeding .

Index: 1356, GT Entity Name: vomiting

hi ! yesterday , i ate barbecue and then had a boba drink about 45 minutes
later . i then went to the movies where my stomach was feeling quite queasy
but i didn ’ t think much of it . it wasn ’ t until i got home , where i
starting vomiting profusely . i woke up today with a headache and and a sore
stomach . i then stood up from my bed and instantly had to vomit again . i had
only had water since my last vomiting session . now , whenever i get up from a
chair or bed , i become light headed and have to vomit . i was wondering what
should i do ? i haven ’ t been able to keep down anything .

Index: 1356, Constrastive-NER, F-Score: 0.889

hi ! yesterday , i ate barbecue and then had a boba drink about 45 minutes
later . i then went to the movies where my stomach was feeling quite queasy
but i didn ’ t think much of it . it wasn ’ t until i got home , where i
starting vomiting profusely . i woke up today with a headache and and a sore
stomach . i then stood up from my bed and instantly had to vomit again . i had
only had water since my last vomiting session . now , whenever i get up from a
chair or bed , i become light headed and have to vomit . i was wondering what
should i do ? i haven ’ t been able to keep down anything .

Index: 1356, Rule-Based-NER, F-Score: 0.4

hi ! yesterday , i ate barbecue and then had a boba drink about 45 minutes
later . i then went to the movies where my stomach was feeling quite queasy
but i didn ’ t think much of it . it wasn ’ t until i got home , where i
starting vomiting profusely . i woke up today with a headache and and a sore
stomach . i then stood up from my bed and instantly had to vomit again . i had
only had water since my last vomiting session . now , whenever i get up from a
chair or bed , i become light headed and have to vomit . i was wondering what
should i do ? i haven ’ t been able to keep down anything .

Index: 1408, GT Entity Name: purulent sputum

i had bronchitis and finished my antibiotics 2 weeks ago . my mucus went from
green to be clear . now my mucus is green again and i cough up couple times a
day green mucus without any other symptoms except a dry cough once or twice a
day . is the infection still there or is green mucus bad and should i go back
to the doctor ?

Index: 1408, Constrastive-NER, F-Score: 0.533

i had bronchitis and finished my antibiotics 2 weeks ago . my mucus went from
green to be clear . now my mucus is green again and i cough up couple times a
day green mucus without any other symptoms except a dry cough once or twice a
day . is the infection still there or is green mucus bad and should i go back
to the doctor ?

Index: 1408, Rule-Based-NER, F-Score: 0.4

i had bronchitis and finished my antibiotics 2 weeks ago . my mucus went from
green to be clear . now my mucus is green again and i cough up couple times a
day green mucus without any other symptoms except a dry cough once or twice a
day . is the infection still there or is green mucus bad and should i go back
to the doctor ?

Index: 1416, GT Entity Name: periumbilical abdominal pain radiating to right
lower quadrant

woken up around 330 am with abdomen pain . now it is on the lower right .
constant and stabby . diagonal from the belly button instead of a straight
line below if that makes sense . no other symptoms . it ’ s been constant pain
for 10 hours . gets a bit better with motrin . gets worse when moving ,
pushing on the area .

Index: 1416, Constrastive-NER, F-Score: 0.0

woken up around 330 am with abdomen pain . now it is on the lower right .
constant and stabby . diagonal from the belly button instead of a straight
line below if that makes sense . no other symptoms . it ’ s been constant pain
for 10 hours . gets a bit better with motrin . gets worse when moving ,
pushing on the area .

Index: 1416, Rule-Based-NER, F-Score: 0.0

woken up around 330 am with abdomen pain . now it is on the lower right .
constant and stabby . diagonal from the belly button instead of a straight
line below if that makes sense . no other symptoms . it ’ s been constant pain
for 10 hours . gets a bit better with motrin . gets worse when moving ,
pushing on the area .

Index: 1420, GT Entity Name: hematochezia

three weeks ago i had a little blood in my stool . i monitored it , and had a
little bit of blood in my stool again wednesday evening and thursday afternoon
this week , about three weeks after the first time . i have since pooped twice
and gotten no blood . i scheduled an appointment with a new doctor but he can
not see me for two months . i want to know if this should be looked at more
urgently .

Index: 1420, Constrastive-NER, F-Score: 0.588

three weeks ago i had a little blood in my stool . i monitored it , and had a
little bit of blood in my stool again wednesday evening and thursday afternoon
this week , about three weeks after the first time . i have since pooped twice
and gotten no blood . i scheduled an appointment with a new doctor but he can
not see me for two months . i want to know if this should be looked at more
urgently .

