Results in Mathematics Vol. 12 (1987)

A CLASS OF FIBERED LOOPS

Elena Zizioli *

0378-6218/87/040459-10$1.50+0.20/0 (c) 1987 Birkhauser Verlag, Basel

Summary. In this note we introduce a constructive method to obtain fibered loops either of any order n~4 or of countable order. Moreover we investi-gate some properties of this class of loops.

§ 1 INTRODUCTION

As it is well known a fibered group (or a group with partition) (p,jO,.)

is a group (p,.) together with a set ~ of subgroups of (p,.) such that

for any a liP " {1} there is exactly one element FE jO with aEF. The set

so is called fibration (or partition) of (p,.). We can always provide a

fibered group (p,SO,·) with the geometric structure of incidence space (p,SP)

(see §2) where ~:= {a'X aEP , X E Y}. Since for any aE P the map ai, : P-P;

x- ax is an element of Aut(P,5/!) then (p,5/!,.) is a so-called incidence

group. Moreover if we assume that the fibration jO is kinematic (i.e.

\! ad' , "Ix E jO a.X.a-1 E jO then (p,5/!,.) is a kinematic space.

Owing to results of Baer,Kegel and Suzuki (e.g. [1],[4],[6] ) an almost complete

classification of finite fibered groups is available.

Over the last years the notions of fibered group, incidence group and kinematic

space have been generalized by many authors (cf. [3],[7]);for instance if

we allow the group (p,.) to be a loop we obtain the notions respectively of

fibered,incidence and kinematic loop (§2). In this note we present a method

to construct fibered loops of any finite order n ~ 4 or of countable order

(§4) by means of a suitable class of latin squares (§3). In §4 we give

some examples for the minimal orders and in §5 we discuss some properties of

* Research supported by the Italian Ministry of Education (M.P.I. 40%,60%)

460 Zizioli

the so constructed loops in order to check whether it is possible to provide

them with a geometric structure of incidence space as in the fibered group

case.

§ 2 FIBERED INCIDENCE LOOPS

Let (p,.) be a loop with neutral element 1 and let P*:=P ,{I} .For any aEP

we denote as usually by at and a~ respectively the maps: at: P --. P;

x '_ ax and a'l: P --. P; x --. xa

An incidence loop is a triple (p,if,·) where (p,~) is an incidence space '),

(p,.) is a loop and for any aEP a EAut(P, if). t

A kinematic loop (p,if,') is an incidence loop such that:

i)

with lEX (X, .) is a subloop of (p,.) .

A fibered loop (p, §,.) is a loop (p, .) with a set § (I§ I ~2) of subloops

of (p,.) such that pEF .

The set §will be called a fibration of (p,.) and its elements will be called

fibers •

By definition,in a kinematic loop (p, y,.) the set .l!"(l) :={ XE if I lEX} is

a fibration of (p, .), (P,jf(l),.) is a fibered loop and aEP ,

Conversely let (p,§,.) be a fibered loop. In [8] we stated the following con-

ditions:

(Fl) {a·X aEP , XE§}= { Y·b bEP YE Y; }

(F2) Va,bEP VXE§ 3YE§ such that a(bX) (ab)Y

(F2' ) Va,bEP VXe:§ 3Ye:§ such that (Xa)b Y(ab)

(F3) VaEP VXE§ VXEX (ax)X = aX

and we proved,by setting if·-.- { a·X aEP , XE§ },:

') By an incidence space (p,if) we mean a set P and a set ifof subset of P such that: i) Va,bEP ii) VLEif

a f b

ILl ~2 a,bEL

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(2.1) (p,~,.) is an incidence loop if and only if (F2) holds.

(2.2) (p,~,.) is a kinematic loop if and only if (Fl,2,2') hold.

By introducing the following relation "II" on ~:

a·X b·Y : ..... 3ZE§, 3a',b'EP such that a·X a'· Z and b·Y b'.Z

we proved ([8]) :

(2.3) (p,~, II) is an incidence space with parallelism 2) if and only if

(F3) holds.

