This document introduces hyperpolyadic structures, which are n-ary analogs of binary division algebras like the reals, complexes, quaternions, and octonions. It proposes two constructions: 1) A matrix polyadization procedure that increases the dimension of a binary algebra to obtain a corresponding n-ary algebra by using cyclic shift block matrices. 2) An "imaginary tower" construction on subsets of binary division algebras that gives nonderived ternary division algebras of half the original dimension, called "half-quaternions" and "half-octonions."

1. arXiv:2312.01366v2
[math.RA]
5
Dec
2023
HYPERPOLYADIC STRUCTURES
STEVEN DUPLIJ
Center for Information Technology (CIT), Universität Münster, Röntgenstrasse 7-13
D-48149 Münster, Deutschland
ABSTRACT. We introduce a new class of division algebras, hyperpolyadic algebras, which correspond to the
binary division algebras R, C, H, O without considering new elements. First, we use the proposed earlier
matrix polyadization procedure which increases the algebra dimension. The obtained algebras obey the
binary addition and nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For
each invertible element we define a new norm which is polyadically multiplicative and the corresponding map
is n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds
to the consequent embedding of monomial matrices from the polyadization procedure. Then we obtain
another series of n-ary algebras corresponding to the binary division algebras which have more dimension,
that is proportional to intermediate arities, and they are not isomorphic to those obtained by the previous
constructions. Second, we propose a new iterative process (we call it “imaginary tower”), which leads to
nonunital nonderived ternary division algebras of half dimension, we call them “half-quaternions” and “half-
octonions”. The latter are not subalgebras of the binary division algebras, but subsets only, since they have
different arity. Nevertheless, they are actually ternary division algebras, because allow division, and their
nonzero elements are invertible. From the multiplicativity of the introduced “half-quaternion” norm we
obtain the ternary analog of the sum of two squares identity. We prove that the introduced unitless ternary
division algebra of imaginary “half-octonions” is ternary alternative.
CONTENTS
1. INTRODUCTION 2
2. PRELIMINARIES 2
3. MATRIX POLYADIZATION 3
4. POLYADIZATION OF DIVISION ALGEBRAS 6
5. POLYADIC NORMS 13
6. POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION 15
6.1. Abstract (tuple) approach 15
6.2. Concrete (hyperembedding) approach 16
6.3. Polyadic Cayley-Dickson process 17
7. POLYADIC IMAGINARY DIVISION ALGEBRAS 19
7.1. Complex number ternary division algebra 19
7.2. Half-quaternion ternary division algebra 20
7.3. Half-octonion ternary division algebra 21
REFERENCES 23
E-mail address: douplii@uni-muenster.de; sduplij@gmail.com; https://ivv5hpp.uni-muenster.de/u/douplii.
Date: of start September 23, 2023. Date: of completion December 3, 2023.
Total: 37 references.
2010 Mathematics Subject Classification. 11R52, 17A35, 17A40, 17A42, 20N10, 20N15 .
Key words and phrases. n-ary algebra, querelement, quaternion, octonion, Cayley-Dickson construction, division algebra.

2. 1. INTRODUCTION
The field extension is a fundamental concept of algebra MCCOY [1972], ROTMAN [2010], LOVETT
[2016] and number theory BOREVICH AND SHAFAREVICH [1966], NEURKICH [1999], SAMUEL [1972].
Informally, the main idea is to enlarge a given structure by a special way (using elements not from the
underlying set) and try to obtain the resulting algebraic structure with “good” properties. One of the
first well known examples is the field of complex numbers C which is a simple field extension of real
numbers R. The direct generalization of this construction leads to the hypercomplex numbers (see, e.g.
WEDDERBURN [1908], KANTOR AND SOLODOVNIKOV [1989]) defined as finite D-dimensional alge-
bras A over reals with the special basis (which squares restrict to 0, ˘1). Among numerous versions of
hypercomplex number systems HAWKES [1902], TABER [1904] (for modern review, see, e.g. YAGLOM
[1968], BURLAKOV AND BURLAKOV [2020]), only complex numbers A “ C (D “ 2), quaternions
A “ H (D “ 4) and octonions A “ O (D “ 8) are classical division algebras (with no zero divisors and
nilpotents) SCHAFER [1966], SALTMAN [1992], GUBARENI [2021], and the two latter can be obtain by
the Cayley-Dickson doubling procedure DICKSON [1919], ALBERT [1942], SCHAFER [1954].
In this paper we construct nonderived hyperpolyadic structures corresponding to the above di-
vision algebras without introducing new elements (as in, e.g., DUBROVSKI AND VOLKOV [2007],
LIPATOV ET AL. [2008]). First, we use the matrix polyadization procedure proposed by the author DUPLIJ
[2022]. We show that the polyadic analog of Cayley-Dickson construction can lead only to non-division
of more higher dimensions than initial division algebras. For the obtained n-ary algebras we introduce a
new norm which is polyadically multiplicative and is well-defined for invertible elements.
Second, on the subsets of the binary division algebras we propose another new construction (we call it
“imaginary tower”) and the iterative process which gives naturally the corresponding nonderived ternary
division algebras of half dimension. The latter are not subalgebras, because they have another multi-
plication and different arity than initial algebras. We call the obtained nonunital ternary algebras “half-
quaternions” and “half-octonions”. They are actually division algebras, because nonzero elements are
invertible and allow division. From the multiplicity of “half-quaternions” norm we obtain the ternary
analog of the sum of two squares identity. Finally, we show that the introduced unitless ternary division
algebra of imaginary “half-octonions” satisfies the ternary alternativity.
2. PRELIMINARIES
Here we briefly remind notation from DUPLIJ [2022]. A (one-set) polyadic algebraic structure A is a
set A closed with respect to polyadic operations (or n-ary multiplication) µrns
: An
Ñ A (n-ary magma).
We denote polyads POST [1940] as ~
a “ ~
apkq
“ pa1, . . . , akq, ai P A, and ak
“
k
´hkkikkj
a, . . . , a
¯
, a P A (usually,
the value of k follows for the context). A (positive) polyadic power is
axℓµy
“
`
µrns
˘˝ℓµ “
aℓµpn´1q`1
‰
, a P A, ℓµ P N. (2.1)
A polyadic ℓµ-idempotent (or idempotent for ℓµ “ 1) is defined by axℓµy
“ a. A polyadic zero is defined
by µrns
r~
a, zs “ z, z P A, ~
a P An´1
, where z can be on any place. An element of a polyadic algebraic
– 2 –

