ma hema ics
A icle
Pseudococyclic Pa ial Hadama d Ma ices o e
La in Rec angles
Raúl M. Falcón * , Víc o Ál a ez , Ma ía Dolo es F au , Félix Gudiel and Ma ía Belén Güemes
Ci a ion: Falcón, R.M.; Ál a ez, V.;
Falcón, M.D.; Gudiel, F.;
Güemes, M.B. Pseudococyclic Pa ial
Hadama d Ma ices o e La in
Rec angles. Ma hema ics 2021,9, 113.
h p://doi.o g/10.3390/ma h9020113
Recei ed: 2 Decembe 2020
Accep ed: 4 Janua y 2021
Published: 6 Janua y 2021
Publishe ’s No e: MDPI s ays neu-
al wi h ega d o ju isdic ional clai-
ms in published maps and ins i u io-
nal a ilia ions.
Copy igh : © 2021 by he au ho s. Li-
censee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and con-
di ions o he C ea i e Commons A -
ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
Depa men o Applied Ma hema ics I, Uni e si y o Se ille, 41004 Se illa, Spain; [email p o ec ed] (V.Á.);
[email p o ec ed] (M.D.F.); [email p o ec ed] (F.G.); [email p o ec ed] (M.B.G.)
*Co espondence: [email p o ec ed]
Abs ac :
The classical design o cocyclic Hadama d ma ices has ecen ly been gene alized by means
o bo h he no ions o he cocycle o Hadama d ma ices o e La in ec angles and he pseudococycle
o Hadama d ma ices o e quasig oups. This pape del es in o his opic by in oducing he concep
o he pseudococycle o a pa ial Hadama d ma ix o e a La in ec angle, whose undamen als a e
comp ehensi ely s udied and illus a ed.
Keywo ds: Hadama d ma ix; La in ec angle; pseudocobounda y; pseudococycle; quasig oup
MSC: 05B20; 05B15; 20N05
1. In oduc ion
A (bina y) Hadama d ma ix is a squa e ma ix
H
o o de
n
wi h en ies in he se
{−
1,1
}
such ha
HH =nIn
. As such, all i s ows (equi alen ly, columns) a e pai wise o -
hogonal, and hence, i s o de mus be 1, 2, o a mul iple o 4. The Hadama d conjec u e [
1
]
ensu es he exis ence o Hadama d ma ices o e e y o de mul iple o 4. I has emained
open o mo e han a cen u y [2].
In 1993, as a new way o gene a ing combina o ial designs ha gene alizes he g oup
de elopmen me hod, in combina o ial design heo y, Ho adam and de Launey [
3
] (see
also [
4
,
5
]) in oduced he undamen als o he so-called cocyclic de elopmen o e ini e
g oups. In his con ex , a ma ix wi h en ies in he se
{−
1,1
}
is said o be cocyclic o e
a ini e g oup
(G
,
·)
i he e exis s a map
φ:G×G→ {−
1,1
}
sa is ying he so-called
cocycle equa ion:
φ(i·j,k)φ(i,j)φ(j,k)φ(i,j·k) = 1, (1)
o all
i
,
j
,
k∈G
, so ha he ma ix unde conside a ion is Hadama d equi alen o he
cocyclic ma ix
Mφ:= (φ(i
,
j))i,j∈G
. Tha is, hey a e equal up o pe mu a ion o nega ion
o ows and columns. The map
φ
is a cocycle [
3
,
6
] o e he g oup. A cocyclic ma ix
necessa ily has a cons an ow and a cons an column. Acco ding o he cocyclic es [
6
],
i is Hadama d whene e he summa ion o all he en ies o each ow is ze o, excep o
he ones in i s cons an ow. As such, de e mining whe he a cocyclic ma ix is Hadama d
is compu a ionally much as e han checking he de ini ion o a Hadama d ma ix.
In 1995, Ho adam and de Launey [
6
] p o ed ha his cocyclic amewo k p o ides
an excellen s uc u al app oach o dealing wi h he Hadama d conjec u e, which would
be a consequence o he so-called cocyclic Hadama d conjec u e [
3
], o which a cocyclic
Hadama d ma ix o o de 4
exis s o e e y posi i e in ege
. I is so ha many known
amilies o Hadama d ma ices a e cocyclic o e ce ain g oups: Syl es e ma ices [
7
],
Paley ma ices [
1
], Williamson ma ices [
8
], o I o’s ype Q ma ices [
9
] (see also [
2
,
10
–
14
]
o some cons uc ions in his ega d). Ne e heless, he cocyclic amewo k u ned ou o
ail [
12
] o wo o he mos p oli ic amilies o Hadama d ma ices: he wo-ci culan co e
Hadama d ma ices [15] and he Goe hals–Seidel a ays [16].
Ma hema ics 2021,9, 113. h ps://doi.o g/10.3390/ma h9020113 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2021,9, 113 2 o 20
Ve y ecen ly, a new app oach in oduced by he au ho s o [
17
] has success ully deal
wi h a cocyclic de elopmen o Goe hals–Seidel a ays, no o e a g oup, bu o e a amily
o Mou ang loops. This app oach is comp ehended in he new heo y o cocyclic de el-
opmen o e quasig oups and La in ec angles, which has also ecen ly been in oduced
by he au ho s in [
18
]. Mo e speci ically, a cocycle
φ
o e a quasig oup
(Q
,
·)
is a map
φ:Q×Q→ {−
1,1
}
sa is ying he cocycle Equa ion (1) o all
i
,
j
,
k∈Q
. I an o de ing o
he elemen s o
Q
is es ablished, hen he cocycle
φ
is uniquely ep esen ed by he cocyclic
ma ix
Mφ:= (φ(i
,
j))i,j∈Q
. In pa icula , he quasig oup
(Q
,
·)
mus be a loop whene e
he ma ix
Mφ
is Hadama d. Mo eo e , he cocyclic Hadama d es also holds in his case.
The main aspec o his new app oach is he ac ha associa i i y is no longe a
necessa y condi ion o dealing wi h any o he concep s and esul s ha a e usually
in ol ed in he cocyclic de elopmen o e ini e g oups. I is so ha he exis ence o
cobounda ies o e non-associa i e loops has al eady been p o ed [
17
]. In his ega d, we
emind he eade ha a cocycle
φ
o e a quasig oup
(Q
,
·)
is called a cobounda y i he e
exis s a map ∂:Q→ {−1,1}such ha
φ(i,j) = ∂(i)∂(j)∂(ij), o all i,j∈Q. (2)
This cobounda y
φ
is said o be elemen a y i he e exis s an elemen
h∈Q
such ha
∂=∂h
, whe e
∂h(i) = −
1, i
i=h
, and
∂h(i) =
1 o he wise. F om he cocycle equa ion,
i is equi alen o say ha
i(jk) = h⇔(ij)k=h,
holds o all
i
,
j
,
k∈Q
. I is s aigh o wa dly sa is ied in he case whe e
(Q
,
·)
is a g oup.