We explicitly notice that since by [9] the conditions (F2) and (F3) are inde-

pendent,(p,~,.) can be an incidence loop but not an incidence space with

parallelism and conversely.

§ 3 A CLASS OF LATIN SQUARES

A latin square of order n is a nxn matrix with entries from the set

N:= {1,2, ••. ,n} such that each row and each column contains every element

of N exactly once.

A latin square of order n [a .. ] is said to be semi-diagonal if 1.J

a.. i 1.1.

for any iE {1,2, ... ,n}

The notion of latin square has been generalized to the infinite case (cf. [5] ).

By an infinite latin square is meant a countably infinite array of rows

and columns of nAtural numbers such that eAch of them occurs exactly once

in each row and column.

We are now going to show that it is possible to construct semi-diagonal la-

tin squares of any order n E~ (n ? 3). As it is known latin squares and quasi-

groups are closely related.In fact given a quasi-group of order n its Cayley

table is a latin square of the same order and conversely. Hence there exist

latin squares of any order n E~ . Moreover in this bijection semi-diagonal

latin squares correspond to idempotent quasi-group.

The existence of idempotent quasi-groups of order n,for any n?3,is proved

in [2] .Here we complete this proposition by exhibiting a method to construct

semi-diagonal latin squares for any order n? 3.

2) By an incidence space (P,~) with

VaEP YLE~

space with parallelism an equivalence relation

::ilL' E~ aEL' and

(p,~,II) we "II" defined

L'II L .

mean an incidence on ~ such that:

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for m+1 Si s2m

(4) where 1 1

s'-.- 2m-i+2

if

if

i= m+1

i > m+1

463

(3.2) The matrix [aij J defined by (2), (3) and (4) fulfils the properties;

i) a .. i for iE{1,2, ... ,n} , 11

ii) [aij J is a latin square of order n.

Proof. (i) By (2) a .. 11

~(i+i) i for i~m. By (4) and (2) we have

a = a = m+1. If i>m+1 then a = a m+1,m+1 1,m ii h+1,h

with h;= 2m-i+1E{1,2,.

.. ,m-1} and by (2) a.. 2m-h+1 = i 11

ii) By (2) and (3) we can see that the first m rows of [a .. J are filled up 1J

according with the following pattern;

1 2m 2 2m-1 a ;=m+1 x m m+2 1,m

2m 2 2m-1 a ;=x 1 m+1 m 2,m-1

2 2m-1 2m 1 m+1

2

2m-1 m':l

x m-1 m+2

m+1 m-1 m+2 a ;=m m,m

where x:= h+1 if m=2h+1 and x:= 3h+1 if m=2h .

m-1 m+3

m+2

m+1

a ;=x m,2m

We can see that the first row of [a .. J is composed by all the numbers 1,2, ... , 1J

2m therefore a '" a for j '" k. The subsequent (m-1) rows are permu-1,j 1,k

tat ions of the first row hence their elements are all distinct. Since,by (4),

the elements of the k-th row with k > m are those of a sui table j-th row

with j ~m written in the opposite direction, they are all distinct.

We now consider the k-th column of [a .. J • If k ~ m the k-th column is a per-1J

mutation of the k-th row and if k> m then the k-th column is a permutation

of the j-th row with j=k+1 for k~2m-1 and j=m+1 for k=2m. Therefore

[a .. J is a latin square. 1J

With (3.1) and (3.2) we have proved;

(3.3) For any nE ~ ,n~ 3 it is possible to determine a semi-diagonal

latin-square of order n.

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for m+1 Si s2m

(4) where 1 1

s'-.- 2m-i+2

if

if

i= m+1

i > m+1

463

(3.2) The matrix [aij J defined by (2), (3) and (4) fulfils the properties;

i) a .. i for iE{1,2, ... ,n} , 11

ii) [aij J is a latin square of order n.

