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The Hadamard multiary quasigroup product
Ra´ul M. Falc´on
Department of Applied Maths I.
Universidad de Sevilla.
rafalgan@us.es
Kongunadu Arts and Science College.
Tamil Nadu, India.
January 30, 2024.
Joint work with:
Lorenzo Mella and Petr Vojtˇechovsk´y.
(Spanish Strategic R+D Project TED2021-130566B-I00)
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 1 / 45
Multiary operations.
Let m≥2 be a positive integer.
Ωm(X):= {m-ary operations on a set X}
An m-ary groupoid over Xis any pair (X,f), with f∈Ωm(X).
Example
Let X={1,2,3}. We consider the binary groupoid (X,f) described by
its Cayley table
A≡
1 2 3
2 1 1
3 1 1
so that f(i,j) := A[i,j], for all i,j∈X.
f(1,2) = 2
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 4 / 45
Multiary operations.
Example
Let X={1,2,3}. We consider the ternary groupoid (X,f) described by
the following arrays.
A1≡
123
211
311
A2≡
2 1 2
1 2 3
2 3 2
A3≡
3 3 1
3 3 3
1 3 1
so that f(i,j,k) := Ak[i,j], for all i,j,k∈X.
f(1,2,1) = 2 f(1,2,2) = 1 f(1,2,3) = 3
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 5 / 45
Multiary operations.
An m-ary groupoid (X,f) is an m-ary monoid if it has an identity
element e∈X. That is,
f(e,...,e
| {z }
i
,a,e,...,e
| {z }
m−i−1
) = a
for all a∈Xand every non-negative integer i<m.
Example
1 is the unit element of the binary monoid
123
211
311
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 6 / 45
Multiary operations.
Example
1 and 2 are the unit elements of the ternary monoid (X,f) described by
the following arrays.
A1≡
123
211
311
A2≡
2 1 2
1 2 3
2 3 2
A3≡
3 3 1
3 3 3
1 3 1
so that f(i,j,k) := Ak[i,j], for all i,j,k∈X.
f(1,2,1) = 2 f(1,2,2) = 1 f(1,2,3) = 3
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 7 / 45
Multiary operations.
An element a∈Xis invertible in an m-ary monoid (X,f) if there
exist a−1∈Xand an identity element ein (X,f) such that
f(a,...,a
| {z }
i
,a−1,a,...,a
| {z }
m−i−1
) = e(1)
for every non-negative integer i<m.
The element a−1is an inverse of ain (X,f).
The set of inverses of ain (X,f) is Inv(a,f)
Example
In the binary monoid
123
211
311
it is Inv(1,f) = {1}and Inv(2,f) = Inv(3,f) = {2,3}.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 8 / 45
Generalizing the classical Hadamard product.
+Z30 1 2
0 0 1 2
1 1 2 0
2 2 0 1
Example
A≡
112
201
022
∈ M3(Z3)
B≡
0 2 1
2 1 2
1 1 0
∈ M3(Z3)
⇒A+Z3B≡
1 0 0
1 1 0
1 0 2
∈ M3(Z3).
The relevant aspect here is the Cayley table under consideration.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 13 / 45
Generalizing the classical Hadamard product.
+Z30 1 2
0 0 1 2
1 1 2 0
2 2 0 1
Example
A≡
112
201
022
∈ M3(Z3)
B≡
0 2 1
2 1 2
1 1 0
∈ M3(Z3)
⇒A+Z3B≡
1 0 0
1 1 0
1 0 2
∈ M3(Z3).
The relevant aspect here is the Cayley table under consideration.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 13 / 45
Generalizing the classical Hadamard product.
L≡
0 1 2
1 2 0
2 0 1
Example
A≡
112
201
022
∈ M3(Z3)
B≡
021
212
110
∈ M3(Z3)
⇒ALB≡
1 0 0
1 1 0
1 0 2
∈ M3(Z3).
In this study, we focus on Cayley tables of multiary quasigroups. That is,
on Latin hypercubes.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 14 / 45
Generalizing the classical Hadamard product.