Index: 1420, Rule-Based-NER, F-Score: 0.0

three weeks ago i had a little blood in my stool . i monitored it , and had a
little bit of blood in my stool again wednesday evening and thursday afternoon
this week , about three weeks after the first time . i have since pooped twice
and gotten no blood . i scheduled an appointment with a new doctor but he can
not see me for two months . i want to know if this should be looked at more
urgently .

Index: 1422, GT Entity Name: dysphagia

hi so on the floor of my mouth just today i ’ ve noticed some pain and the
back of my throat noticed a little bump on the one side . no white dots or
anything just a little lump . no trouble breathing or swallowing with the
throat lump

Index: 1422, Constrastive-NER, F-Score: 0.667

hi so on the floor of my mouth just today i ’ ve noticed some pain and the
back of my throat noticed a little bump on the one side . no white dots or
anything just a little lump . no trouble breathing or swallowing with the
throat lump

Index: 1422, Rule-Based-NER, F-Score: 0.0

hi so on the floor of my mouth just today i ’ ve noticed some pain and the
back of my throat noticed a little bump on the one side . no white dots or
anything just a little lump . no trouble breathing or swallowing with the
throat lump

Index: 1427, GT Entity Name: eyelid edema

i went to urgent care and they said i have a corneal abrasion , small like a
pin size . my eye feels like there ’ s something in it , my lid is very
swollen and it ’ s red and my eye won ’ t stop watering . i was wondering if
there ’ s anything i can do to ease symptoms .

Index: 1427, Constrastive-NER, F-Score: 0.5

i went to urgent care and they said i have a corneal abrasion , small like a
pin size . my eye feels like there ’ s something in it , my lid is very
swollen and it ’ s red and my eye won ’ t stop watering . i was wondering if
there ’ s anything i can do to ease symptoms .

Index: 1427, Rule-Based-NER, F-Score: 0.0

i went to urgent care and they said i have a corneal abrasion , small like a
pin size . my eye feels like there ’ s something in it , my lid is very
swollen and it ’ s red and my eye won ’ t stop watering . i was wondering if
there ’ s anything i can do to ease symptoms .

Index: 1428, GT Entity Name: unilateral eye redness

i went to urgent care and they said i have a corneal abrasion , small like a
pin size . my eye feels like there ’ s something in it , my lid is very
swollen and it ’ s red and my eye won ’ t stop watering . i was wondering if
there ’ s anything i can do to ease symptoms .

Index: 1428, Constrastive-NER, F-Score: 0.4

i went to urgent care and they said i have a corneal abrasion , small like a
pin size . my eye feels like there ’ s something in it , my lid is very
swollen and it ’ s red and my eye won ’ t stop watering . i was wondering if
there ’ s anything i can do to ease symptoms .

Index: 1428, Rule-Based-NER, F-Score: 0.0

i went to urgent care and they said i have a corneal abrasion , small like a
pin size . my eye feels like there ’ s something in it , my lid is very
swollen and it ’ s red and my eye won ’ t stop watering . i was wondering if
there ’ s anything i can do to ease symptoms .

Index: 1430, GT Entity Name: lower abdominal pain

my lower abdomen has been hurting for a few days and now is painful to have
sex . this stomach pain seems to be my only symptom so i am wondering if i
should go to doctors or just wait a few days

Index: 1430, Constrastive-NER, F-Score: 0.857

my lower abdomen has been hurting for a few days and now is painful to have
sex . this stomach pain seems to be my only symptom so i am wondering if i
should go to doctors or just wait a few days

Index: 1430, Rule-Based-NER, F-Score: 0.0

my lower abdomen has been hurting for a few days and now is painful to have
sex . this stomach pain seems to be my only symptom so i am wondering if i
should go to doctors or just wait a few days

Index: 1438, GT Entity Name: delayed menstruation

my period was two weeks late after i took a plan b & now it ’ s the week of my
period i ’ m just barely spotting

Index: 1438, Constrastive-NER, F-Score: 0.8

my period was two weeks late after i took a plan b & now it ’ s the week of my
period i ’ m just barely spotting

Index: 1438, Rule-Based-NER, F-Score: 0.0

my period was two weeks late after i took a plan b & now it ’ s the week of my
period i ’ m just barely spotting

Index: 1444, GT Entity Name: throat pain

i have a tonsil that is larger on the right side it has been like that for a
while although currently it has been hurting for past few weeks . i went to a
urgent clinic and was told it was a possible acute infection that should of
cleared up in the next few days . and a week after that i started spitting up
mucus with blood usually in the morning . what would you advice me to do about
this ? thank you .