Mo eo e , he cocycle equa ion has u ned ou no o be necessa y in he quasig oup
de elopmen heo y. In his ega d, a pseudocobounda y o e he quasig oup
(Q
,
·)
is
de ined as any map
ψh:Q×Q→Q
wi h
h∈Q
, sa is ying Equa ion (2) o some
∂h
desc ibed as abo e. By ex ension, a pseudococycle is any map
ψ=∏h∈H⊆Qψhφ
ha is
ob ained as he p oduc o some pseudocobounda ies
ψh
wi h
h∈H⊆Q
and a cocycle
φ
, all o hem o e a gi en quasig oup
(Q
,
·)
. I is ep esen ed by he pseudococyclic
ma ix
Mψ:= (ψ(i
,
j))i,j∈Q
. I i is Hadama d equi alen o a gi en ma ix, hen he la e
is called a pseudococyclic Hadama d ma ix. Unlike he cocyclic amewo k o e ini e
g oups, e e y Goe hals–Seidel a ay cons i u es a pseudococyclic Hadama d ma ix o e a
Mou ang loop [17].
This las asse ion co obo a es he ele an ole ha non-associa i e quasig oups play
in he gene aliza ion o he cocyclic amewo k o e g oups. This pape del es in o his
opic by ocusing on he undamen als o he pseudococyclic amewo k no only o e
quasig oups, bu also o e La in ec angles. I enables us o gene alize he classical no ion
o he cocycle o Hadama d ma ices o e g oups o ha o he pseudococycle o pa ial
Hadama d ma ices o e La in ec angles. We emind he eade in his ega d ha a pa ial
Hadama d ma ix is an
×n
(bina y) ma ix
H
wi h
≤n
such ha
HH =nI
. The ecen
implemen a ion o hese ypes o ma ices in c yp og aphy [
19
], expe imen al design [
20
],
and quan um in o ma ion [
21
] has awakened he in e es in desc ibing di e en ways o
cons uc ing hem [
22
–
25
]. In addi ion, La in ec angles may be implemen ed in In e ne
o Things (IoT) s udies [
26
], coding heo y [
27
,
28
], and mode n 5G wi eless ne wo ks [
29
].
O pa icula in e es in ou s udy, he ele an ole ha quasig oups wi h ew associa i e
iples play in c yp og aphy [
30
,
31
] is ema kable. I is so ha quasig oups wi h a high
amoun o non-associa i e iples a e ecei ing pa icula a en ion [32–36].
The pape is o ganized as ollows. In Sec ion 2, we e iew some p elimina y concep s
and esul s on quasig oups and La in ec angles ha a e used h oughou he pape .
In
Sec ion 3
, we in oduce and illus a e he no ions o bo h he pseudocobounda y and
pseudococycle o e La in ec angles. Then, we deal wi h he ollowing wo open p oblems
conce ning he pseudocobounda y amewo k o e La in ec angles. Bo h o hem a e
comple ely answe ed in Sec ion 4.
Ma hema ics 2021,9, 113 3 o 20
P oblem 1.
Unde which condi ions may we ensu e he exis ence o a pa ial Hadama d ma ix
ha is a pseudocobounda y o e a gi en La in ec angle?
P oblem 2.
Unde which condi ions is a gi en pa ial Hadama d ma ix a pseudocobounda y o e
a La in ec angle?
We also deal wi h he p oblem o de e mining unde which condi ions we may
ensu e he exis ence o a pa ial Hadama d ma ix ha is pseudococyclic o e a gi en
La in ec angle. In his ega d, Sec ions 5and 6 ocus espec i ely on he pseudococyclic
amewo k associa ed wi h i ial cocycles and he pseudococyclic amewo k ela ed o
non- i ial cocycles. Finally, since his pape has a high dependence on no a ion, a glossa y
o symbols is shown in Appendix A.
2. P elimina ies
Le us e iew some o he basic concep s and esul s on quasig oups and La in ec -
angles ha a e used h oughou he pape . We e e he eade o [
18
,
37
] o mo e de ails
abou hese opics.
A quasig oup [
38
] o o de
n
is a pai
(Q
,
·)
o med by a ini e se
Q
o
n
elemen s ha
is endowed wi h a p oduc
·
so ha any wo o he h ee elemen s
i
,
j
,
k∈Q
in he equa ion
i·j=k
uniquely de e mine he hi d elemen . Tha is o say, he p oduc
·
makes possible
bo h he le and he igh di ision in
Q
. A loop is a quasig oup
(Q
,
·)
wi h a uni elemen
e
such ha i·e=e·i=i o all i∈Q. E e y associa i e quasig oup is a g oup.
The Cayley able o a quasig oup o o de
n
cons i u es a La in squa e o he same
o de ; ha is, an
n×n
a ay wi h en ies in a se o
n
dis inc symbols so ha each symbol
occu s exac ly once pe ow and exac ly once in each column. The emo al o a leas
one ow o a La in squa e cons i u es a La in ec angle. Mo e speci ically, an
×n
La in
ec angle, wi h
≤n
, is an
×n
a ay wi h en ies in a se o
n
dis inc symbols so ha
each symbol occu s exac ly once pe ow and a mos once in each column. F om he e
on, le
R ,n
deno e he se o
×n
La in ec angles wi h en ies in he se
[n]:={
1,
. . .
,
n}
.
Fu he , L[i,j]deno es he symbol con ained in he cell (i,j)o a La in ec angle L∈ R ,n.
Le L∈ R ,n. I one de ines he subse o symbols
S(L):= [ ]∪ {L[i,j]|1≤i,j≤ } ⊆ [n],
hen a cocycle o e
L
is any unc ion
φ:S(L)×[n]→ {−
1,1
}
sa is ying he cocycle
equa ion
φ(L[i,j],k)φ(i,j)φ(j,k)φ(i,L[j,k]) = 1, (3)
o all posi i e in ege s
i
,
j≤
and
k≤n
. I is e med i ial i
φ(i
,
j) =
1 o all
(i
,
j)∈ S(L)×[n]
. No ice ha he nega ion -
φ
o a cocycle
φ
o e
L
is also a cocycle o e
L
.
Fu he , e e y cocycle φo e Lis uniquely ep esen ed by he cocyclic ma ix
Mφ:= (φ(i,j))(i,j)∈S(L)×[n].