L≡
0 1 2
1 2 0
2 0 1
Example
A≡
112
201
022
∈ M3(Z3)
B≡
021
212
110
∈ M3(Z3)
⇒ALB≡
1 0 0
1 1 0
1 0 2
∈ M3(Z3).
In this study, we focus on Cayley tables of multiary quasigroups. That is,
on Latin hypercubes.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 14 / 45
n-ary quasigroups and Latin hypercubes.
Ruth Moufang (1935)
Aquasigroup is a pair (X,f)formed by
•a set X, and
•a binary operation f:X×X→X,
such that both equations
f(a,x) = band f(y,a) = b
have unique solutions x,y∈X, for all a,b∈X.
Its Cayley table is a Latin square.
1 2 3
2 3 1
3 1 2
Entry set:Ent(L) := {(row,column,symbol)}={(i,j,L[i,j])}.
Ent(L) = {(1,1,1),(1,2,2),(1,3,3)
(2,1,2),(2,2,3),(2,3,1),
(3,1,3),(3,2,1),(3,3,2)}.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 15 / 45
n-ary quasigroups and Latin hypercubes.
Ruth Moufang (1935)
Aquasigroup is a pair (X,f)formed by
•a set X, and
•a binary operation f:X×X→X,
such that both equations
f(a,x) = band f(y,a) = b
have unique solutions x,y∈X, for all a,b∈X.
Its Cayley table is a Latin square.
1 2 3
2 3 1
3 1 2
Entry set:Ent(L) := {(row,column,symbol)}={(i,j,L[i,j])}.
Ent(L) = {(1,1,1),(1,2,2),(1,3,3)
(2,1,2),(2,2,3),(2,3,1),
(3,1,3),(3,2,1),(3,3,2)}.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 15 / 45
n-ary quasigroups and Latin hypercubes.
Ruth Moufang (1935)
Aquasigroup is a pair (X,f)formed by
•a set X, and
•a binary operation f:X×X→X,
such that both equations
f(a,x) = band f(y,a) = b
have unique solutions x,y∈X, for all a,b∈X.
Its Cayley table is a Latin square.
1 2 3
2 3 1
3 1 2
An n-ary quasigroup is a pair (X,f)formed by a set Xand an n-ary
operation f:Xn→Xsuch that the equation
f(x1,...,xn) = y
has unique solution in X, whenever n−1 variables in {x1,...,xn}, and
also the variable y, are fixed.
Its Cayley table is a Latin hypercube of dimension n.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 16 / 45
n-ary quasigroups and Latin hypercubes.
Ruth Moufang (1935)
Aquasigroup is a pair (X,f)formed by
•a set X, and
•a binary operation f:X×X→X,
such that both equations
f(a,x) = band f(y,a) = b
have unique solutions x,y∈X, for all a,b∈X.
Its Cayley table is a Latin square.
1 2 3
2 3 1
3 1 2
An n-ary quasigroup is a pair (X,f)formed by a set Xand an n-ary
operation f:Xn→Xsuch that the equation
f(x1,...,xn) = y
has unique solution in X, whenever n−1 variables in {x1,...,xn}, and
also the variable y, are fixed.
Its Cayley table is a Latin hypercube of dimension n.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 16 / 45
n-ary quasigroups and Latin hypercubes.
Ruth Moufang (1935)
Aquasigroup is a pair (X,f)formed by
•a set X, and
•a binary operation f:X×X→X,
such that both equations
f(a,x) = band f(y,a) = b
have unique solutions x,y∈X, for all a,b∈X.
Its Cayley table is a Latin square.
1 2 3
2 3 1
3 1 2
An n-ary quasigroup is a pair (X,f)formed by a set Xand an n-ary
operation f:Xn→Xsuch that the equation
f(x1,...,xn) = y
has unique solution in X, whenever n−1 variables in {x1,...,xn}, and
also the variable y, are fixed.
Its Cayley table is a Latin hypercube of dimension n.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 16 / 45
Isotopisms of multiary quasigroups.