Proof. (i) By (2) a .. 11

~(i+i) i for i~m. By (4) and (2) we have

a = a = m+1. If i>m+1 then a = a m+1,m+1 1,m ii h+1,h

with h;= 2m-i+1E{1,2,.

.. ,m-1} and by (2) a.. 2m-h+1 = i 11

ii) By (2) and (3) we can see that the first m rows of [a .. J are filled up 1J

according with the following pattern;

1 2m 2 2m-1 a ;=m+1 x m m+2 1,m

2m 2 2m-1 a ;=x 1 m+1 m 2,m-1

2 2m-1 2m 1 m+1

2

2m-1 m':l

x m-1 m+2

m+1 m-1 m+2 a ;=m m,m

where x:= h+1 if m=2h+1 and x:= 3h+1 if m=2h .

m-1 m+3

m+2

m+1

a ;=x m,2m

We can see that the first row of [a .. J is composed by all the numbers 1,2, ... , 1J

2m therefore a '" a for j '" k. The subsequent (m-1) rows are permu-1,j 1,k

tat ions of the first row hence their elements are all distinct. Since,by (4),

the elements of the k-th row with k > m are those of a sui table j-th row

with j ~m written in the opposite direction, they are all distinct.

We now consider the k-th column of [a .. J • If k ~ m the k-th column is a per-1J

mutation of the k-th row and if k> m then the k-th column is a permutation

of the j-th row with j=k+1 for k~2m-1 and j=m+1 for k=2m. Therefore

[a .. J is a latin square. 1J

With (3.1) and (3.2) we have proved;

(3.3) For any nE ~ ,n~ 3 it is possible to determine a semi-diagonal

latin-square of order n.

464 Zizioli

§ 4 LOOPS WITH FIBRATION

Let n ~3 be a fixed natural number and let la .. J be the semi-diagonal latin 1J

square of order n obtained following the definitions (1) and (2),(3),(4) of

the previous §3.

For any kE IN ,k ~ 1 let ~N:={1,2, ... ,n} be n distinct

loops of order k+l where is the neutral element (the loops may be also 1 2

all isomorphic). Without loss of generality we may identify

=bon:=l and we may assume that for any i,jEN, i # j, bi # bj r s

b := b := ••. -

2, ... ,k }=:K. Hence the set P:=u{ r. 1

iEN}

o 0

for any

has cardinality m=nk+l .

Moreover for any (i,j)E N"N, i # j let [b .. J (r,sEK:={1,2, ... ,k}) be J.J,rs

a latin square of order k the elements of which are all the members different

with b E f * from 1 of the loop h:= a .. ,i.e. for any 1J

(r,s)EKx:K ij,rs a . .

= f * h

We can now define on the set P the following composition law:

b .. if i # j

bi.b j 1J,rs

(5) .-r s i i

(product f. , . )) b ·b in if i=j r s 1.

(4.1) (p, .)is ~ fibered loop of order m = nk+l .

Proof. By definition, 1 is the neutral element of (p, .).

Now we must solve the i

equation b ·X = bj Let i#j and r,s> 0 (if r=O or r s

by our assumptions ,we have i=j) . In the i-th row of the latin square la . . J 1J

1J

s=O,

j

occurs exactly once;let ZEN such that a. = j (z # i = a ). We consider 1Z ii

now the latin square lb. J ,the elements of which are all the elements of

f ~ = f* J a. 1Z

1Z,pq . In the r-th row the element b j occurs exactly once;let yEK

s the unique index of K such that b.