The ollowing example illus a es all hese concep s. F om he e on, we ep esen ,
espec i ely, he symbols −1 and 1 in any gi en bina y a ay wi h he symbols +and −.
Example 1. Le us conside he 2×4La in ec angle
L≡1 2 4 3
3 1 2 4
whe e we ha e highligh ed hose cells ha a e used o de ine he subse o symbols
S(L) = {
1,2,3
}
.
The e exis exac ly ou cocycles o e he La in ec angle
L
: he i ial one and he unc ion
φ:S(L)×[
4
]→{−1,1}
, which a e ep esen ed by he ollowing ma ix, oge he wi h hei
espec i e nega ions.
Ma hema ics 2021,9, 113 4 o 20
Mφ≡
+ + + +
+++−
++−+
In pa icula , le us check ha he unc ion φsa is ies he cocycle Equa ion (3).
•
I
i=
1, hen
L[i
,
j] = j
o all
j∈ {
1,2
}
, and hence, he cocycle equa ion holds eadily om
he ac ha he i s ow o he ma ix Mφis cons an .
•
I
(i
,
j)=(
2,1
)
, hen
φ(
3,
k)φ(
2,1
)φ(
1,
k)φ(
2,
L[
1,
k]) = φ(
3,
k)φ(
2,
L[
1,
k]) =
1 o all
k≤4.
•
I
(i
,
j)=(
2,2
)
, hen
φ(
1,
k)φ(
2,2
)φ(
2,
k)φ(
2,
L[
2,
k]) = φ(
2,
k)φ(
2,
L[
2,
k]) =
1 o all
k≤4.
3. Pseudocobounda ies and Pseudococycles o e La in Rec angles
In his sec ion, we in oduce he no ions o bo h he pseudocobounda y and pseudo-
cocycle o e a La in ec angle as a na u al gene aliza ion o he simila concep s desc ibed
o e quasig oups in [
17
] by keeping in mind, o his end, he concep s in oduced in [
18
].
Fi s ly, le us de ine he ypes o La in ec angles whe e such a gene aliza ion is easible.
Le and nbe wo posi i e in ege s such ha ≤n, and le L∈ R ,nbe such ha
L[L[i,j],k]6=L[i,L[j,k]], (4)
o some iple
(i
,
j
,
k)∈[ ]×[ ]×[n]
such ha
L[i
,
j]≤
. We apply he e m “non-
associa i e” o any such iple sa is ying Condi ion (4). Le
NS(L)
deno e om he e on
he se o such non-associa i e iples wi hin he La in ec angle
L
. The ca dinali y o
his se is he index o non-associa i i y o
L
, which is deno ed by
ns(L)
. I
=n
, hen
Condi ion (4) implies ha he associa i e p ope y does no hold o he iple
(i
,
j
,
k)
in
he non-associa i e quasig oup wi h
L
as i s Cayley able. In his case, he index
ns(L)
measu es he associa i i y o ha quasig oup. This index has been s udied o di e en
ypes o algeb aic
s uc u es [39–42]
since i was in oduced in 1947 by Climescu [
43
] o
any gi en mul iplica i e sys em. Pa icula ly, i is easily e i ied [
44
] ha
ns(L)≤n3−n
o e e y La in squa e
L
o o de
n
. This uppe bound has ecen ly been p o ed [
32
] o be
sha p o o de
n>
1. Fu he mo e, i is also known [
45
] ha 16
n−
64
≤ns(L)
o e e y
La in squa e o e en o de
n≥
168. The eade is also e e ed o [
46
,
47
] o some o he
s udies dealing wi h he numbe o non-associa i e iples o a La in squa e.
In his pape , we a e in e es ed in he La in ec angles
L∈ R ,n
such ha
ns(L)>
0.
The ollowing lemma cha ac e izes he case o =1.
Lemma 1.
Le
L∈ R1,n
. Then,
ns(L)>
0i and only i
L[
1,1
] =
1and he e exis s a posi i e
in ege k ≤n such ha L[1, k]6=k.
P oo .
No ice om Condi ion (4) ha e e y non-associa i e iple o he 1
×n
La in
ec angle
L
would be o he o m
(
1,1,
k)
o some posi i e in ege
k≤n
sa is ying ha
L[L[
1,1
]
,
k]6=L[
1,
L[
1,
k]]
. In addi ion, Condi ion (4) also implies ha
L[
1,1
] =
1 and,
hence, L[1, k]6=L[1, L[1, k]]. As a consequence, L[1, k]6=k.
Le
L∈ R ,n
be such ha
ns(L)>
0. E e y non-associa i e iple
(i
,
j
,
k)∈NS(L)
is
ela ed o wo dis inc posi i e in ege s
h1
,
h2≤n
such ha
h1=L[L[i
,
j]
,
k]6=L[i
,
L[j
,
k]] =
h2. F om he e on, le H(L)deno e he se o posi i e in ege s h≤nsuch ha
{h} ⊂ {L[L[i,j],k],L[i,L[j,k]]}, (5)
o some
(i
,
j
,
k)∈NS(L)
. I is eadily e i ied ha
ns(L) =
0 whene e
n≤
2. So, om
now on, we suppose ha
n>
2 h oughou he pape . No ice also ha e e y La in squa e
in
Rn,n
wi h
ns(L)>
0 is he Cayley able o a non-associa i e quasig oup o o de
n
.
The case <nis illus a ed by he ollowing example.
Ma hema ics 2021,9, 113 5 o 20
Example 2. Le us conside he La in ec angle L ha is desc ibed in Example 1. Then,
NS(L) = {(1, 1,3),(1,1,4),(1,2,1),(1,2,4),(2,2,1),(2, 2, 2),(2,2,3),(2, 2,4)}.
Hence,
ns(L) =
8. In addi ion,
H(L) = [
4
]
. To p o e i , ake, o ins ance, he iples
(
2,2,1
)
and (1,2,1)in NS(L).
Le
L∈ R ,n
be such ha
ns(L)>
0 and le
h∈ H(L)
. We de ine he
h
-pseudocoboun-
da y o e he La in ec angle
L
as he map
ψL;h:[ ]×[n]→ {−
1,1
}
, which is desc ibed
so ha
ψL;h(i,j):=∂h(i)∂h(j)∂h(L[i,j]), (6)
o all posi i e in ege s i≤ and j≤n, whe e
∂h(k):=(−1, i k=h,
1, o he wise.