SX≡Symmetric group on the set X.
Two m-ary quasigroups (X,f) and (X,g) are isotopic if there exists
m+ 1 permutation π1, . . . , πm+1 ∈SXsuch that
g(π1(a1), . . . , πm(am)) = πm+1 (f(a1,...,am))
for all a1,...,am∈X.
This is an isomorphism if π1=. . . =πm+1.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 18 / 45
The operator .
Ωn(X):= {n-ary operations on a set X}
x:= (x1,...,xn)∈Xn.
: Ωm(X)→Ωm(Ωn(X))
f→f: (Ωn(X))m→Ωn(X)
(g1,...,gm)→f(g1,...,gm)
f(g1,...,gm)(x) := f(g1(x),...,gm(x))
m=n= 2:
f(g1,g2)(x,y) := f(g1(x,y),g2(x,y))
In the usual notation for binary operations:
g1≡ 4 g2≡f≡
4 ?→x4 ?y:= (x4y)(xy)
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 20 / 45
The operator .
Ωn(X):= {n-ary operations on a set X}
x:= (x1,...,xn)∈Xn.
: Ωm(X)→Ωm(Ωn(X))
f→f: (Ωn(X))m→Ωn(X)
(g1,...,gm)→f(g1,...,gm)
f(g1,...,gm)(x) := f(g1(x),...,gm(x))
m=n= 2:
f(g1,g2)(x,y) := f(g1(x,y),g2(x,y))
In the usual notation for binary operations:
g1≡ 4 g2≡f≡
4 ?→x4 ?y:= (x4y)(xy)
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 20 / 45
The operator .
Ωn(X):= {n-ary operations on a set X}
x:= (x1,...,xn)∈Xn.
: Ωm(X)→Ωm(Ωn(X))
f→f: (Ωn(X))m→Ωn(X)
(g1,...,gm)→f(g1,...,gm)
f(g1,...,gm)(x) := f(g1(x),...,gm(x))
m=n= 2:
f(g1,g2)(x,y) := f(g1(x,y),g2(x,y))
In the usual notation for binary operations:
g1≡ 4 g2≡f≡
4 ?→x4 ?y:= (x4y)(xy)
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 20 / 45
The operator .
γc:Xn→X
a7→ γc(a) := c
Lemma
(X,f)embeds into (Ωn(X),f)via the homomomorphism
(X,f)→(Ωn(X),f)
c→Xn→X
x→c
Proof
Let c1,...,cm∈Xand a∈Xn. Then,
γf(c1,...,cm)(a) = f(γc1(a), . . . , γcm(a)) = f(γc1, . . . , γcm)(a).
Hence, γf(c1,...,cm)=f(γc1, . . . , γcm).
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 25 / 45
The operator .
Lemma
(X,f)holds an algebraic identity ϕ=ψif and only if (Ωn(X),f)does.
Necessary condition: If (X,f) holds ϕ=ψ, then
ϕ(g1(a),...,gt(a)) = ψ(g1(a),...,gt(a))
for all g1,...,gm∈Ωn(X) and a∈Xn. Then,
ϕ(g1,...,gt) = ψ(g1,...,gt)
Hence, (Ωn(X),f) also holds ϕ=ψ.
Sufficient condition:
If (Ωn(X),f) holds ϕ=ψ, then
ϕ(γa1, . . . , γat) = ψ(γa1, . . . , γat)
for all a1,...,at∈X. Thus, ϕ(a1,...,at) = ψ(a1,...,at). Hence, (X,f)
also holds ϕ=ψ.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 27 / 45
The operator .
The image of fis img(f):= {f(a): a∈Xm}.
Lemma
Ii(Ωn(X),f) = {g∈Ωn(X) : img(g)⊆Ii(X,f)}.
Example
123
1 1 2 3
2 1 2 3
3 1 1 1
.
Since I0(X, ) = {1,2}, the binary operation
◦123
1 1 2 1
2 2 2 1
3 1 2 2
is a left identity element in (Ω2(X),?).