3. PRELIMINARIES
structure a is called ℓµ-nilpotent (or nilpotent for ℓµ “ 1), if there exist ℓµ such that axℓµy
“ z. A polyadic
(or n-ary) identity (or neutral element) is defined by
µrns
“
a, en´1
‰
“ a, @a P A, (2.2)
where a can be on any place in the l.h.s. of (2.2). In addition, there exist neutral polyads (usually not
unique) satisfying
µrns
ra,~
ns “ a, @a P A. (2.3)
A one-set polyadic algebraic structure
@
A | µrns
D
is totally associative, if
`
µrns
˘˝2
”
~
a,~
b,~
c
ı
“ µrns
”
~
a, µrns
”
~
b
ı
,~
c
ı
“ invariant, (2.4)
with respect to placement of the internal multiplication on any of the n places, and ~
a,~
b,~
c are polyads of
the necessary sizes.
A polyadic semigroup Srns
is a one-set and one-operation structure in which µrns
is totally associative.
A polyadic structure is (totally) commutative, if µrns
“ µrns
˝ σ, for all σ P Sn. A polyadic structure is
solvable, if for all polyads b, c and an element x, one can (uniquely) resolve the equation (with respect
to x) for µrns
”
~
b, x,~
c
ı
“ a, where x can be on any place, and ~
b,~
c are polyads of the needed lengths.
A solvable polyadic structure is called n-ary quasigroup BELOUSOV [1972]. An associative polyadic
quasigroup is called a n-ary (or polyadic) group Grns
(for review, see, e.g. GAL’MAK [2003]). In an n-ary
group the only solution of
µrns
“
an´1
, ã
‰
“ a, a, ã P A, (2.5)
is called a querelement (polyadic analog of inverse) of a and denoted by ã DÖRNTE [1929], where ã
can be on any place. The relation (2.5) can be considered as a definition of the unary queroperation
µ̄p1q
ras “ ã GLEICHGEWICHT AND GŁAZEK [1967].
For further details and references, see DUPLIJ [2022].
3. MATRIX POLYADIZATION
Let us briefly (just to establish notation and terminology) remind that the 2, 4, 8-dimensional algebras
are the only hypercomplex extensions of reals A “ R (Hurwitz’s theorem for composition algebras)
A “ D “ C, H, O which are normed division algebras. The first two ones are associative (and can
be represented by matrices), and only C is commutative (being a field). We use the unified notation
z P D, and if we need to distinguish and concretize, the standard parametrization will be exploited
C Qz “ zp2q “ a ` bi and H Qz “ zp4q “ a ` bi ` cj ` dk, etc., a, d, c, d P R. The standard Euclidean
norm (2-norm) }z} “
?
z˚z (where the conjugate is z˚
p2q “ a´bi, etc.), for z P D (which for C coincides
with the modulus
›
›zp2q
›
› “
ˇ
ˇzp2q
ˇ
ˇ “
?
a2 ` b2, and
›
›zp4q
›
› “
?
a2 ` b2 ` c2 ` d2, etc.) has the properties
}1} “ 1, (3.1)
}λz} “ |λ| }z} , (3.2)
}z1
` z2
} ď }z1
} ` }z2
} , 1, λ P R, 1, z, z1
, z2
P D, (3.3)
and is multiplicative
}z1z2} “ }z1} }z2} P Rě0, z P D, (3.4)
such that the corresponding mapping D Ñ Rě0 is a homomorphism. Each nonzero element of the above
normed unital algebra has the multiplicative inverse z´1
z “ 1, because the norm vanishes only for z “ 0
– 3 –

4. MATRIX POLYADIZATION
and there are no zero divisors, therefore from }z}2
“ z˚
z P Rě0, it follows
z´1
“
z˚
}z}2 , z P Dz t0u . (3.5)
To construct polyadic analogs of the binary hypercomplex algebras A and, in particular, of the binary
division algebras D (over R), we use the polyadization procedure proposed in DUPLIJ [2022] (called
there block-matrix polyadization). That is based on the general structure theorem for polyadic rings (a
generalization of the Wedderburn theorem): any simple p2, nq-ring is isomorphic to the ring of special
cyclic shift block-matrices (of the shape (3.6)) over a division ring NIKITIN [1984].
Let us introduce the pn ´ 1q ˆ pn ´ 1q cyclic shift (weighted) block-matrix with elements from the
algebra A
Z “ Zrns
“
¨
˚
˚
˚
˚
˚
˝
0 z1 . . . 0 0
0 0 z2 . . . 0
0 0
...
...
.
.
.
.
.
.
.
.
.
... 0 zn´2
zn´1 0 . . . 0 0
˛
‹
‹
‹
‹
‹
‚
, zi P A. (3.6)
The set of monomial matrices of the form (3.6) is closed with respect to the ordinary product p¨q of
exactly n matrices, but not less. Therefore, we can define the n-ary multiplication DUPLIJ [2022]
µ
rns
Z
»
–
n
hkkkkkkikkkkkkj
Z1
, Z2
. . . Z3
fi
fl “ Z1
¨ Z2
¨ . . . ¨ Z3
“ Z “ Zrns
, (3.7)
which is nonderived in the sense that the binary (and all ď n ´ 1) product of Zrns
’s is out of the set (3.6).
Remark 3.1. The binary addition in the algebra A transfers in the standard way to the matrix addition (as
componentwise addition), and so we will pay attention on the multiplicative part mostly, implying that
the addition of Z-matrices (3.6) is always binary.
Definition 3.2. We call the following new algebraic structure
Arns
“
A!
Zrns
)
| p`q, µ
rns
Z
E
(3.8)
a n-ary hypercomplex algebra Arns
which corresponds to the binary hypercomplex algebra A by the
matrix polyadization procedure.
Assertion 3.3. The dimension Drns
of n-ary hypercomplex algebra Arns
is
Drns
“ dim Arns
“ D pn ´ 1q . (3.9)
Proof. It obviously follows from (3.6).
– 4 –

5. MATRIX POLYADIZATION
Proposition 3.4. If the binary hypercomplex algebra A is associative (for the dimensions D “ 1, D “ 2
and D “ 4), the n-ary multiplication (3.7) in components has the cyclic product form DUPLIJ [2022]
n
hkkkkkkkkikkkkkkkkj
z1
1z2
2 . . . z3
n´1z4
1 “ z1,
n
hkkkkkkkikkkkkkkj
z1
2z2
3 . . . z3
1 z4
2 “ z2,
.
.
.
n
hkkkkkkkkkkkikkkkkkkkkkkj
z1
n´1z2
1 . . . z3
n´2z4
n´1 “ zn´1, zi, z1
i, . . . , z3
i , z4
i P A. (3.10)
Proof. It follows from (3.6) and (3.7).
Remark 3.5. The cycled product (3.10) can be treated as some n-ary extension of the Jordan pair LOOS
[1975, 1974], which is different from FAULKNER [1995].
Assertion 3.6. If A is unital, then Arns
contains n-ary unit (polyadic identity (2.2)) being the permutation
(cyclic shift) matrix of the form
Erns
“
¨
˚
˚
˚
˚
˚
˝
0 1 . . . 0 0
0 0 1 . . . 0
0 0
...
...
.
.
.
.
.
.
.
.
.
... 0 1
1 0 . . . 0 0
˛
‹
‹
‹
‹
‹
‚
P Arns
, 1 P A. (3.11)
Proof. It follows from (3.6), (3.7) and (3.10).
Consider the polyadization of the hypercomplex two-dimensional algebra of dual numbers.
Example 3.7 (4-ary dual numbers). The commutative and associative two-dimensional algebra Adual “
xtzu | p`q , p¨qy is defined by the element z “ a ` bε with ε2
“ 0, a, b P R. The binary multiplication
µ
r2s
v “ µ
r2s
dual of pairs v2 “
ˆ
a
b
˙
P Atuple
dual “
@
tv2u | p`q , µr2s
D
(addition is componentwise, see Remark
3.1) is
µr2s
v
„ˆ
a1
b1
˙
,
ˆ
a2
b2
˙
“
ˆ
a1
a2
a1
b2
` b1
a2
˙
P Atuple
dual , a1
, a2
, b1
, b2
P R. (3.12)
It follows from (3.12) that Atuple
dual (and so is Adual) is a non-division algebra, because it contains idempo-
tents and zero divisors (e.g.
ˆ
0
b
˙2
“
ˆ
0
0
˙
).
Using the matrix polyadization procedure, we construct 4-ary algebra Ar4s
dual “
A!
Zr4s
)
| p`q , µr4s
E
of the dimension Dr4s
“ 6 (see (3.9)) by introducing the following 3ˆ3 cyclic shift weighted matrix (3.6)
Zr4s
“
¨
˝
0 z1 0
0 0 z2
z3 0 0
˛
‚“
¨
˝
0 a1 ` b1ε 0
0 0 a2 ` b2ε
a3 ` b3ε 0 0
˛
‚P Ar4s
dual, z1, z2, z3 P Adual. (3.13)
– 5 –