In addi ion, we apply he e m “
h
-pseudocobounda y ma ix” o e
L
o he
×n
ma ix
MψL;h:= (ψL;h(i
,
j))(i,j)∈[ ]×[n]
. When we wan o e e o any
h
-pseudocobounda y
(ma ix) o e
L
, we omi he p e ix
h
. As such, he concep o he pseudocobounda y o e
a La in ec angle cons i u es a gene aliza ion o ha o e a quasig oup [
17
], which a ises
when
=n
. In any case, he ollowing esul es ablishes ha he pseudococyclic amewo k
o e La in ec angles is no included in he cocyclic amewo k o e such a ays. Hence,
i cons i u es a new p oposal ha has o be independen ly s udied.
Lemma 2.
Le
L∈ R ,n
be such ha
ns(L)>
0and le
h∈ H(L)
. The
h
-pseudococycle
ψL;h
is
no a cocycle o e L.
P oo .
Le us see ha he
h
-pseudocycle
ψL;h
does no hold he cocycle Equa ion (3). To
his end, le (i,j,k)∈NS(L)be such ha Condi ion (5) holds. Then,
ψL;h(L[i,j],k)ψL;h(i,j)ψL;h(j,k)ψL;h(i,L[j,k]) = ∂h(L[L[i,j],k])∂h(L[i,L[j,k]]) = −1.
Le us illus a e all o hese concep s wi h a se ies o examples.
Example 3.
Le
L
be he La in ec angle desc ibed in Example 1. Acco ding o Example 2, we can
de ine ou pseudocobounda ies o e L, which a e ep esen ed by he ollowing ma ices.
MψL;1 ≡− − − −
−−+ + MψL;2 ≡++++
−++−
MψL;3 ≡++− −
−+−+MψL;4 ≡++− −
++++
The ollowing example enables us o ensu e ha , unlike he cocyclic de elopmen
o e quasig oups, he e exis Hadama d ma ices ha a e pseudocobounda y ma ices
o e quasig oups ha a e no loops.
Ma hema ics 2021,9, 113 6 o 20
Example 4. Le us conside he La in squa e
L≡
1 2 4 3
2 1 3 4
3 4 1 2
4 3 2 1
.
We ha e ha
ns(L) =
32 and
H(L) = [
4
]
. In o de o p o e his las end, ake, o ins ance,
he subse
{(
1,1,3
)
,
(
1,3,3
)} ⊂ NS(L)
. I is simply e i ied ha e e y
h
-pseudocobounda y
ma ix o L wi h 1≤h≤4is Hadama d.
MψL;1 ≡
−−−−
−−+ +
−+−+
−++−
MψL;2 ≡
++++
++− −
+−+−
+−−+
MψL;3 =MψL;4 ≡
+ + − −
+ + + +
+−+−
+− − +
Obse e ha all he pseudocobounda y ma ices shown in Examples 3and 4cons i u e
(pa ial) Hadama d ma ices. P oposi ion 2desc ibed in Sec ion 4enables us o ensu e ha
his condi ion does no hold in gene al. Finally, he ollowing example enables us o ensu e
he exis ence o Hadama d ma ices ha a e no cocyclic o e any La in ec angle, bu ha
a e pseudocobounda y ma ices o e a La in squa e.
Example 5.
I is known ([
18
] Example 41) ha he ollowing Hadama d ma ix is no cocyclic
o e any La in ec angle.
M≡
++++
−+ + −
− − + +
+−+−
Ne e heless, i cons i u es a 2-pseudocobounda y ma ix o e he La in squa e
L≡
1 2 3 4
3 4 2 1
2 1 4 3
4 3 1 2
.
Le us inish his sec ion by in oducing he no ion o a pseudococycle o e a La in
ec angle as a gene aliza ion o bo h he concep s o a cocycle o e a La in
ec angle [18]
and a pseudococycle o e a quasig oup [
17
]. To his end, we ake in o accoun he p e i-
ously desc ibed no ion o a pseudocobounda y o e La in ec angles. Thus, we de ine a
pseudococycle o e a gi en La in ec angle L∈ R ,nwi h ns(L)>0 as any map
ψ=
∏
h∈S⊆H(L)
ψh
φ
ha is ob ained as he p oduc o some
h
-pseudocobounda ies wi h
h∈S⊆ H(L)
and a
cocycle
φ
, all o hem o e he La in ec angle
L
. I is ep esen ed by he pseudococyclic
ma ix
Mψ:= (ψ(i
,
j))(i,j)∈[ ]×[n]
. In pa icula , no ice om his de ini ion ha e e y
pseudocobounda y o e a La in ec angle is a pseudococycle o e he la e by means
o he i ial cocycle. Fu he , i
S=∅
, hen all o hese concep s e e o he cocyclic
amewo k o e La in ec angles, whose undamen als we e comp ehensi ely s udied
in [18].
In a simila way, i
=n
, hen hey e e o he pseudococyclic amewo k o e
quasig oups, which has only been b ie ly deal wi h in [
17
]. This pape ocuses, he e o e,
on he undamen als o he case
S6=∅
, wha e e he posi i e in ege
≤n
is. The
ollowing example illus a es his case.
Ma hema ics 2021,9, 113 7 o 20
Example 6.
Le
L
be he La in ec angle desc ibed in Example 1. Then, he ollowing asse ions
a e eadily e i ied om he cocyclic ma ix
Mφ
desc ibed in ha example, oge he wi h he
pseudocobounda y ma ices MψL;3 and MψL;4 desc ibed in Example 3.
•
The pseudococyclic ma ix o e
L
ha is associa ed wi h he pseudococycle
ψL;3ψL;4
is pa ial
Hadama d.
MψL;3ψL;4 ≡+ + + +
−+−+
•
The pseudococyclic ma ix o e
L
ha is associa ed wi h he pseudococycle
ψL;3ψL;4φ
is no a
pseudococyclic pa ial Hadama d ma ix.
MψL;3ψL;4φ≡++++
−+− −
4. Pseudocobounda y Pa ial Hadama d Ma ices o e La in Rec angles
Le us s a ou s udy by dealing wi h P oblem 1conce ning he condi ions unde
which we can ensu e he exis ence o pseudocobounda y pa ial Hadama d ma ices o e
a gi en La in ec angle L∈ R ,nwi h ns(L)>0. Fi s ly, we ocus on he case =1.
P oposi ion 1.
The e always exis s a pseudocobounda y pa ial Hadama d ma ix o e a La in
ec angle L ∈ R1,nsa is ying ha ns(L)>0.
P oo .
Le
L∈ R1,n
be such ha
ns(L)>
0. F om Lemma 1, i mus be
L[
1,1
] =
1 and
L[
1,
j] = h
o some posi i e in ege s
j
,
h≤n
such ha 1
6=j6=h6=
1. Hence, he La in
ec angle condi ion o no epe i ion o symbols in each ow implies ha
L[L[
1,1
]
,
j] = h6=
L[
1,
h] = L[
1,
L[
1,
j]]
. Thus,
(
1,1,
j)∈NS(L)
and
h∈ H(L)
. The ma ix
ψL;h
is i ially
pa ial Hadama d o e L.