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 29 / 45
The operator .
Proposition
It is satisfied that
I(Ωn(X),f) = {g∈Ωn(X) : img(g)⊆I(X,f)}.
In particular, the following two statements hold.
1If I(X,f) = {e}for some e∈X, then I(Ωn(X),f) = {γe}.
2If I(Ωn(X),f) = {g}for some g∈Ωn(X), then g=γefor some
e∈X. Moreover, I(X,f) = {e}.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 30 / 45
The operator .
The operator also preserves inverse elements.
Proposition
1If gis an invertible n-ary operator in (Ωn(X),f),
Inv (g,f) = {h∈Ωn(X): h(a)∈Inv(g(a),f)for all a∈Xn}.
2Let c∈X. If c−1∈Inv(c,f), then γc−1∈Inv (γc,f).
3Let c∈X. If (γc)−1∈Inv (γc,f), then img (γc)−1⊆Inv(c,f).
4If (X,f)has unique inverses, then (Ωn(X),f)also has unique
inverses. More precisely, the inverse g−1of g∈Ωn(X)is given by
g−1(a)=(g(a))−1for all a∈Xn.
5If (Ωn(X),f)has unique inverses, then (X,f)also has unique
inverses.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 31 / 45
The operator .
Lemma
If (X,f)is isotopic to (X,g), then (Ωn(X),f)is isotopic to (Ωn(X),g).
Proof
If (π1, . . . , πm+1) is an isotopism from (X,f) to (X,g), we define for each
positive integer i≤m+ 1 the bijection
ρi: Ωn(X)→Ωn(X)
h7→ πih
If h1,...,hm∈Ωn(X) and a∈Xn
g(ρ1(h1), . . . , ρm(hm)) (a) = g(π1(h1(a)), . . . , πm(hm(a))) =
=πm+1(f(h1(a),...,hm(a))) =
=πm+1(f(h1,...,hm)(a)) =
=ρm+1(f(h1,...,hm))(a).
Thus, g(ρ1(h1), . . . , ρm(hm)) = ρm+1(f(h1,...,hm)).
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 32 / 45
The Hadamard quasigroup product.
Qm(X):= {f∈Ωm(X): (X,f) is an m-ary quasigroup}.
Proposition
f∈ Qm(X)⇔ f∈ Qm(Ωn(X)).
Lemma (m=n)
Let (g1,...,gn)∈(Ωn(X))n. The map
Ωn(X)→Ωn(X)
f→ f(g1,...,gn)
is a permutation if and only iff the set {g1,...,gn}is orthogonal. That is,
iff the map
Xn→Xn
x→(g1(x),...,gn(x))
is a permutation.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 33 / 45
The Hadamard quasigroup product.
Qm(X):= {f∈Ωm(X): (X,f) is an m-ary quasigroup}.
Proposition
f∈ Qm(X)⇔ f∈ Qm(Ωn(X)).
Lemma (m=n)
Let (g1,...,gn)∈(Ωn(X))n. The map
Ωn(X)→Ωn(X)
f→ f(g1,...,gn)
is a permutation if and only iff the set {g1,...,gn}is orthogonal. That is,
iff the map
Xn→Xn
x→(g1(x),...,gn(x))
is a permutation.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 33 / 45
The Hadamard quasigroup product.
O(g1,...,gn):={g∈Ωn: ({g1,...,gn}\{gi})∪ {g}orthogonal, for alli}
Proposition
If {g1,...,gn}is orthogonal, then the map
Qn(X)→ O(g1,...,gn)
f→ f(g1,...,gn)
is a bijection.
Under which conditions f(g1,...,gn)∈ Qn(X)?
(m=n= 2)
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 34 / 45
The Hadamard quasigroup product.
O(g1,...,gn):={g∈Ωn: ({g1,...,gn}\{gi})∪ {g}orthogonal, for alli}
Proposition
If {g1,...,gn}is orthogonal, then the map
Qn(X)→ O(g1,...,gn)
f→ f(g1,...,gn)
is a bijection.