Index: 1444, Constrastive-NER, F-Score: 0.0

i have a tonsil that is larger on the right side it has been like that for a
while although currently it has been hurting for past few weeks . i went to a
urgent clinic and was told it was a possible acute infection that should of
cleared up in the next few days . and a week after that i started spitting up
mucus with blood usually in the morning . what would you advice me to do about
this ? thank you .

Index: 1444, Rule-Based-NER, F-Score: 0.0

i have a tonsil that is larger on the right side it has been like that for a
while although currently it has been hurting for past few weeks . i went to a
urgent clinic and was told it was a possible acute infection that should of
cleared up in the next few days . and a week after that i started spitting up
mucus with blood usually in the morning . what would you advice me to do about
this ? thank you .

Index: 1472, GT Entity Name: nephrolithiasis

i had my gallbladder removed in 2008 . some of the stones spilled into my body
now i keep getting kidney stones could it be possible that the hall stones got
in my kidney ?

Index: 1472, Constrastive-NER, F-Score: 0.6

i had my gallbladder removed in 2008 . some of the stones spilled into my body
now i keep getting kidney stones could it be possible that the hall stones got
in my kidney ?

Index: 1472, Rule-Based-NER, F-Score: 0.571

i had my gallbladder removed in 2008 . some of the stones spilled into my body
now i keep getting kidney stones could it be possible that the hall stones got
in my kidney ?

Index: 1483, GT Entity Name: fever

my son is sick and i want to know if i should take him to the er . he has a
fever of 103 . 7 ( fever has been since last night ) , is still drinking but
not really eating . sleeping all day .

Index: 1483, Constrastive-NER, F-Score: 0.571

my son is sick and i want to know if i should take him to the er . he has a
fever of 103 . 7 ( fever has been since last night ) , is still drinking but
not really eating . sleeping all day .

Index: 1483, Rule-Based-NER, F-Score: 0.667

my son is sick and i want to know if i should take him to the er . he has a
fever of 103 . 7 ( fever has been since last night ) , is still drinking but
not really eating . sleeping all day .

Index: 1493, GT Entity Name: hot flashes

i forgot to eat today , because i slept through all the meals , woke up , now
i have heart palpitations , tremors , chest pain , my arm feels weird right
now , my blood pressure is 96 , 68 , pulse is 120 - 122 , keep stuttering , am
lightheaded , dizzy , and stuff like that also really tired and getting hot
and cold flashes

Index: 1493, Constrastive-NER, F-Score: 0.667

i forgot to eat today , because i slept through all the meals , woke up , now
i have heart palpitations , tremors , chest pain , my arm feels weird right
now , my blood pressure is 96 , 68 , pulse is 120 - 122 , keep stuttering , am
lightheaded , dizzy , and stuff like that also really tired and getting hot
and cold flashes

Index: 1493, Rule-Based-NER, F-Score: 0.0

i forgot to eat today , because i slept through all the meals , woke up , now
i have heart palpitations , tremors , chest pain , my arm feels weird right
now , my blood pressure is 96 , 68 , pulse is 120 - 122 , keep stuttering , am
lightheaded , dizzy , and stuff like that also really tired and getting hot
and cold flashes

Index: 1544, GT Entity Name: odynophagia

about two weeks ago i had trouble with a sore throat , trouble swallowing and
a dry cough . the symptoms started to die down with otc medicines but the
terrible swallowing continued . it got so bad that when i would yawn it would
feel like i was ripping a hole in my throat . now that feeling is gone and i
have this pain on one side of my throat and when i swallow the pain goes
directly to that area .

Index: 1544, Constrastive-NER, F-Score: 0.222

about two weeks ago i had trouble with a sore throat , trouble swallowing and
a dry cough . the symptoms started to die down with otc medicines but the
terrible swallowing continued . it got so bad that when i would yawn it would
feel like i was ripping a hole in my throat . now that feeling is gone and i
have this pain on one side of my throat and when i swallow the pain goes
directly to that area .