Le us ocus now on he case
>
1. Since
ns(L) =
0 o all
L∈ R2,2
, we also suppose
ha he numbe
n≥
o columns is a mul iple o 4. We s a wi h a p elimina y lemma
ha desc ibes he en ies wi hin each ow and column o any pseudocobounda y pa ial
Hadama d ma ix o e a gi en La in ec angle. Pa icula ly, i cha ac e izes he ows and
columns ha a e uni o mly signed.
Lemma 3.
Le
and
n
be wo posi i e in ege s such ha 2
≤ ≤n
. Fu he , le
ψL;h
be he
h
-pseudocobounda y o e a La in ec angle
L∈ R ,n
wi h
ns(L)>
0and
h∈ H(L)
. Then, he
ollowing asse ions hold.
1. Le i ≤ be such ha L[i,h]6=h. Then,
ψL;h(i,j) = (−∂h(i),i ei he j =h o L[i,j] = h,
∂h(i),o he wise.
2. Le j ≤n. Then,
ψL;h(i,j) =
−∂h(j),i (h≤ ,L[h,j]6=h and ei he i =h o L[i,j] = h,
h> and L[i,j] = h,
∂h(j),o he wise.
3.
The
i
h ow o he
h
-pseudocobounda y ma ix
MψL;h
wi h
i≤
is uni o mly signed i and
only i
L[i
,
h] = h
. In such a case,
ψL;h(i
,
j) = ∂h(i)
o all
j≤n
. As a consequence, he e
always exis s a mos one uni o mly signed ow.
4.
Le
j≤n
. I
h>
, hen he
j
h column o
Mψh
is uni o mly signed i and only i
L[i
,
j]6=h
o e e y posi i e in ege
i≤
. O he wise, i
h≤
, hen he
j
h column o
Mψh
is uni o mly
signed i
L[h
,
j] = h
. I
>
2, hen his su icien condi ion is also necessa y. In any case,
Ma hema ics 2021,9, 113 8 o 20
ψL;h(i
,
j) = ∂h(j)
o all
i≤
. Fu he mo e, he e exis s exac ly one uni o mly signed
column i h ≤ and >2.
P oo .
The i s wo asse ions and he su icien condi ions o he las wo asse ions ollow
om he De ini ion (6). Le us ocus now on he p oo o he necessa y condi ion o he
hi d asse ion ( ha one o he ou s a emen s ollows simila ly). Thus, le us suppose
he exis ence o a posi i e in ege
i≤
such ha he
i
h ow o he
h
-pseudocobounda y
ma ix
MψL;h
is uni o mly signed. Then, he men ioned De ini ion (6) implies ha ei he
∂h(L[i
,
j]) = ∂h(j)
o
∂h(L[i
,
j]) = −∂h(j)
o all
j≤n
. Ne e heless, since
n>
2, he
de ini ion o he map
∂h
, oge he wi h he La in ec angle condi ion o no epe i ions
o symbols pe ow, implies ha he second op ion is no possible. Hence, i mus be
L[i
,
h] = h
. The inal consequence desc ibed in he hi d asse ion holds s aigh o wa dly
om he La in ec angle condi ion o no epe i ions o symbols in each column.
Conce ning he las sen ence o he ou h asse ion, he de ini ion o he map
∂h
,
oge he wi h (6) and he La in ec angle condi ion o no epe i ions o symbols pe ow,
implies he exis ence o exac ly one uni o mly signed column when >2.
Example 7.
Le
L
be he La in ec angle desc ibed in Example 1. The hi d asse ion o Lemma 3
explains, o ins ance, he uni o mi y o signs o he i s ow o bo h ma ices
MψL;1
and
MψL;2
, and
also o he second ow o he ma ix
MψL;4
, all o hem desc ibed in Example 3. In addi ion, i also
explains ha he e does no exis any uni o mly signed ow in he ma ix MψL;3.
The ou h asse ion o Lemma 3explains, o ins ance, he uni o mi y o signs o he i s
column o
MψL;1
and he hi d column o
MψL;2
. I also explains he wo uni o mly signed columns
o bo h ma ices
MψL;3
and
MψL;4
. Ne e heless, his ou h asse ion o Lemma 3does no explain
he uni o mi y o signs o he second columns o
MψL;1
and
MψL;2
, which ollows indeed om he
second asse ion o his lemma. I illus a es, in pa icula , he excep ional case
=
2 ha was
disca ded he ein. The case
>
2is illus a ed by he exis ence o exac ly one uni o mly signed
column in any o he La in squa es desc ibed in Examples 4and 5.
The ollowing esul cha ac e izes he La in ec angles o e which a pseudocobound-
a y pa ial Hadama d ma ix exis s. As such, i cons i u es, oge he wi h P oposi ion 1,
he answe o P oblem 1.
P oposi ion 2.
Le
and
n
be wo posi i e in ege s such ha 2
≤ ≤n
. Fu he , le
ψL;h
be
he
h
-pseudocobounda y o e a La in ec angle
L∈ R ,n
wi h
ns(L)>
0and
h∈ H(L)
. Then,
he pseudocobounda y ma ix MψL;his pa ial Hadama d i and only i n =4.
P oo . Lemma 3enables us o ensu e ha he h-pseudocobounda y ψL;hhas a leas −2
ows wi h p ecisely wo nega i e en ies. Hence, he pseudocobounda y ma ix
MψL;h
canno be Hadama d i
n>
4. Conce ning he case
n=
4, le us emind he eade ha
he e exis 576 La in squa es o o de ou , om which only 16 o hem cons i u e he Cayley
able o an associa i e quasig oup. A simple and exhaus i e compu a ion enables us o
ensu e ha
H(L) = [
4
]
o all o he 560 emaining La in squa es
L∈ R4,4
, and also ha all
o hei ela ed
h
-pseudocobounda y ma ices a e pa ial Hadama d, wha e e he posi i e
in ege
h≤
4 is. As a consequence, e e y
h
-pseudocobounda y ma ix o an
×
4 La in
ec angle is pa ial Hadama d, wha e e he wo posi i e in ege s h, ≤4 a e.
Fo La in squa es o any gi en o de , he ollowing esul holds as an immedia e
consequence o Lemma 3, once i is no iced ha i s wo las asse ions always hold in he
case o
L
being a La in squa e. I is illus a ed by any o he pseudocobounda y ma ices
desc ibed in Examples 4and 5.