Under which conditions f(g1,...,gn)∈ Qn(X)?
(m=n= 2)
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 34 / 45
The Hadamard quasigroup product (m=n= 2).
The Hadamard quasigroup product does not preserve the Latin square
property in general.
f≡
123
312
231
⇒f2:= f(f,f)≡
1 1 1
1 1 1
1 1 1
Lemma
Let f∈ Q2(X)and let g1,g2∈Ω2(X). Then, f(g1,g2)∈ Q2(X)iff
{(g1(x,y),g2(x,y)) : y∈X}
and
{(g1(y,x),g2(y,x)) : y∈X}
are Latin transversals in (X,f), for all x∈X.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 35 / 45
The Hadamard quasigroup product (m=n= 2).
2
`f:= 2
ρf:= f2.
k
`f:= ff,k−1
`fand k
ρf:= fk−1
ρf,f
Proposition
The minimum positive integers (f)and ρ(f)such that
`(f)+1
`f=ρ(f)+1
ρf=f
are quasigroup isomorphism invariants. They satisfy that
(f) = ρ(ft).
2 4 1 3 5
1 3 5 2 4
5 2 4 1 3
4 1 3 5 2
3 5 2 4 1
(ρ, `) = (3,5)
21543
43215
15432
32154
54321
(ρ, `) = (5,3)
25314
53142
31425
14253
42531
(ρ, `) = (5,5)
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 37 / 45
The Hadamard quasigroup product (m=n= 2).
2
`f:= 2
ρf:= f2.
k
`f:= ff,k−1
`fand k
ρf:= fk−1
ρf,f
Proposition
The minimum positive integers (f)and ρ(f)such that
`(f)+1
`f=ρ(f)+1
ρf=f
are quasigroup isomorphism invariants. They satisfy that
(f) = ρ(ft).
2 4 1 3 5
1 3 5 2 4
5 2 4 1 3
4 1 3 5 2
3 5 2 4 1
(ρ, `) = (3,5)
21543
43215
15432
32154
54321
(ρ, `) = (5,3)
25314
53142
31425
14253
42531
(ρ, `) = (5,5)
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 37 / 45
The Hadamard quasigroup product (m=n= 2).
2 5 12 4 8 7 13 14 15 16 3 6 10 11 1 9
3 1 7 8 11 6 14 15 5 13 16 2 9 12 10 4
4 6 5 2 10 15 7 13 11 8 14 3 16 1 9 12
5 15 9 3 14 1 6 8 2 4 7 13 11 16 12 10
1 3 8 9 6 16 11 4 13 12 15 14 5 10 7 2
10 2 11 12 7 4 3 5 14 15 9 16 1 13 8 6
9 12 3 10 1 2 8 6 16 14 13 15 4 5 11 7
15 14 2 1 5 13 4 7 3 6 8 10 12 9 16 11
13 7 14 15 16 3 5 9 10 11 6 12 2 8 4 1
14 10 15 7 13 5 16 2 12 9 11 1 8 4 6 3
16 13 4 5 15 14 1 11 7 10 12 9 3 6 2 8
12 4 16 13 3 8 15 1 9 7 10 11 6 2 14 5
6 8 10 11 9 12 2 16 1 5 4 7 14 3 15 13
7 9 6 16 12 11 10 3 8 1 2 4 13 15 5 14
11 16 1 6 2 9 12 10 4 3 5 8 7 14 13 15
8 11 13 14 4 10 9 12 6 2 1 5 15 7 3 16
ρ= 30
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 38 / 45
The Hadamard quasigroup product of orthogonal binary operations.
MO(k,X):= n(f1,...,fk)∈(Ω2(X))k:fi⊥fj,for all i,jo.
MO2(k,X):= ((f1,f2,g1,...,gk−2)∈MO(k,X): (f1,f2∈Ω2(X),
g1,...,gk−2∈ Q2(X)).
Lemma
The following map is an involution.