6. MATRIX POLYADIZATION
The cyclic product of the components (3.10) becomes
z1
1z2
2z3
3 z4
1 “ z1,
z1
2z2
3z3
1 z4
2 “ z2,
z1
3z2
1z3
2 z4
3 “ z3, zi, z1
i, z2
i , z3
i , z4
i P Adual. (3.14)
In terms of 6-tuples v6 over R (cf. (3.12)) the 4-ary multiplication µr4s
in Ar4s,tuple
dual “
A
tv6u | p`q , µ
r4s
v
E
has the form
µr4s
v
»
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˝
a1
1
b1
1
a1
2
b1
2
a1
3
b1
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a2
1
b2
1
a2
2
b2
2
a2
3
b2
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a3
1
b3
1
a3
2
b3
2
a3
3
b3
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a4
1
b4
1
a4
2
b4
2
a4
3
b4
3
˛
‹
‹
‹
‹
‹
‚
fi
ffi
ffi
ffi
ffi
ffi
fl
“
¨
˚
˚
˚
˚
˚
˝
a1
1a2
2a3
3 a4
1
a1
1a2
2a3
3 b4
1 ` a1
1a2
2b3
3 a4
1 ` a1
1b2
2a3
3 a4
1 ` b1
1a2
2a3
3 a4
1
a1
2a2
3a3
1 a4
2
a1
2a3
1 a2
3b4
2 ` a1
2a2
3b3
1 a4
2 ` a1
2b2
3a3
1 a4
2 ` b1
2a2
3a3
1 a4
2
a1
3a2
1a3
2 a4
3
a1
3a2
1a3
2 b4
3 ` a1
3a2
1b3
2 a4
3 ` a1
3b2
1a3
2 a4
3 ` b1
3a2
1a3
2 a4
3
˛
‹
‹
‹
‹
‹
‚
,
(3.15)
which is nonderived and noncommutative due to braidings in the cyclic product (3.14). The polyadic unit
(4-ary unit) in the 4-ary algebra of 6-tuples is (see (2.2))
e6 “
¨
˚
˚
˚
˚
˚
˝
1
0
1
0
1
0
˛
‹
‹
‹
‹
‹
‚
P Ar4s,tuple
dual . (3.16)
It follows from (3.15) that Ar4s,tuple
dual (and so is Ar4s
dual) is a non-division 4-ary algebra, not a field (similar
to Adual), because it contains 4-ary idempotents and zero divisors, for instance,
µr4s
v
»
—
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˝
0
b1
0
b2
a3
b3
˛
‹
‹
‹
‹
‹
‚
4fi
ffi
ffi
ffi
ffi
ffi
ffi
fl
“ µr4s
v
»
—
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˝
a1
b1
0
b2
0
b3
˛
‹
‹
‹
‹
‹
‚
4fi
ffi
ffi
ffi
ffi
ffi
ffi
fl
“ µr4s
v
»
—
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˝
0
b1
a2
b2
0
b3
˛
‹
‹
‹
‹
‹
‚
4fi
ffi
ffi
ffi
ffi
ffi
ffi
fl
“
¨
˚
˚
˚
˚
˚
˝
0
0
0
0
0
0
˛
‹
‹
‹
‹
‹
‚
“ 0v. (3.17)
Thus, the application of the matrix polyadization procedure to the commutative, associative, unital, two-
dimensional, non-division algebra of dual numbers Adual gives noncommutative, 4-ary nonderived totally
associative, unital, 6-dimensional, non-division algebra Ar4s
dual over R, we call that 4-ary dual numbers.
4. POLYADIZATION OF DIVISION ALGEBRAS
Let us consider the matrix polyadization procedure for the division algebras D “ R, C, H, O in more
details paying attention on invertibility and norms.
Recall that the binary division algebra D (without zero element) forms a group (with respect to binary
multiplication) having the inverse (3.5). The polyadic counterpart of the binary inverse is the querelement
(2.5).
– 6 –

7. POLYADIZATION OF DIVISION ALGEBRAS
Theorem 4.1. The nonderived n-ary algebra (3.8) constructed from the binary division algebra D is the
n-ary algebra
Drns
“
A!
Zrns
)
| p`q, µ
rns
Z , r
Z
rns
E
, (4.1)
where r
Z
rns
is the querelement
r
Z “ r
Z
rns
“
¨
˚
˚
˚
˚
˚
˝
0 r
z1 . . . 0 0
0 0 r
z2 . . . 0
0 0
...
...
.
.
.
.
.
.
.
.
.
... 0 r
zn´2
r
zn´1 0 . . . 0 0
˛
‹
‹
‹
‹
‹
‚
, (4.2)
which satisfies (if D is associative)
µ
rns
Z
»
–
n´1
hkkkkikkkkj
Z, Z . . . Z r
Z
fi
fl “
n´1
hkkkkkkkikkkkkkkj
Z ¨ Z ¨ . . . ¨ Z ¨ r
Z “ Z, @Z P Drns
, (4.3)
and r
Z can be on any place, such that
r
zi “ z´1
i´1z´1
i´2 . . . z´1
2 z´1
1 z´1
n´1z´1
n´1 . . . z´1
i`2z´1
i`1, zi P Dz t0u . (4.4)
Proof. The main relation (4.4) follows from (4.3) in components (3.10) as the following cycle products
n
hkkkkkkkkikkkkkkkkj
z1z2 . . . zn´1r
z1 “ z1,
n
hkkkkkkkkkikkkkkkkkkj
z2z3 . . . zn´1z1r
z2 “ z2,
.
.
.
n
hkkkkkkkkkkkikkkkkkkkkkkj
zn´1z1 . . . zn´2r
zn´1 “ zn´1, zi, r
zi P Dz t0u , (4.5)
by applying z´1
i (which exists in D for nonzero z (3.5)) from the left pn ´ 1q times (with suitable indices)
to both sides of each equation in (4.5) to get r
zi.
Corollary 4.2. Each D-dimensional division algebra over reals D “ C, H, O (including R itself as one-
dimensional case) has its n polyadic counterparts (where n is arbitrary), the nonderived n-ary non-division
algebras Drns
(4.1) of the dimension Dpn ´ 1q (having the polyadic unit (3.11) and the querelement (4.2)
for invertible zi P Dz t0u) constructed by the matrix polyadization procedure.
Theorem 4.3. The matrix polyadization changes invertibility property of initial algebra, that is polyadiza-
tion of a binary division algebra D leads to n-ary non-division algebra Drns
with arbitrary n.
Proof. The polyadization procedure is provided by monomial matrices which have determinant which is
proportional to the product of nonzero entries. The nonzero elements of Drns
having some of zi “ 0 are
noninvertible, and therefore Drns
is not a field.
Nevertheless, a special subalgebra of Drns
can be a division n-ary algebra.
Theorem 4.4. The elements of Drns
which have all invertible zi P Dz t0u (or set of invertible Z-matrices
det Z ‰ 0) form a subalgebra Drns
div Ă Drns
which is the division n-ary algebra corresponding to the
division algebra D “ R, C, H, O.
– 7 –

8. POLYADIZATION OF DIVISION ALGEBRAS
The simplest case is polyadization of reals.
Example 4.5 (5-ary real numbers). The 5-ary algebra of real numbers is Rr5s
“
A!
Rr5s
)
| p`q , µ
r5s
R
E
,
where Rr5s
is the cyclic 4 ˆ 4 block-shift matrix (3.6) with real entries
Rr5s
“
¨
˚
˚
˝
0 a1 0 0
0 0 a2 0
0 0 0 a3
a4 0 0 0
˛
‹
‹
‚, (4.6)
det Rr5s
“ a1a2a3a4, ai P R, (4.7)
and the multiplication µ
r5s
R is an ordinary product of 5 matrices. Only with respect to product of 5 ele-
ments Rr5s
the algebra is closed, and therefore Rr5s
is nonderived. In components we have the braiding
cyclic products (3.10) for ai. If ai P Rz t0u, the component equations for the querelement (4.5) (after
cancellation of nonzero ai) become
a2a3a4r
a1 “ 1, (4.8)
a3a4a1r
a2 “ 1, (4.9)
a4a1a2r
a3 “ 1, (4.10)
a1a2a3r
a4 “ 1. (4.11)
The querelement for invertible (4.6) is
r
R
r5s
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
0
1
a2a3a4
0 0
0 0
1
a3a4a1
0
0 0 0
1
a4a1a2
1
a1a2a3
0 0 0
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
, ai P Rz t0u , (4.12)
and therefore, the algebra Rr5s
of 5-ary real numbers is the non-division algebra, because the elements
with some ai “ 0 are in Rr5s
, but they are noninvertible (due to (4.7)), and so Rr5s
is not a field. But the
subalgebra Rr5s
div Ă Rr5s
of invertible matrices Rr5s
(det Rr5s
‰ 0, with all ai ‰ 0) is a division n-ary
algebra of reals (see Theorem 4.4).
Now we provide the example of 4-ary algebra of complex numbers to compare it with the dual numbers
of the same arity in Example 3.7.
Example 4.6 (4-ary complex numbers). First, we establish notations, as before. The commutative and
associative two-dimensional algebra of complex numbers is C “ xtzu | p`q , p¨qy, where z “ a ` bi with
i2
“ ´1, a, b P R. The binary multiplication µ
r2s
v “ µ
r2s
compl of pairs
v2 “
ˆ
a
b
˙
P Ctuple
“
@
tv2u | p`q , µr2s
v
D
(4.13)
(addition is componentwise, see Remark 3.1) is
µr2s
v
„ˆ
a1
b1
˙
,
ˆ
a2
b2
˙
“
ˆ
a1
a2
´ b2
b1
b2
a1
` b1
a2
˙
P Ctuple
, a1
, a2
, b1
, b2
P R. (4.14)
– 8 –