P oposi ion 3.
Le
ψL;h
be he
h
-pseudocobounda y o e a La in squa e o o de
n>
2wi h
ns(L)>
0and
h∈ H(L)
. Then, he
h
-pseudocobounda y ma ix
MψL;h
con ains exac ly one uni o mly
signed ow and exac ly one uni o mly signed column.
Ma hema ics 2021,9, 113 9 o 20
Le us inish his sec ion by ocusing on P oblem 2conce ning he condi ions unde
which a gi en pa ial Hadama d ma ix is a pseudocobounda y o e some La in ec angle
L∈ R ,n
wi h
ns(L)>
0. F om P oposi ion 2, we may assume
n=
4. Fi s ly, we ocus on
he case
=
1. No ice in his ega d ha e e y 1
×n
bina y a ay i ially cons i u es a
pa ial Hadama d ma ix by i sel .
Lemma 4.
Le
M= (m1j)
be a 1
×
4pa ial Hadama d ma ix. I is a pseudocobounda y ma ix
o e a La in ec angle i and only i m11 =1and i con ains exac ly wo nega i e en ies.
P oo .
In o de o p o e he necessa y condi ion, le us suppose ha he pa ial Hadama d
ma ix
M
is an
h
-pseudocobounda y o e a La in ec angle
L∈ R1,4
wi h
h∈ H(L)6=∅
.
F om Lemma 1, i mus be
L[
1,1
] =
1, and hen, he La in ec angle condi ion o no
epe i ions o symbols in each ow implies ha
(
1,1,1
)6∈ NS(L)
and 1
6∈ H(L)
. Hence,
h6=
1 and
m11 =
1. In addi ion, since e e y non-associa i e iple in
NS(L)
is o he o m
(
1,1,
k)
wi h
k∈ {
2,3,4
}
and
h∈ H(L)
, i should be
{h}⊂{L[L[
1,1
]
,
k0]
,
L[
1,
L[
1,
k0]]}=
{L[
1,
k0]
,
L[
1,
L[
1,
k0]]}
o some posi i e in ege
k0∈ {
2,3,4
}
. I
k0=h
, hen we ge
{h}⊂{h}
, which is a con adic ion. So,
L[
1,
h]6=h
, and hence, he ma ix
M
con ains
exac ly wo nega i e en ies. Mo e speci ically, m1k0=m1h=−1.
Now, in o de o p o e he su icien condi ion, le us suppose ha
m11 =
1 and le
h
,
i
,
j∈[
4
] {
1
}
be h ee dis inc posi i e in ege s such ha
m1,h=m1,i=−
1 and
m1,j=
1.
Then, le
L∈ R1,4
be de ined so ha
L[
1,1
] =
1,
L[
1,
h] = j
,
L[
1,
i] = h
, and
L[
1,
j] = i
.
Then,
L[L[
1,1
]
,
h] = j6=i=L[
1,
L[
1,
h]]
. Hence,
(
1,1,
h)∈NS(L)
and
h∈ H(L)
. I is
s aigh o wa dly e i ied ha he pa ial Hadama d ma ix
M
is an
h
-pseudocobounda y
o e L.
Le us ocus now on he case 2
≤ ≤
4. The ollowing p elimina y lemma holds
s aigh o wa dly om he de ini ion (6) o a pseudocobounda y.
Lemma 5.
Le
∈ {
2,3,4
}
and le
M= (mij)
be an
×
4pa ial Hadama d ma ix such ha
he e exis s a La in ec angle
L∈ R ,4
wi h
ns(L)>
0, o e which
M
is an
h
-pseudocobounda y
ma ix o some h ∈ H(L). The ollowing asse ions hold.
1.
The
i
h ow o he pa ial Hadama d ma ix
M
wi h
i≤
is uni o mly signed i and only i
L[i,h] = h. In such a case, mij =∂h(i) o all j ≤4.
2.
I
h≤
and
L[h
,
h] = h
, hen
mih =−
1 o all
i≤
. Mo eo e , i
mij =−
1wi h
i6=h6=j
hen L[i,j] = h.
3.
I
L[i
,
h] = h
o some
i∈[ ] {h}
, hen
mjh =−∂h(j)
o all
j∈[ ] {i}
. Mo eo e ,
i mjk =−∂h(j) o some j ∈[ ] {i}and k ∈[4] {h}, hen L[j,k] = h.
Example 8. Le us conside he ollowing ou pa ial Hadama d ma ices.
M1≡− − − −
+− − +M2≡−−−−
−−+ +
M3≡+ + + +
++− − M4≡++++
−+−+
Le
N= (nij)∈ {M1
,
M2}
. The i s s a emen o Lemma 5enables us o ensu e ha , i he
ma ix
N
we e an
h
-pseudocobounda y o e some La in ec angle
L∈ R2,4
wi h
h∈ H(L)6=∅
,
hen i should be
h=
1and
L[
1,1
] =
1. Howe e , hen, he second s a emen o he men ioned
lemma implies ha
n2,1 =−
1, which is no he case when
N=M1
. As a consequence, he pa ial
Hadama d ma ix
M1
is no a pseudocobounda y o e any La in ec angle. Fu he , conce ning
he case
N=M2
, he second s a emen o Lemma 5also enables us o ensu e ha
L[
2,2
] =
1.
Thus, o ins ance, i is eadily e i ied ha he ma ix
M2
is a 1-pseudocobounda y o e he
La in ec angle
Ma hema ics 2021,9, 113 16 o 20
MψL;1ψL;2ψL;3 ≡−−+ +
− − − − .
Thus, i we conside he subse
S={
1,2,3
}
, hen
D−
L(S
,2
) = S
,and hence,
S∆D−
L(S
,2
) =
∅
. In addi ion,
D−
L(S
,1
) = {
1,2,4
}
, and hus,
S∆D−
L(S
,1
) = {
3,4
}
. Tha is,
|S∆D−
L(S,1)|=2
.
Now, in o de o illus a e he sha pness o he lowe bound desc ibed in Theo em 2, we
can make use o any o he
h
-pseudocobounda y ma ices o e
L
wi h
h∈ {
1,2,4
}
ha a e
desc ibed in Example 3. Thus, o ins ance, i we conside
S={
1
}
, hen
D−
L(S
,1
) = S
and
S∆D−
L(S,2) = {1, 2}.
In o de o illus a e he sha pness o his lowe bound, bu now a oiding he pu ely pseudo-
cobounda y amewo k, le us conside he ollowing 2×8La in ec angle.