1z,ry b j . Then by (5) the element

s bZ is the

y

unique solution of

Let now i=j ; by (5) there exists a x E fj uniquely determined in fj

f j ,') is a loop;this solution is unique because XE fh # fi

bix E f with a1' h # a .. = 1. The equation x.b i b j r a ih 11 r s

can be solved analogously. Hence (p,1F,.) is a fibered loop with respect to

because

would imply

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the fibration ~= { r , iE:N }. 1

Therefore we have proved:

(4.2) For any mE IN ,m ~4, it is possible to determine a fibered loop

(p, ~,.) of order m having n (? 3) fibers of the same fixed cardi-

nality (k+l)~2 such that m = nk+l

Examples .

1. For m=4 the fibered loop constructed with this method is the Klein

4-group.

2. For m=5 we must set n=4 and k=l . Hence by (2),(3) and (4) the

semi-diagonal latin square of order 4 is given by

1 3 4 2 4 2 1 3 2 4 3 1 3 1 2 4

465

Let r, 1

.- U,a, for i=1,2,3,4.In this case the latin squares 1

[bij,rs J have order 1 that is for any i,jE{1,2,3,4} b :=a with ij,l1 h

h:= a ij

Therefore,by means of (5) , the set p ={ 1,a, ,a, ,a, ,a.} becomes

a fibered loop with the following multiplication table:

1 a, a, a, a.

a, 1 a, a. a,

a, a. 1 a, a,

a, a 2 a. 1 a,

a. a, a, a, 1

(p,.) is the proper fibered loop of minimum order of this type.

3. For k=2 the minimum order for these fibered loops is m= 3·2+1 7.

By (1) the semi-diagonal latin square of order n=3 is given by

[a, ,J lJ

132 321 213

. Let r, : ={ 1

2 1,a , ,a,

1 1

2 a,a ,

1 1

2 a,a,

1 1 1} i=1,2,3.

466 zizioli

2 Hence the set P = { 1,a. ,a. i=1,2,3 } becomes a loop with respect to the

~ ~

following multiplication table:

2 2 2 1 a 1 a 1 a 2 a 2 a, a,

2 2 2 a 1 ~ 1 a, a, a2 a2

2 2 2 a 1 1 a 1 a, a, a 2 a 2

2 2 2 a 2 a, a, a 2 1 a 1 a 1

2 2 2 a 2 a, a, 1 a 2 a a

1 1

2 2 2 a, a 2 a 2 a a a, 1

1 1

2 2 2 a, a 2 a 2 a 1 a 1 1 a,

We can immediately generalize (4.2) to the countable case by means of the notion

of infinite latin square. In this case we always fix the semi-diagonal latin

square [a .. 1 . . ~J ~ ~

of order n and moreover we fix n countable loops i

ri:={bo' 1 2 n

b 1 , .. ·,bk , .. } ie:N:= {1,2, ... ,n} with the same neutral element b =b = ..• =b =: 1 o 0 0

and such that r. n r. ={1} for i,je: N, i;ofj. ~ J

For any (i,j)e:NxN ,i;ofj, let [b 1 r,se: IN ij,rs

,be an infinite latin square the

elements of which are all the members of r* . If we provide the countable a

set P: = u {r. 1 i e:N} wi th the operation " ;i;j defined in (5) then, wi th the same ~

arguments used for (4.1),one can prove that (p,.) is a loop with a fibration

§ := { r. ~

So we have:

ie:N} of n fibers of countable order.

(4.3) It is possible to determine countable fibered loops (p, §,.) having n

fibers of countable order for any n e: N.

§ 5 GEOMETRIC PROPERTIES

In this section we shall prove the following:

(5.1) Let (p, §, 0) be a fibered loop constructed as in (4.1) or (4.3) and let

.!l':={x·r. 1 xe:P, r.e:§}. Then ~ ~

i) (p,.!l',.) fulfils (Fl),

ii) (P,.!l',·) is an incidence loop if and only if for any r.e:§ 1 r.1 ~ ~

2 .

In this case (P,.!l',.) is a kinematic loop,

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iii) (p. Y. II) is an incidence space with parallelism if and only if

Ipl=4 • i.e. P is the Klein 4-group.