9. POLYADIZATION OF DIVISION ALGEBRAS
The algebra of pairs Ctuple
does not contain idempotents and zero divisors, its multiplication agrees with
one of C (z1
¨ z2
“ z), and therefore it is a division algebra being isomorphic to C
Ctuple
– C. (4.15)
Using the matrix polyadization procedure, we construct the nonderived 4-ary algebra of complex num-
bers Cr4s
“
A!
Zr4s
)
| p`q , µ
r4s
Z
E
of the dimension Dr4s
“ 6 (see (3.9)) by introducing the following
3 ˆ 3 matrix (3.6)
Z “ Zr4s
“
¨
˝
0 z1 0
0 0 z2
z3 0 0
˛
‚“
¨
˝
0 a1 ` b1i 0
0 0 a2 ` b2i
a3 ` b3i 0 0
˛
‚P Cr4s
, z1, z2, z3 P C. (4.16)
The 4-ary product of Z-matrices (4.16) is
µ
r4s
Z rZ1
, Z2
, Z3
, Z4
s “ Z1
Z2
Z3
Z4
. (4.17)
The corresponding to (4.17) cyclic product in the components (3.10) becomes
z1
1z2
2z3
3 z4
1 “ z1,
z1
2z2
3z3
1 z4
2 “ z2,
z1
3z2
1z3
2 z4
3 “ z3, zi, z1
i, z2
i , z3
i , z4
i P C. (4.18)
In terms of 6-tuples v6 over R (cf. (3.12)) the 4-ary multiplication µ
r4s
v in Cr4s,tuple
“
A
tv6u | p`q , µ
r4s
v
E
has the form (cf. (4.14))
µr4s
v
»
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˝
a1
1
b1
1
a1
2
b1
2
a1
3
b1
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a2
1
b2
1
a2
2
b2
2
a2
3
b2
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a3
1
b3
1
a3
2
b3
2
a3
3
b3
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a4
1
b4
1
a4
2
b4
2
a4
3
b4
3
˛
‹
‹
‹
‹
‹
‚
fi
ffi
ffi
ffi
ffi
ffi
fl
(4.19)
“
¨
˚
˚
˚
˚
˚
˝
a1
1a2
2a3
3 a4
1 ´ a1
1b2
2b3
3 a4
1 ´ b1
1a2
2b3
3 a4
1 ´ b1
1b2
2a3
3 a4
1 ´ a1
1a2
2b3
3 b4
1 ´ a1
1b2
2a3
3 b4
1 ´ b1
1a2
2a3
3 b4
1 ` b1
1b2
2b3
3 b4
1
a1
1a2
2b3
3 a4
1 ` a1
1b2
2a3
3 a4
1 ` b1
1a2
2a3
3 a4
1 ` a1
1a2
2a3
3 b4
1 ´ b1
1b2
2b3
3 a4
1 ´ a1
1b2
2b3
3 b4
1 ´ b1
1a2
2b3
3 b4
1 ´ b1
1b2
2a3
3 b4
1
a1
2a2
3a3
1 a4
2 ´ a1
2b2
3b3
1 a4
2 ´ b1
2b2
3a3
1 a4
2 ´ b1
2a2
3b3
1 a4
2 ´ a1
2b2
3a3
1 b4
2 ´ a1
2a2
3b3
1 b4
2 ´ b1
2a2
3a3
1 b4
2 ` b1
2b2
3b3
1 b4
2
a1
2b2
3a3
1 a4
2 ` a1
2a2
3b3
1 a4
2 ` b1
2a2
3a3
1 a4
2 ` a1
2a2
3a3
1 b4
2 ´ b1
2b2
3b3
1 a4
2 ´ a1
2b2
3b3
1 b4
2 ´ b1
2b2
3a3
1 b4
2 ´ b1
2a2
3b3
1 b4
2
a1
3a2
1a3
2 a4
3 ´ b1
3a2
1b3
2 a4
3 ´ a1
3b2
1b3
2 a4
3 ´ b1
3b2
1a3
2 a4
3 ´ a1
3a2
1b3
2 b4
3 ´ b1
3a2
1a3
2 b4
3 ´ a1
3b2
1a3
2 b4
3 ` b1
3b2
1b3
2 b4
3
a1
3a2
1b3
2 a4
3 ` b1
3a2
1a3
2 a4
3 ` a1
3b2
1a3
2 a4
3 ` a1
3a2
1a3
2 b4
3 ´ b1
3b2
1b3
2 a4
3 ´ b1
3a2
1b3
2 b4
3 ´ a1
3b2
1b3
2 b4
3 ´ b1
3b2
1a3
2 b4
3
˛
‹
‹
‹
‹
‹
‚
,
(4.20)
which is nonderived and noncommutative due to braidings in the cyclic product (3.14). The polyadic unit
(4-ary unit) in the 4-ary algebra of 6-tuples is (see (2.2))
e6 “
¨
˚
˚
˚
˚
˚
˝
1
0
1
0
1
0
˛
‹
‹
‹
‹
‹
‚
P Cr4s,tuple
, (4.21)
µr4s
v re6, e6, e6, v6s “ µr4s
v re6, e6, v6, e6s “ µr4s
v re6, v6, e6, e6s “ µr4s
v rv6, e6, e6, e6s “ v6. (4.22)
– 9 –

10. POLYADIZATION OF DIVISION ALGEBRAS
The querelement (2.5), (4.2) for invertible elements of 4-ary algebra of complex numbers Cr4s
has the
matrix form which follows from the equations (4.5)
r
Z
r4s
“
¨
˝
0 1
z2z3
0
0 0 1
z1z3
1
z1z2
0 0
˛
‚P Cr4s
, z1, z2, z3 P Cz t0u . (4.23)
Thus, Cr4s
“
A!
Zr4s
)
| p`q , µr4s
, Ă
p q
E
is the nonderived 4-ary non-division algebra over R obtained
by the matrix polyadization procedure from the algebra C of complex numbers. The 4-ary algebra of
complex numbers Cr4s
is not a field, because it contains noninvertible nonzero elements (with some zi “
0).
In Cr4s,tuple
the querelement is given by the following 6-tuple
r
v6 “
Č
¨
˚
˚
˚
˚
˚
˝
a1
b1
a2
b2
a3
b3
˛
‹
‹
‹
‹
‹
‚
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
a2a3 ´ b2b3
pa2b3 ` a3b2q2
` pa2a3 ´ b2b3q2
´
a2b3 ` a3b2
pa2b3 ` a3b2q2
` pa2a3 ´ b2b3q2
a1a3 ´ b1b3
pa1b3 ` a3b1q2
` pa1a3 ´ b1b3q2
´
a1b3 ` a3b1
pa1b3 ` a3b1q2
` pa1a3 ´ b1b3q2
a1a2 ´ b1b2
pa1b2 ` a2b1q2
` pa1a2 ´ b1b2q2
´
a1b2 ` a2b1
pa1b2 ` a2b1q2
` pa1a2 ´ b1b2q2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
, ai, bi P R. (4.24)
Therefore, Cr4s,tuple
is a division nonderived 4-ary algebra isomorphic to Cr4s
.
Thus, the application of the matrix polyadization procedure to the commutative, associative, unital,
two-dimensional, division algebra of complex numbers C gives noncommutative, 4-ary nonderived totally
associative, unital, 6-dimensional, non-division algebra Cr4s
over R (with the corresponding isomorphic
4-ary non-division algebra of 6-tuples Cr4s,tuple
), we call that 4-ary complex numbers. The subalgebra
Cr4s
div Ă Cr4s
of invertible matrices Zr4s
(det Zr4s
‰ 0, with all zi ‰ 0) is the division 4-ary algebra of
complex numbers (by Theorem 4.4).
Consider now polyadization of the noncommutative quaternion algebra H.
Example 4.7 (Ternary quaternions). The associative four-dimensional algebra of quaternions is
H “ xtqu | p`q , p¨qy , q “ a ` bi ` cj ` dk,
ij “ k, ij “ ´ji, p+cycledq , i2
“ j2
“ k2
“ ijk “ ´1, a, b, c, d P R. (4.25)
The binary multiplication µ
r4s
v of the quadruples
v4 “
¨
˚
˚
˝
a
b
c
d
˛
‹
‹
‚P Htuple
“
@
tv4u | p`q , µr2s
v
D
(4.26)
– 10 –