L0≡1 2 3 4 5 6 7 8
4 3 1 2 6 5 8 7
I is easily e i ied ha
NS(L0) = {(
1,1,3
)
,
(
1,1,4
)
,
(
1,2,1
)
,
(
1,2,2
)}
, and
H(L0) =
{
3,4
}
. I we conside he subse
S={
3,4
}
, hen we ha e ha
D−
L0(S
,1
) = S
and
S∆D−
L0(S
,2
) =
{
1,2,3,4
}
. Then, he pseudococyclic ma ix o e
L0
associa ed wi h he pseudococycle
ψL0;3ψL0;4
is
pa ial Hadama d.
MψL0;3ψL0;4 ≡++++++++
−−−−+ + + +
6. Pseudococyclic Pa ial Hadama d Ma ices Associa ed wi h Non-T i ial Cocycles
Le us inish ou s udy by ocusing on he pseudococycles
∏h∈S⊆H(L)ψL;hφ
o e a
gi en La in ec angle
L∈ R ,n
wi h
ns(L)>
0 whe e he cocycle
φ
o e
L
is no i ial. The
ollowing esul gene alizes P oposi ion 7by cha ac e izing he La in ec angles o e which
one such pseudococycle exis s so ha i s ela ed pseudococyclic ma ix is pa ial Hadama d.
P oposi ion 8.
Le
ψ=∏h∈S⊆H(L)ψL;hφ
be a pseudococycle o e a La in ec angle
L∈ R ,n
wi h
ns(L)>
0 o some cocycle
φ
o e
L
. The pseudococyclic ma ix
Mψ
is pa ial Hadama d i
and only i , o each pai o posi i e in ege s i1,i2≤ , he se
j∈ D−
L(S,i1)∆D−
L(S,i2):φ(i1,j) = φ(i2,j)∪j6∈ D−
L(S,i1)∆D−
L(S,i2):φ(i1,j) = −φ(i2,j)(7)
has ca dinali y n
2.
P oo .
The esul ollows in a simila way o he p oo o P oposi ion 7, once i is obse ed
ha ∑j≤nψ(i1,j)ψ(i2,j) = 0 i and only i
∑
j∈D−
L(S,i1)∆D−
L(S,i2)
φ(i1,j)φ(i2,j) = ∑
j6∈D−
L(S,i1)∆D−
L(S,i2)
φ(i1,j)φ(i2,j).
Keeping in mind he obse a ion made jus a e P oposi ion 7, he ime complexi y o
he implici algo i hm desc ibed in P oposi ion 8is
O(n5)
, which co esponds once mo e
o he La in squa e case.
Example 14.
Le
L
be he La in ec angle de ined in Example 1and le us conside he pseu-
dococyclic ma ix
MψL2;3ψL2;4 φ2
desc ibed in Example 6, which is no pa ial Hadama d. I we
again ake
S={
3,4
}
, hen we ha e ha
D−
L(S
,1
) = {
3,4
}
and
D−
L(S
,2
) = {
1,4
}
. Hence,
D−
L(S
,1
)∆D−
L(S
,2
) = {
1,3
}
. In his case, he se de ined in (7) conce ning he pseudococycle
ψL;3ψL;4φ2
is he se
{
2,3,4
}
, which is no o med by wo elemen s, as is equi ed by
P oposi ion 8
.
Ma hema ics 2021,9, 113 17 o 20
Example 15.
Le us conside he La in ec angle
L∈ R2,12
ha is desc ibed in Example 12, and
le us de ine he cocycle φo e L ha is ep esen ed by he ma ix
Mφ≡
++++++++++++
++++−−++++++
++++++++++++
No ice he e ha he hi d ow o his cocyclic ma ix co esponds o he posi i e in ege
4∈ S(L) = {1, 2,4}. The pseudococyclic ma ix associa ed wi h he pseudococycle ψL;1ψL;4φis
MψL;1ψL;4φ≡− − ++−−−−−−−−
− − ++−−++++++
which is pa ial Hadama d. I we conside he subse
S={
1,4
}
, hen we ha e ha
D−
L(S
,1
)∆D−
L
(S
,2
) = [
4
]
. Mo eo e , he se de ined in (7) conce ning he pseudococycle
ψL;1ψL;4φ
is he se
{1,2,3,4,5,6}, which is o med by six elemen s, as is equi ed by P oposi ion 8.
The pseudococyclic pa ial Hadama d ma ix in Example 15 shows ha Theo em 1
canno be gene alized o pseudococycles associa ed wi h non- i ial cocycles. Conce ning
he possible gene aliza ion o Theo em 2, he ollowing esul deals wi h he case o
a pseudocycle ela ed o a non- i ial cocycle whose pseudococyclic pa ial Hadama d
ma ix con ains a uni o mly signed ow.
P oposi ion 9.
Le
ψ=∏h∈S⊆H(L)ψL;hφ
be a pseudococycle o e a La in ec angle
L∈ R ,n
wi h
ns(L)>
0 o some cocycle
φ
o e
L
. I he pseudococyclic ma ix
Mψ
is pa ial Hadama d,
hen he e exis s a mos one posi i e in ege i ≤ and an in ege a ∈ {−1,1}such ha
φ(i,j) = (a,i j ∈S∆D−
L(S,i),
−a,i j 6∈ S∆D−
L(S,i).
P oo . Since he pseudococyclic ma ix Mψis pa ial Hadama d, i can only ha e a mos
one uni o mly signed ow. Hence, he e exis s a mos one posi i e in ege
i≤
such ha
ψ(i
,
j1) = ψ(i
,
j2)
o all pai s o posi i e in ege s
j1
,
j2≤n
. The esul hen ollows om
he hi d s a emen o Lemma 6, oge he wi h he de ini ion o he cocycle φ.
Example 16.
Le us conside he La in ec angle
L∈ R2,12
ha is desc ibed in Example 12 and
le us conside he cocycle φo e L ha is ep esen ed by he ma ix
Mφ≡
− − ++−−−−−−−−
+ + −−−−−−−−−−
++++++++++++
Fu he , le us conside he subse
S={
1,4
}
. Then,
S∆D−
L(S
,1
) = {
3,4
}
and
S∆D−
L(S
,2
) =
{
1,2
}
. Acco ding o P oposi ion 9, he pseudococyclic ma ix
MψL;1ψL;4φ
o e
L
is no pa ial
Hadama d. In ac ,
MψL;1ψL;4φ≡++++++++++++
−−−−−−−−−−−−.