Index: 1544, Rule-Based-NER, F-Score: 0.0

about two weeks ago i had trouble with a sore throat , trouble swallowing and
a dry cough . the symptoms started to die down with otc medicines but the
terrible swallowing continued . it got so bad that when i would yawn it would
feel like i was ripping a hole in my throat . now that feeling is gone and i
have this pain on one side of my throat and when i swallow the pain goes
directly to that area .

Index: 1579, GT Entity Name: viral upper respiratory tract infection

i am recovering from an ear infection in both ears and a diagnosed upper
respiratory infection . i was prescribed antibiotics ( amoxicillin ) for my
ears , and told liquids and bed rest for my uri . i have expelled the mucus
from my lungs , but now i have a deep , crunchy , wheezy sound in my upper
chest when i exhale . i have tried to cough up whatever it might be but
nothing is coming up . i can breath “ around ” the noise , but whenever i take
a shallow breath it sets it off and i go into a cough .

Index: 1579, Constrastive-NER, F-Score: 0.889

i am recovering from an ear infection in both ears and a diagnosed upper
respiratory infection . i was prescribed antibiotics ( amoxicillin ) for my
ears , and told liquids and bed rest for my uri . i have expelled the mucus
from my lungs , but now i have a deep , crunchy , wheezy sound in my upper
chest when i exhale . i have tried to cough up whatever it might be but
nothing is coming up . i can breath “ around ” the noise , but whenever i take
a shallow breath it sets it off and i go into a cough .

Index: 1579, Rule-Based-NER, F-Score: 0.0

i am recovering from an ear infection in both ears and a diagnosed upper
respiratory infection . i was prescribed antibiotics ( amoxicillin ) for my
ears , and told liquids and bed rest for my uri . i have expelled the mucus
from my lungs , but now i have a deep , crunchy , wheezy sound in my upper
chest when i exhale . i have tried to cough up whatever it might be but
nothing is coming up . i can breath “ around ” the noise , but whenever i take
a shallow breath it sets it off and i go into a cough .

Index: 1587, GT Entity Name: pregnancy

i was afraid this girl was pregnant , from me , i was very stressed , she told
me she had her period now , but i ’ m still stressed even though she isn ’ t
pregnant . it ’ s been over a month now .

Index: 1587, Constrastive-NER, F-Score: 0.571

i was afraid this girl was pregnant , from me , i was very stressed , she told
me she had her period now , but i ’ m still stressed even though she isn ’ t
pregnant . it ’ s been over a month now .

Index: 1587, Rule-Based-NER, F-Score: 0.667

i was afraid this girl was pregnant , from me , i was very stressed , she told
me she had her period now , but i ’ m still stressed even though she isn ’ t
pregnant . it ’ s been over a month now .

Index: 1612, GT Entity Name: delayed menstruation

my period is 10 days late but i got my period last month . i haven ’ t had
protected sex for 2 months . also i took a pregnancy test yesterday but it
resulted negative . yet i ’ ve been experiencing acne and cramps sometimes . i
don ’ t know why i ’ m late i ’ m usually regular or very little late .

Index: 1612, Constrastive-NER, F-Score: 0.5

my period is 10 days late but i got my period last month . i haven ’ t had
protected sex for 2 months . also i took a pregnancy test yesterday but it
resulted negative . yet i ’ ve been experiencing acne and cramps sometimes . i
don ’ t know why i ’ m late i ’ m usually regular or very little late .

Index: 1612, Rule-Based-NER, F-Score: 0.0

my period is 10 days late but i got my period last month . i haven ’ t had
protected sex for 2 months . also i took a pregnancy test yesterday but it
resulted negative . yet i ’ ve been experiencing acne and cramps sometimes . i
don ’ t know why i ’ m late i ’ m usually regular or very little late .

Index: 1615, GT Entity Name: gingival pain

the roof of my mouth has a small hole and around one tooth is super sore .

Index: 1615, Constrastive-NER, F-Score: 0.75

the roof of my mouth has a small hole and around one tooth is super sore .

Index: 1615, Rule-Based-NER, F-Score: 0.0

the roof of my mouth has a small hole and around one tooth is super sore .

Index: 1619, GT Entity Name: fear

hello , i have an std . i think . there ’ s definitely something wrong and i
have an appointment scheduled . the appointment is in over a week , however i
am currently in a lot of pain and discomfort . i ’ m also living in a foreign
country and i ’ ve never had something like this before . i ’ m terrified . my
question is , what can i do to relieve the pain and itchiness until my
appointment ? it ’ s very hard coming into work and i ’ m constantly on the
verge of tears because i ’ m freaking out about it . honestly i noticed ths
ages ago and i could have avoided this all if i made an appointment when i
first realized . but fear got in the way , the fear of how much this will cost
, the fear of doing this on my own , and the fear that there something
seriously wrong with me . and i literally have no one to talk to about this ,
no friends or family in this country . i ’ m only 19 . any help would be
highly appreciated .