11. POLYADIZATION OF DIVISION ALGEBRAS
is (we present the binary product in our notation for completeness to compare with the ternary case below)
µr4s
v
»
—
—
–
¨
˚
˚
˝
a1
b1
c1
d1
˛
‹
‹
‚,
¨
˚
˚
˝
a2
b2
c2
d2
˛
‹
‹
‚
fi
ffi
ffi
fl “
¨
˚
˚
˝
a1
a2
´ b1
b2
´ c1
c2
´ d1
d2
a1
b2
` b1
a2
´ d1
c2
` c1
d2
a1
c2
` c1
a2
` d1
b2
´ b1
d2
a1
d2
` b1
c2
´ c1
b2
` d1
a2
˛
‹
‹
‚P Htuple
, a1
, a2
, b1
, b2
, c1
, c2
, d1
, d2
P R.
(4.27)
The binary algebra of quadruples Htuple
is a noncommutative division algebra (isomorphic to H), without
idempotents and zero divisors.
By the above matrix polyadization procedure, we construct the nonderived ternary algebra of quater-
nions Hr3s
“
A!
Qr3s
)
| p`q , µ
r3s
Q
E
of the dimension Dr3s
“ 8 (see (3.9)) by introducing the following
2 ˆ 2 matrix (3.6)
Q “ Qr4s
“
ˆ
0 q1
q2 0
˙
“
ˆ
0 a1 ` b1i ` c1j ` d1k
a2 ` b2i ` c2j ` d2k 0
˙
P Hr3s
, q1, q2 P H.
(4.28)
The nonderived ternary product of Q-matrices (4.16) is
µ
r4s
Q rQ1
, Q2
, Q3
s “ Q1
Q2
Q3
. (4.29)
The corresponding to (4.17) cyclic product in the components (3.10) becomes
q1
1q2
2q3
1 “ q1,
q1
2q2
1q3
2 “ q2, qi, q1
i, q2
i , q3
i P H. (4.30)
In terms of 8-tuples v8 over R (cf. (3.12)) the ternary multiplication µ
r3s
v in Hr3s,tuple
“
A
tv8u | p`q , µ
r3s
v
E
has the form (cf. (4.14))
µ
r3s
v
“
v1
8, v2
8, v3
8
‰
“ µ
r3s
v
»
—
—
—
—
—
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
a1
1
b1
1
c1
1
d1
1
a1
2
b1
2
c1
2
d1
2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
a2
1
b2
1
c2
1
d2
1
a2
2
b2
2
c2
2
d2
2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
a3
1
b3
1
c3
1
d3
1
a3
2
b3
2
c3
2
d3
2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
`
a1
1a2
2 ´ b1
1b2
2 ´ c1
1c2
2 ´ d1
1d2
2
˘
a3
1 ´
`
a1
1b2
2 ` b1
1a2
2 ` c1
1d2
2 ´ d1
1c2
2
˘
b3
1 ´
`
a1
1c2
2 ` c1
1a2
2 ` d1
1b2
2 ´ b1
1d2
2
˘
c3
1 ´
`
a1
1d2
2 ` d1
1a2
2 ` b1
1c2
2 ´ c1
1b2
2
˘
d3
1
`
a1
1b2
2 ` b1
1a2
2 ` c1
1d2
2 ´ d1
1c2
2
˘
a3
1 `
`
a1
1a2
2 ´ b1
1b2
2 ´ c1
1c2
2 ´ d1
1d2
2
˘
b3
1 ´
`
a1
1d2
2 ` d1
1a2
2 ` b1
1c2
2 ´ c1
1b2
2
˘
c3
1 `
`
a1
1c2
2 ` c1
1a2
2 ` d1
1b2
2 ´ b1
1d2
2
˘
d3
1
`
a1
1c2
2 ` c1
1a2
2 ` d1
1b2
2 ´ b1
1d2
2
˘
a3
1 `
`
a1
1d2
2 ` d1
1a2
2 ` b1
1c2
2 ´ c1
1b2
2
˘
b3
1 `
`
a1
1a2
2 ´ b1
1b2
2 ´ c1
1c2
2 ´ d1
1d2
2
˘
c3
1 ´
`
a1
1b2
2 ` b1
1a2
2 ` c1
1d2
2 ´ d1
1c2
2
˘
d3
1
`
a1
1d2
2 ` d1
1a2
2 ` b1
1c2
2 ´ c1
1b2
2
˘
a3
1 ´
`
a1
1c2
2 ` c1
1a2
2 ` d1
1b2
2 ´ b1
1d2
2
˘
b3
1 `
`
a1
1b2
2 ` b1
1a2
2 ` c1
1d2
2 ´ d1
1c2
2
˘
c3
1 `
`
a1
1a2
2 ´ b1
1b2
2 ´ c1
1c2
2 ´ d1
1d2
2
˘
d3
1
`
a1
2a2
1 ´ b1
2b2
1 ´ c1
2c2
1 ´ d1
2d2
1
˘
a3
2 ´
`
a1
2b2
1 ` b1
2a2
1 ` c1
2d2
1 ´ d1
1c2
2
˘
b3
2 ´
`
a1
2c2
1 ` c1
2a2
1 ` d1
2b2
1 ´ b1
2d2
1
˘
c3
2 `
`
a1
2d2
1 ` d1
2a2
1 ` b1
2c2
1 ´ c1
2b2
1
˘
d3
2
`
a1
1b2
2 ` b1
1a2
2 ` c1
1d2
2 ´ d1
1c2
2
˘
a3
2 ´
`
a1
2b2
1 ` b1
2a2
1 ` c1
2d2
1 ´ d1
2c2
1
˘
b3
2 ´
`
a1
2d2
1 ` d1
2a2
1 ` b1
2c2
1 ´ c1
2b2
1
˘
c3
2 `
`
a1
2c2
1 ` c1
2a2
1 ` d1
2b2
1 ´ b1
2d2
1
˘
d3
2
`
a1
2c2
1 ` c1
2a2
1 ` d1
2b2
1 ´ b1
2d2
1
˘
a3
2 `
`
a1
2d2
1 ` d1
2a2
1 ` b1
2c2
1 ´ c1
2b2
1
˘
b3
2 `
`
a1
2a2
1 ´ b1
2b2
1 ´ c1
2c2
1 ´ d1
2d2
1
˘
c3
2 ´
`
a1
2b2
1 ` b1
2a2
1 ` c1
2d2
1 ´ d1
2c2
1
˘
d3
2
`
a1
2d2
1 ` d1
2a2
1 ` b1
2c2
1 ´ c1
2b2
1
˘
a3
2 ´
`
a1
2c2
1 ` c1
2a2
1 ` d1
2b2
1 ´ b1
2d2
1
˘
b3
2 `
`
a1
2b2
1 ` b1
2a2
1 ` c1
2d2
1 ´ d1
2c2
1
˘
c3
2 `
`
a1
2a2
1 ´ b1
2b2
1 ´ c1
2c2
1 ´ d1
2d2
1
˘
d3
2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
(4.31)
where a1
i, a2
i , a3
i , b1
i, b2
i , b3
i , c1
i, c2
i , c3
i , d1
i, d2
i , d3
i P R.
Remark 4.8. It is important to note, that the ternary multiplication of 8-tuples (4.31) is nonderived and
noncommutative, however it is not the ordinary product of three quaternion pairs, but it corresponds to
the nontrivial cyclic braided products (4.30).
– 11 –