Φ : MO2(k,X)→MO2(k,X)
(f1,f2,g1,...,gk−2)→fΦ
1,fΦ
2,gΦ
1,...,gΦ
k−2
where
fΦ
1(f1(x,y),f2(x,y)) := x
fΦ
2(f1(x,y),f2(x,y)) := y
gΦ
s(f1(x,y),f2(x,y)) := gs(x,y),
for all s∈ {1,...,k−2}.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 39 / 45
The Hadamard quasigroup product of orthogonal binary operations.
f1f2g1. . . gk−2
x y f1(x,y)f2(x,y)g1(x,y). . . gk−2(x,y)
l
f1f2g1. . . gk−2
f1(x,y)f2(x,y)x y g1(x,y). . . gk−2(x,y)
l
fΦ
1fΦ
2gΦ
1. . . gΦ
k−2
f1(x,y)f2(x,y)x y g1(x,y). . . gk−2(x,y)
If f1,f2∈ Q2(X), then (f1,f2,g1, . . . gk−2) and fΦ
1,fΦ
2,gΦ
1,...,gΦ
k−2are
two paratopic k-MOLS [Egan, Wanless’16].
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 40 / 45
The Hadamard quasigroup product of orthogonal binary operations.
f1f2g1. . . gk−2
x y f1(x,y)f2(x,y)g1(x,y). . . gk−2(x,y)
l
f1f2g1. . . gk−2
f1(x,y)f2(x,y)x y g1(x,y). . . gk−2(x,y)
l
fΦ
1fΦ
2gΦ
1. . . gΦ
k−2
f1(x,y)f2(x,y)x y g1(x,y). . . gk−2(x,y)
If f1,f2∈ Q2(X), then (f1,f2,g1, . . . gk−2) and fΦ
1,fΦ
2,gΦ
1,...,gΦ
k−2are
two paratopic k-MOLS [Egan, Wanless’16].
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 40 / 45
The Hadamard quasigroup product of orthogonal binary operations.
fΦ
1(f1(x,y),f2(x,y)) := x
fΦ
2(f1(x,y),f2(x,y)) := y
gΦ
s(f1(x,y),f2(x,y)) := gs(x,y),
Lemma
For each s∈ {1,...,k−2},
gΦ
s(f1,f2) = gs
and
gsfΦ
1,fΦ
2=gΦ
s.
f1f2g1. . . gk−2
lΦ
fΦ
1fΦ
2g1fΦ
1,fΦ
2. . . gsfΦ
1,fΦ
2
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 41 / 45
The Hadamard quasigroup product of orthogonal binary operations.
fΦ
1(f1(x,y),f2(x,y)) := x
fΦ
2(f1(x,y),f2(x,y)) := y
gΦ
s(f1(x,y),f2(x,y)) := gs(x,y),
Lemma
For each s∈ {1,...,k−2},
gΦ
s(f1,f2) = gs
and
gsfΦ
1,fΦ
2=gΦ
s.
f1f2g1(f1,f2). . . gk−2(f1,f2)
lΦ
fΦ
1fΦ
2g1. . . gk−2
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 42 / 45
The Hadamard quasigroup product of orthogonal binary operations.
Theorem
If {g1,g2}is orthogonal, then the following map is a bijection.
O(g1,g2)∩ Q2(X)→ O gΦ
1,gΦ
2∩ Q(X)
f→ fgΦ
1,gΦ
2
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gΦ
1gΦ
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Ra´ul M. Falc´on The Hadamard multiary quasigroup product 43 / 45
REFERENCES
J Egan, IM Wanless, Enumeration of MOLS of small order. Math. Comput. 85 (2016),
799–824.
RM Falc´on, V ´
Alvarez, JA Armario, MD Frau, F Gudiel, MB G¨uemes, A computational
approach to analyze the Hadamard quasigroup product, Electron. Res. Arch. 31 (2023),
3245–3263.
RM Falc´on, L Mella, Petr Vojtˇechovsk´y, The multiary quasigroup product, Submitted.
Ra´ul M. Falc´on The Hadamard multiary quasigroup product 44 / 45