Index: 1619, Constrastive-NER, F-Score: 1.0

hello , i have an std . i think . there ’ s definitely something wrong and i
have an appointment scheduled . the appointment is in over a week , however i
am currently in a lot of pain and discomfort . i ’ m also living in a foreign
country and i ’ ve never had something like this before . i ’ m terrified . my
question is , what can i do to relieve the pain and itchiness until my
appointment ? it ’ s very hard coming into work and i ’ m constantly on the
verge of tears because i ’ m freaking out about it . honestly i noticed ths
ages ago and i could have avoided this all if i made an appointment when i
first realized . but fear got in the way , the fear of how much this will cost
, the fear of doing this on my own , and the fear that there something
seriously wrong with me . and i literally have no one to talk to about this ,
no friends or family in this country . i ’ m only 19 . any help would be
highly appreciated .

Index: 1619, Rule-Based-NER, F-Score: 0.333

hello , i have an std . i think . there ’ s definitely something wrong and i
have an appointment scheduled . the appointment is in over a week , however i
am currently in a lot of pain and discomfort . i ’ m also living in a foreign
country and i ’ ve never had something like this before . i ’ m terrified . my
question is , what can i do to relieve the pain and itchiness until my
appointment ? it ’ s very hard coming into work and i ’ m constantly on the
verge of tears because i ’ m freaking out about it . honestly i noticed ths
ages ago and i could have avoided this all if i made an appointment when i
first realized . but fear got in the way , the fear of how much this will cost
, the fear of doing this on my own , and the fear that there something
seriously wrong with me . and i literally have no one to talk to about this ,
no friends or family in this country . i ’ m only 19 . any help would be
highly appreciated .

Index: 1620, GT Entity Name: fear

hello , i have an std . i think . there ’ s definitely something wrong and i
have an appointment scheduled . the appointment is in over a week , however i
am currently in a lot of pain and discomfort . i ’ m also living in a foreign
country and i ’ ve never had something like this before . i ’ m terrified . my
question is , what can i do to relieve the pain and itchiness until my
appointment ? it ’ s very hard coming into work and i ’ m constantly on the
verge of tears because i ’ m freaking out about it . honestly i noticed ths
ages ago and i could have avoided this all if i made an appointment when i
first realized . but fear got in the way , the fear of how much this will cost
, the fear of doing this on my own , and the fear that there something
seriously wrong with me . and i literally have no one to talk to about this ,
no friends or family in this country . i ’ m only 19 . any help would be
highly appreciated .

Index: 1620, Constrastive-NER, F-Score: 1.0

hello , i have an std . i think . there ’ s definitely something wrong and i
have an appointment scheduled . the appointment is in over a week , however i
am currently in a lot of pain and discomfort . i ’ m also living in a foreign
country and i ’ ve never had something like this before . i ’ m terrified . my
question is , what can i do to relieve the pain and itchiness until my
appointment ? it ’ s very hard coming into work and i ’ m constantly on the
verge of tears because i ’ m freaking out about it . honestly i noticed ths
ages ago and i could have avoided this all if i made an appointment when i
first realized . but fear got in the way , the fear of how much this will cost
, the fear of doing this on my own , and the fear that there something
seriously wrong with me . and i literally have no one to talk to about this ,
no friends or family in this country . i ’ m only 19 . any help would be
highly appreciated .

Index: 1620, Rule-Based-NER, F-Score: 0.333

hello , i have an std . i think . there ’ s definitely something wrong and i
have an appointment scheduled . the appointment is in over a week , however i
am currently in a lot of pain and discomfort . i ’ m also living in a foreign
country and i ’ ve never had something like this before . i ’ m terrified . my
question is , what can i do to relieve the pain and itchiness until my
appointment ? it ’ s very hard coming into work and i ’ m constantly on the
verge of tears because i ’ m freaking out about it . honestly i noticed ths
ages ago and i could have avoided this all if i made an appointment when i
first realized . but fear got in the way , the fear of how much this will cost
, the fear of doing this on my own , and the fear that there something
seriously wrong with me . and i literally have no one to talk to about this ,
no friends or family in this country . i ’ m only 19 . any help would be
highly appreciated .