12. POLYADIZATION OF DIVISION ALGEBRAS
The polyadic unit (ternary unit) e8 in the ternary algebra of 8-tuples is (see (2.2))
e8 “
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
1
0
0
0
1
0
0
0
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
P Hr3s,tuple
, (4.32)
µr3s
v re8, e8, v8s “ µr3s
v re8, v8, e8s “ µr3s
v rv8, e8, e8s “ v8. (4.33)
The querelement (2.5), (4.2) of nonderived noncommutative 8-dimensional ternary algebra of quaternions
Hr3s
has the matrix form which follows from the general equations (4.5)
r
Q
r3s
“
ˆ
0 q´1
2
q´1
1 0
˙
P Hr3s
, q1, q2 P Hz t0u . (4.34)
Therefore, Hr3s
“
A!
Qr3s
)
| p`q , µ
r3s
Q , Ă
p q
E
is the nonderived noncommutative ternary non-division
algebra obtained by the matrix polyadization procedure from the algebra H of quaternions, because the
elements Qr3s
with q1 “ 0 or q2 “ 0 are nonzero, but noninvertible, such that Hr3s
is not a field.
In Hr3s,tuple
the querelement is given by the following 8-tuple
r
v8 “
Č
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
a1
b1
c1
d1
a2
b2
c2
d2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
a1
a2
1 ` b2
1 ` c2
1 ` d2
1
´
b1
a2
1 ` b2
1 ` c2
1 ` d2
1
´
c1
a2
1 ` b2
1 ` c2
1 ` d2
1
´
d1
a2
1 ` b2
1 ` c2
1 ` d2
1
a2
a2
2 ` b2
2 ` c2
2 ` d2
2
´
b2
a2
2 ` b2
2 ` c2
2 ` d2
2
´
c2
a2
2 ` b2
2 ` c2
2 ` d2
2
´
d2
a2
2 ` b2
2 ` c2
2 ` d2
2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
, a2
1,2 ` b2
1,2 ` c2
1,2 ` d2
1,2 ‰ 0, ai, bi, ci, di P R. (4.35)
Therefore, Hr3s,tuple
is a non-division ternary algebra isomorphic to Hr3s
.
To conclude, the application of the matrix polyadization procedure to the noncommutative, associative,
unital, four-dimensional, division algebra of quaternions H gives noncommutative, nonderived ternary to-
tally associative, unital, 8-dimensional, non-division algebra Hr3s
over R (with the corresponding isomor-
phic ternary non-division algebra of 8-tuples Hr3s,tuple
), we call that ternary quaternions. The subalgebra
Hr3s
div Ă Hr3s
of invertible matrices Qr3s
(det Qr3s
‰ 0, with all qi ‰ 0) is the division ternary algebra of
quaternions (see Theorem 4.4).
– 12 –

13. POLYADIC NORMS
5. POLYADIC NORMS
The division algebras D “ R, C, H, O are normed as vector spaces, and the corresponding Euclidean
2-norm is multiplicative (3.4) such that the corresponding mapping is the binary homomorphism. It would
be worthwhile to define a polyadic analog of the binary norm } } which would have similar properties.
Definition 5.1. We define the polyadic (n-ary) norm } }rns
in the n-ary algebra Drns
, that is obtained from
D by the matrix polyadization procedure (3.6), as the product (in R) of the component norms
}Z}rns
“ }z1} }z2} . . . }zn´1} P R, Z P Drns
, zi P D. (5.1)
Corollary 5.2. The polyadic norm (5.1) is zero for noninvertible elements of Drns
, having some zi “ 0.
Therefore, it is worthwhile to consider polyadic norm for invertible elements of Drns
only.
Proposition 5.3. The division n-ary subalgebras Drns
div Ă Drns
are normed n-ary algebras with respect to
the polyadic norm } }rns
(5.1).
Let us consider properties of the polyadic norm (5.1).
Assertion 5.4. The introduced polyadic norm } }rns
(5.1) has the properties (for invertible Z P Drns
)
›
›
›Erns
›
›
›
rns
“ 1 P R, Erns
P Drns
, (5.2)
}λZ}rns
“ |λ|n´1
}Z}rns
, λ P R, (5.3)
}Z1
` Z2
}
rns
ď }Z}rns
` }Z}rns
, Z P Drns
, (5.4)
where Erns
is the polyadic unit in n-ary algebra Drns
(3.11).
Proof. The first property is obvious, the second one follows from the definition (5.1) and linearity of the
ordinary Euclidean norm in D (3.2).The polyadic triangle inequality (5.4) follows from the binary triangle
inequality (3.3), because of the binary addition of Z-matrices of the cyclic block-shift form (3.6).
The norms satisfying (5.3) are called norms of higher degree, and they were investigated for the binary
case in PUMPLÜN [2011].
The most important property of any (binary) norm is its multiplicativity (3.4).
Theorem 5.5. The polyadic norm } }rns
defined in (5.1) is n-ary multiplicative (such that the correspond-
ing map Drns
Ñ R is an n-ary homomorphism)
›
›
›
›
›
›
µ
rns
Z
»
–
n
hkkkkkkikkkkkkj
Z1
, Z2
. . . Z3
fi
fl
›
›
›
›
›
›
rns
“
n
hkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkj
}Z1
}
rns
¨ }Z2
}
rns
¨ . . . ¨ }Z3
}
rns
, Z1
, Z2
, Z3
P Drns
div. (5.5)
Proof. Consider the component form (3.10) of each multiplier Z in (5.5), then use the definition (5.1) and
commutativity (as they are in R) and multiplicativity (3.4) of the ordinary binary norms } } to rearrange
the products of norms from l.h.s. to r.h.s. in (5.5). That is
n´1
hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj
›
›z1
1z2
2 . . . z3
n´1z4
1
›
› }z1
2z2
3 . . . z3
1 z4
2 } . . .
›
›z1
n´1z2
1 . . . z3
n´2z4
n´1
›
›
“
n
hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj
`
}z1
1} }z1
2} . . .
›
›z1
n´1
›
›
˘ `
}z2
1} }z2
2} . . .
›
›z2
n´1
›
›
˘
. . .
`
}z3
1 } }z3
2 } . . .
›
›z3
n´1
›
›
˘ `
}z4
1 } }z4
2 } . . .
›
›z4
n´1
›
›
˘
.
(5.6)
– 13 –

14. POLYADIC NORMS
Remark 5.6. The n-ary multiplicativity (5.5) of the introduced polyadic norm (5.1) is independent of the
concrete form of binary norm } }, only the multiplicativity of the latter is needed.
Proposition 5.7. The polyadic norm of the querelement in n-ary division subalgebra Drns
div is
›
›
›r
Z
›
›
›
rns
“
1
´
}Z}rns
¯n´2 P Rą0, @r
Z P Drns
div. (5.7)
Proof. It follows from the component relations for the querelements }r
zi} (4.5) and multiplicativity of the
binary norm, that
n
hkkkkkkkkkkkkkkikkkkkkkkkkkkkkj
}z1} }z2} . . . }zn´1} }r
z1} “ }z1} ,
n
hkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkj
}z2} }z3} . . . }zn´1} }z1} }r
z2} “ }z2} ,
.
.
.
n
hkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkj
}zn´1} }z1} . . . }zn´2} }r
zn´1} “ }zn´1} , zi, r
zi P D, }zi} , }r
zi} P Rą0. (5.8)
Multiplying all equations in (5.8) and using the definition of the polyadic norm (5.1), together with the
component form (3.6) and the querelement (4.2), we get
´
}Z}rns
¯n´1 ›
›
›r
Z
›
›
›
rns
“ }Z}rns
, (5.9)
from which it follows (5.7).
Example 5.8. In the division 4-ary algebra of complex numbers Cr4s
div from Example 4.6 the polyadic norm
becomes
}Z}r4s
“
b
pa2
1 ` b2
1q pa2
2 ` b2
2q pa2
3 ` b2
3q, ai, bi P R. (5.10)
The polyadic norm of the querelement r
Z in Cr4s
(5.7) is
›
›
›r
Z
›
›
›
r4s
“
1
pa2
1 ` b2
1q pa2
2 ` b2
2q pa2
3 ` b2
3q
, a2
i ` b2
i ‰ 0, ai, bi P R. (5.11)
Example 5.9. In the division ternary algebra of quaternions Hr3s
div from Example 4.7 the polyadic norm
becomes
}Q}r3s
“
b
pa2
1 ` b2
1 ` c2
1 ` d2
1q pa2
2 ` b2
2 ` c2
2 ` d2
2q, ai, bi P R. (5.12)
The polyadic norm of the querelement r
Q in Hr3s
div (5.7) is
›
›
› r
Q
›
›
›
r3s
“
1
a
pa2
1 ` b2
1 ` c2
1 ` d2
1q pa2
2 ` b2
2 ` c2
2 ` d2
2q
, a2
i ` b2
i ` c2
i ` d2
i ‰ 0, ai, bi P R. (5.13)
Further properties of the polyadic norm } }rns
can be investigated for invertible elements of concrete
n-ary algebras.
– 14 –