We consider the following properties of the set Y;={ x . f I ~EP. f.E ji"} i l

(5.2) i E ji"; For any b E P* and for any f. r J

f. if i=j i J i

b . f. f:b r J u {b i} ifj J r

r* with R, ;=a ..• if R, r lJ

By (5.2) we have;

(5.3)

(5.4)

Let ifj and let i i

b .b E f* r s i

i i b ·f.nb ·r.= f*

r J s J R, i

For any b EP and for any r

b i . r r j

with rfs;then

where R,;=a ij

r Eji" there exists a j

if i=j

r*u{b~ with h;=a if ifj h r ji

r Eji" such that; k

Proof. Let ifj.then by (5.2) • R,;=a .. fi . Since [a .. J is a lJ lJ

latin square there is exactly one element kE{1.2 •...• n} iii

So .by (5.2). r·b = r*u{b }=b ·r. kr R, r r J

(5.5) Let bi.bjEP* and ifj. Then r. fl .

i) b.. db lb J ) . I'. n b l . r. lJ.rs r s J r J

ii)

Proof.

with

i j i I(b b )·f.nb ·r.1 ~2

r s J r J

ii) By (5.2) i i

b . f ={b }uf* r j r R,

h;= a By i) and since R,j

with

h=a # R,j

R,;=a ij

and

= a • (5.5)ii) R,R,

such that

b . f. lJ.rs J is proved.

(5.6) Let f. I =2 for all l

f Eji".Then for any i.j d 1.2 •...• n} i

i b . f. if and only if Ipi = 4.

r J

R,=a k.i

= { b ., } u r * lJ .rs h

Proof. The equality in (5.6) holds if and only if the semi-diagonal latin square

[a .. J fulfils the property a . = i for all i.jd1.2 •...• n} .But from the lJ a ..• J

definitions (1) and (2).(3).(4) lJ we can see that this property is verified only

if n=3.i.e. Ipl=4 and so P is the Klein 4-group.

By (5.4).(5.3) and (5.5)ii).(5.6).together with (2.1).(2.2).(2.3).the parts i).

ii) and iii) of (5.1) follow immediately.

468 Zizioli

Remarks.

1. If Ir.I>2 l.

for all r. e: §, (5.3) and (5.5)ii) imply that (p,.) is l.

always a proper loop.

2. If I r. I = 2 for all l.

r.e:§ and Ipi >4,then (5.6) implies that (p,.) l.

is a proper loop also in this case.

Hence this method gives us proper fibered loops of any order m> 4.

REFERENCES

[1 ] BAER,R. : Partitionen endlicher Gruppen, Math. Zeitschr. 75 (1961), 333-372

I DENES,J. and KEEDWELL,A.D. : Latin squares and their applications,

Academic Press, New York - London (1974)

KARZEL,H. and KIST,G.P. Kinematic algebras and their geometries, Rings and Geometries (Kaya et al. eds.) NATO ASI series-C 160 (1985) 437-509

KEGEL,O.K. : Nicht-einfache Partitionen endlicher Gruppen, Arch. der Math. 12 (1961), 170-175

LINDNER,C.C. Extending mutually orthogonal partial latin squares, Acta Sci. Math. (Szeged) 32 (1971), 283-285

[6 1 SUZUKI,M. On a finite group with a partition, Arch. der Math. 12 (1961), 241-245

[7] WAHLING,H. Projektive Inzidenzgruppoide und Fastalgebren ,J.of Geometry ~ (1977), 109-126

[8] ZIZIOLI, E. : Fibered incidence loops and kinematic loops , J. of Geometry (to appear)

[ 9 1 An independence theorem on the condi tions for incidence loops, Proc. of the Conference "Combinatorics '86" (to appear)

Elena Zizioli Dipartimento di Matematica Universita Cattolica via Trieste 17 25121 BRESCIA ITALIA

Eingegangen am 15.1.1987