7. Conclusions and Fu he Wo k
In his pape , we ha e in oduced he concep s o bo h he pseudocobounda y and
pseudococycle o e a La in ec angle (see Sec ion 3) as a na u al gene aliza ion o he
simila no ions ecen ly desc ibed in [
17
] o e quasig oups. To his end, we ha e made
use o he cocyclic amewo k o e La in ec angles p e iously in oduced by he au ho s
in [
18
]. Bo h cocyclic and pseudococyclic de elopmen s o e La in ec angles oge he
Ma hema ics 2021,9, 113 18 o 20
cons i u e a much mo e gene al amewo k han he classical cocyclic amewo k o e
g oups. I s po en ial has al eady been illus a ed in he men ioned pape s by means o
examples o (pseudo)cocyclic Hadama d ma ices o e quasig oups ha a e no cocyclic
o e any g oup. This pape cons i u es a s ep o wa d in his ega d. Thus, o ins ance,
Example 5
illus a es a pseudocobounda y Hadama d ma ix o e a La in squa e ha is
no cocyclic o e any La in ec angle.
Le us ema k ha his pape is concei ed as an in oduc o y s age conce ning he
undamen als o bo h he pseudocobounda y and he pseudococyclic amewo ks o e
La in ec angles. Pa icula ly, in Sec ion 4, we comple ely answe ed bo h
P oblems 1and 2
conce ning he condi ions unde which we may ensu e ei he he exis ence o a pa ial
Hadama d ma ix ha is a pseudocobounda y o e a gi en La in ec angle o , ecip ocally,
he exis ence o a La in ec angle o e which a gi en pa ial Hadama d ma ix is a pseudo-
cobounda y. Mo e speci ically, P oposi ions 1and 2gi e he answe o he i s ques ion,
whe eas he second one is answe ed by Lemma 1, oge he wi h P oposi ions 4–6.
Fu he mo e, we ha e also deal wi h he p oblem o de e mining unde which
condi ions we may ensu e he exis ence o a pa ial Hadama d ma ix ha is pseudococyclic
o e a gi en La in ec angle. To his end, we ha e dis inguished wo dis inc amewo ks
(see Sec ions 5and 6), depending on whe he we make use o i ial cocycles o no .
The ecip ocal p oblem conce ning he condi ions unde which we may ensu e he exis ence
o a La in ec angle o e which a gi en pa ial Hadama d ma ix is pseudococyclic is
es ablished as u u e wo k. Once his las ques ion is sol ed, he nex na u al s age would
be he cons uc ion o pseudococyclic pa ial Hadama d ma ices o highe dimensions in
o de o deal wi h he Hadama d conjec u e desc ibed in he in oduc o y sec ion, which
indeed cons i u es he keys one o he heo y o Hadama d ma ices.
The ollowing open ques ions a e also es ablished as u u e wo k. They gene alize
simila ones desc ibed o he cocyclic de elopmen o Hadama d ma ices o e La in
ec angles [18].
P oblem 3.
Le
M
be an
×n
pa ial Hadama d ma ix ha is no pseudococyclic o e any La in
ec angle. Does he e exis , howe e , a pa ial Hadama d equi alen ma ix in he same equi alence
class o M o which one such La in ec angle can be ound?
P oblem 4.
Le us conside an equi alence class o Hadama d ma ices such ha none o hem a e
cocyclic o e any ini e g oup. Does he e exis , howe e , a Hadama d ma ix wi hin such a class
ha is pseudococyclic o e a La in ec angle?
Pseudococycles o e La in ec angles ha e been in oduced in his pape as he p od-
uc o a cocycle wi h some pseudocobounda ies. A possible gene aliza ion o his no ion
consis s o enabling he p oduc o a cocycle no only wi h pseudocobounda ies ( ela ed
o non-associa i e iples), bu also wi h elemen a y cobounda ies ( ela ed o associa i e
iples). This would cons i u e a mo e gene al amewo k ha pu s oge he bo h co-
cyclic and pseudococyclic amewo ks o e La in ec angles. I s o mal desc ip ion and
cha ac e iza ion is also p oposed as u u e wo k.
Finally, ano he ques ion o ake in o conside a ion o u he s udy is he ollowing
one. Bo h he pseudocobounda y and he pseudococyclic amewo ks o e La in ec angles
desc ibed in his pape a e based on he exis ence o non-associa i e iples wi hin a La in
ec angle. As was al eady indica ed in he in oduc o y sec ion and in Sec ion 3, he s udy
o his ype o iple in he case o dealing wi h La in squa es has ecei ed pa icula
a en ion in he ecen li e a u e [
32
–
36
] because o i s possible applica ion in di e en a eas
as c yp og aphy [
30
,
31
]. A comp ehensi e s udy o non-associa i e iples in he case o
dealing wi h La in ec angles ins ead o La in squa es is es ablished, he e o e, as na u al
u he wo k.
Ma hema ics 2021,9, 113 19 o 20
Au ho Con ibu ions:
Concep ualiza ion, V.Á., R.M.F. and F.G.; Da a cu a ion, M.D.F., F.G. and
M.B.G.; Fo mal analysis, V.Á., R.M.F. and F.G.; In es iga ion, V.Á. and R.M.F.; Me hodology, R.M.F.,
M.D.F. and M.B.G.; So wa e, V.Á. and R.M.F.; Supe ision, F.G.; Valida ion, M.D.F. and M.B.G.;
W i ing—o iginal d a , R.M.Falcón; W i ing— e iew & edi ing, V.Á., R.M.F., M.D.F. and F.G. All au-
ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Acknowledgmen s:
This wo k was pa ially suppo ed by he Resea ch P ojec FQM-016 om Jun a
de Andalucía. In addi ion, he au ho s wan o exp ess hei g a i ude o he anonymous e e ees o
he comp ehensi e eading o he pape and hei pe inen commen s and sugges ions, which helped
imp o e he manusc ip . Pa icula ly, we a e g a e ul o he anonymous e e ee who sugges ed o us
he mo e gene al amewo k desc ibed in he conclusion sec ion conce ning he p oduc o a cocycle
wi h bo h ypes o pseudocobounda ies and elemen a y cobounda ies o a gi en La in ec angle.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Appendix A. Glossa y o Symbols
H(L)The se o posi i e in ege s sa is ying Condi ion (5).
MψL;h
The
h
-pseudocobounda y ma ix associa ed o a La in ec angle
L∈ R ,n
, wi h
h∈ H(L)
.
[n]The se {1, . . . , n}.
ns(L)The non-associa i e index o a La in ec angle L.
NS(L)The se o non-associa i e iples o a La in ec angle L.
R ,nThe se o ×nLa in ec angles wi h en ies in [n].
S(L)The subse o symbols desc ibing he ows o a cocyclic ma ix o e a La in ec angle L.
ψL;hThe h-pseudocobounda y o e a La in ec angle L, wi h h∈ H(L).
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