15. Abstract (tuple) approach POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION
6. POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION
The standard method of obtaining the higher hypercomplex algebras is the Cayley-Dickson construction
SCHAFER [1954], KORNILOWICZ [2012], FLAUT [2019]. It is well known, that all four binary division
algebras D “ R, C, H, O can be built in this way KANTOR AND SOLODOVNIKOV [1989]. Here we
generalize the Cayley-Dickson construction to the polyadic (n-ary) division algebras introduced in the
previous section. As the result, the number of polyadic division algebras becomes infinity (as opposite to
above four in the binary case), because of arbitrary initial and final arities of algebras under consideration.
For illustration, we present several low arity examples, since higher arity cases become too cumbersome
and unobservable. First, we recall in brief (just to install our notation) the ordinary (binary) Cayley-
Dickson doubling process (in our notation convenient for the polyadization procedure).
6.1. Abstract (tuple) approach. Consider the sequence of algebras Aℓ, ℓ ě 0, over reals, starting from
A0 “ R. The main idea is to repeat the doubling process of complex number construction using pairs
(doubles) (4.14) and taking into account the isomorphism (4.15) on each stage Aℓ Ñ Aℓ`1. Let us denote
the binary algebra over R on ℓ-th stage with the underlying set tAℓu as
Aℓ “
A
tAℓu | p`q , µ
r2s
ℓ , p˚ℓq
E
, (6.1)
where p˚ℓq is involution in Aℓ and µ
r2s
ℓ : Aℓ b Aℓ Ñ Aℓ is its binary multiplication, we also will write
µ
r2s
ℓ ” p¨ℓq. The corresponding algebra of doubles (2-tuples)
v2pℓq “
ˆ
apℓq
bpℓq
˙
, apℓq, bpℓq P Aℓ, (6.2)
is denoted by
Atuple
ℓ “
A
v2pℓq
(
| p`q , µ
r2s
vpℓq,
´
˚tuple
ℓ
¯E
, (6.3)
where
´
˚tuple
ℓ
¯
is involution in Atuple
ℓ and µ
r2s
vpℓq : Atuple
ℓ bAtuple
ℓ Ñ Atuple
ℓ is the binary product of doubles.
If ℓ “ 0, then the conjugation is the identical map, as it should be for reals R. The addition and scalar
multiplication are made componentwise in the standard way.
In this notation, the Cayley-Dickson doubling process is defined by the recurrent multiplication formula
µ
r2s
vpℓ`1q
„ˆ
a1
pℓq
b1
pℓq
˙
,
ˆ
a2
pℓq
b2
pℓq
˙
“
¨
˝
µ
r2s
ℓ
”
a1
pℓq, a2
pℓq
ı
´ µ
r2s
ℓ
“`
b2
pℓq
˘˚ℓ
, b1
pℓq
‰
µ
r2s
ℓ
”
b2
pℓq, a1
pℓq
ı
` µ
r2s
ℓ
”
b1
pℓq,
´
a2
pℓq
¯˚ℓ
ı
˛
‚ (6.4)
“
˜
a1
pℓq ¨ℓ a2
pℓq ´
`
b2
pℓq
˘˚ℓ
¨ℓ b1
pℓq
b2
pℓq ¨ℓ a1
pℓq ` b1
pℓq ¨ℓ
´
a2
pℓq
¯˚ℓ
¸
”
ˆ
apℓ`1q
bpℓ`1q
˙
P Atuple
ℓ`1 , (6.5)
and the recurrent conjugation
ˆ
apℓq
bpℓq
˙˚tuple
ℓ`1
“
ˆ
a˚ℓ
pℓq
´bpℓq
˙
P Atuple
ℓ`1 , apℓq, bpℓq P Aℓ. (6.6)
Then, we use the isomorphism (4.15) which is now becomes
Atuple
ℓ`1 – Aℓ`1. (6.7)
To go to the next level of recursion with the obtained Aℓ`1 we use (6.4)–(6.7) with changing ℓ Ñ ℓ ` 1.
The dimension of the algebra Aℓ is
Dpℓq “ 2ℓ
, (6.8)
– 15 –

16. POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION Concrete (hyperembedding) approach
such that each element can be presented in the form of 2ℓ
-tuple of reals
¨
˚
˚
˝
apℓq,1
apℓq,2
.
.
.
apℓq,2ℓ
˛
‹
‹
‚P Aℓ, (6.9)
and the conjugated 2ℓ
-tuple becomes
¨
˚
˚
˝
apℓq,1
´apℓq2
.
.
.
´apℓq,2ℓ
˛
‹
‹
‚, apℓq,k P R. (6.10)
For clarity, we intensionally mark elements and operations of ℓ-th level manifestly, because they are
really different for different ℓ. The example with ℓ “ 0 is obtaining the algebra of complex numbers A1 ”
C from the algebra of reals A0 ” R (4.13)–(4.15). All the binary division algebras D “ R, C, H, O can
be obtained by the Cayley-Dickson construction KANTOR AND SOLODOVNIKOV [1989].
6.2. Concrete (hyperembedding) approach. Alternatively, one can reparametrize the pairs (6.2) satis-
fying the complex-like multiplication (6.4), as the field extension p
Aℓ “ Aℓ piℓq by one complex-like unit
iℓ on each ℓ-th stage of iteration. The Cayley-Dickson doubling process is given by the iterations
p
Aℓ`1 “
A
tAℓ piℓqu | p`q , p
µ
r2s
ℓ`1, p˚ℓ`1q
E
piℓ`1q , i2
ℓ “ ´1, i2
ℓ`1 “ ´1, iℓ`1iℓ “ ´iℓiℓ`1, ℓ ě 0.
(6.11)
In the component form it is
zpℓ`1q “ zpℓq,1 ` zpℓq,2 ¨ℓ`1 iℓ`1 “ zpℓq,1 ` p
µ
r2s
ℓ`1
“
zpℓq,2, iℓ`1
‰
P p
Aℓ`1, zpℓq,1, zpℓq,2 P p
Aℓ. (6.12)
The product in p
Aℓ`1 can be expressed through the product of previous stage from p
Aℓ and the conjuga-
tion by using anticommutation of the imagery units from different stages iℓ`1iℓ “ ´iℓiℓ`1 (see (6.11)).
Thus, we obtain the standard complex-like multiplication (see (4.14) and (6.5)) on each ℓ-th stage of the
Cayley-Dickson doubling process
z1
pℓ`1q ¨ℓ`1 z2
pℓ`1q “
`
z1
pℓq,1 ¨ℓ z2
pℓq,1 ´
`
z2
pℓq,2
˘˚ℓ
¨ℓ z1
pℓq,2
˘
`
`
z2
pℓq,2 ¨ℓ z1
pℓq,1 ` z1
pℓq,2 ¨ℓ
`
z2
pℓq,1
˘˚ℓ
˘
¨ℓ`1 iℓ`1,
(6.13)
or with the manifest form of different stage multiplications p
µ
r2s
ℓ and p
µ
r2s
ℓ`1 (needed to go on higher arities)
p
µ
r2s
ℓ`1
“
z1
pℓ`1q, z2
pℓ`1q
‰
“
´
p
µ
r2s
ℓ
“
z1
pℓq,1, z2
pℓq,1
‰
´ p
µ
r2s
ℓ
“`
z2
pℓq,2
˘˚ℓ
, z1
pℓq,2
‰¯
` p
µ
r2s
ℓ`1
”´
p
µ
r2s
ℓ
“
z2
pℓq,2, z1
pℓq,1
‰
` p
µ
r2s
ℓ
“
z1
pℓq,2,
`
z2
pℓq,1
˘˚ℓ
‰¯
, iℓ`1
ı
. (6.14)
Let us consider the example of quaternion construction in our notation.
Example 6.1. In the case of quaternions ℓ “ 1 and p
Aℓ`1 “ p
A2 “ H, while p
Aℓ “ p
A1 “ C. The recurrent
relations (6.12) for ℓ “ 0, 1 become
zp2q “ zp1q,1 ` zp1q,2 ¨2 i2, zp1q,1, zp1q,2 P p
A1 “ C, zp2q, i2 P H, (6.15)
zp1q,1 “ zp0q,11 ` zp0q,12 ¨1 i1, zp0q,11, zp0q,12 P p
A0 “ R, zp1q,1, i1 P C, (6.16)
zp1q,2 “ zp0q,21 ` zp0q,22 ¨1 i1, zp0q,21, zp0q,22 P p
A0 “ R, zp1q,2, i1 P C. (6.17)
– 16 –

17. Polyadic Cayley-Dickson process POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION
After substitution (6.16)-(6.17) into (6.15) we obtain the expression of quaternion with real coefficients
zp0q,αβ P R, α, β “ 1, 2, and two imagery units (from different parts of H) i1 P CzR Ă H and i2 P HzC
zp2q “ zp1q,1 ` zp1q,2 ¨2 i2 “
`
zp0q,11 ` zp0q,12 ¨1 i1
˘
`
`
zp0q,21 ` zp0q,22 ¨1 i1
˘
¨2 i2
“ zp0q,11 ` zp0q,12 ¨1 i1 ` zp0q,21 ¨2 i2 `
`
zp0q,22 ¨1 i1
˘
¨2 i2. (6.18)
To return to the standard notation (4.25), we put zp2q “ q, zp0q,11 “ a, zp0q,12 “ b, zp0q,21 “ c, zp0q,22 “ d,
i1 “ i P C, i2 “ j P HzC, i1 ¨2 i2 “ k P H and get q “ a ` bi ` cj ` dk. Similarly, the complex-like
multiplication (6.13) can be also applied twice with ℓ “ 0, 1 to obtain the quaternion multiplication in
terms of real coefficients (4.27).
6.3. Polyadic Cayley-Dickson process. Now we provide generalization of the Cayley-Dickson con-
struction to the polyadic case in the framework of the second (embedding) approach using the field ex-
tension formalism (see Subsection 6.2). The main iteration relation (6.11) will now contain, instead of
binary hypercomplex algebras p
Aℓ, n-ary algebras p
Arnℓs
ℓ on each stage.
Definition 6.2. The polyadic Cayley-Dickson process is defined as the iteration of n-ary algebras
p
Arnℓ`1s
ℓ`1 “
A!
Arnℓs
ℓ piℓq
)
| p`q , p
µ
rnℓ`1s
ℓ`1
E
piℓ`1q ,
i2
ℓ “ ´1, i2
ℓ`1 “ ´1, iℓ`1iℓ “ ´iℓiℓ`1, iℓ P Arnℓ`1s
ℓ`1 , nℓ ě 2, ℓ ě 0. (6.19)
The concrete representation of the ℓ-th stage n-ary algebras p
Arnℓs
ℓ is not important for the general
recurrence formula (6.19). Nevertheless, here we will use the matrix polyadization to obtain higher n-ary
non-division algebras (Section 3). First, we need the obvious
Lemma 6.3. The embedding of a block-monomial matrix into a block-monomial matrix gives a block-
monomial matrix.
Corollary 6.4. If the binary Cayley-Dickson construction gives a division algebra, then the corresponding
polyadic Cayley-Dickson process gives a nonderived n-ary non-division algebra, because of noninvertible
elements.
Thus, the structure of general algebra obtained by the polyadic Cayley-Dickson process, is more rich,
than one block-shift monomial matrix (3.6), it is the “tower” of such monomial matrices of size pnℓ ´ 1qˆ
pnℓ ´ 1q on ℓ-th stage embedded one into another. The connection of near arities (and matrix sizes) is
nℓ`1 “ κ pnℓ ´ 1q ` 1, (6.20)
where κ is the polyadic power [ ].
Proposition 6.5. If number of stages of the polyadic Cayley-Dickson process is ℓ, then the dimension of
the final algebra is (cf. (6.8))
DCDpℓq “ DpArn0,n1,...,nℓs
CD q “ 2ℓ
pn0 ´ 1q pn1 ´ 1q . . . pnℓ ´ 1q (6.21)
where ni are arities of the intermediate algebras. The size of the final matrix becomes
pn0 ´ 1q pn1 ´ 1q . . . pnℓ ´ 1q ˆ pn0 ´ 1q pn1 ´ 1q . . . pnℓ ´ 1q . (6.22)
Proof. On the ℓ-th stage of the polyadic Cayley-Dickson process, each block-monomial pnℓ ´ 1q ˆ
pnℓ ´ 1q matrix has pnℓ ´ 1q nonzero blocks of the cycle-shift shape (3.6). The 0-th stage corresponds to
reals, which gives 20
pn0 ´ 1q parameters. Then, the simple field extension ℓ “ 1 (6.12) with blocks of
reals gives pn0 ´ 1q ¨ 2 pn1 ´ 1q parameters and so on. Thus, the ℓ-th stage is given by (6.21).
– 17 –

18. POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION Polyadic Cayley-Dickson process
More concrete, for the nonderived n-ary algebras we have the dimensions
Rrn0s
, DpRrn0s
q “ pn0 ´ 1q ,
Crn0,n1s
CD “ Rrn0s
CD pi1q , DpCrn0,n1s
CD q “ 2 pn0 ´ 1q pn1 ´ 1q ,
Hrn0,n1,n2s
CD “ Crn0,n1s
CD pi2q , DpHrn0,n1,n2s
CD q “ 22
pn0 ´ 1q pn1 ´ 1q pn2 ´ 1q ,
Orn0,n1,n2,n3s
CD “ Hrn0,n1,n2s
CD pi3q , DpOrn0,n1,n2,n3s
CD q “ 23
pn0 ´ 1q pn1 ´ 1q pn2 ´ 1q pn3 ´ 1q ,
.
.
.
.
.
.
Arn0,n1,...,nℓs
CD,ℓ “ Arn0,n1,...,nℓ´1s
CD piℓq , DCDpℓq “ DpArn0,n1,...,nℓs
CD q “ 2ℓ
pn0 ´ 1q pn1 ´ 1q . . . pnℓ ´ 1q .
(6.23)
The starting algebra Rrn0s
is presented by the Z-matrix (3.6) with n “ n0 and real entries (A “ R).
Definition 6.6. The sequence of embedded cyclic shift block matrices in (6.23) can be called a polyadic
block-shift tower.
Remark 6.7. The shape of the final matrices (6.23) are different from the cyclic shift weighted matrices
(3.6), nevertheless, all the intermediate blocks are the cyclic shift block matrices. The nearest (in ℓ)
matrices have not arbitrary size, but connected with (6.20).
Example 6.8. To clarify the above general formulas, we provide the shape of polyadic quaternion matrix
with ℓ “ 2, and n0 “ 5, n1 “ 3, n2 “ 4, as
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
¨ ¨ ¨ ‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
‹ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
(6.24)
We have 24 ˆ 24 monomial matrix, 24 “ p5 ´ 1q p3 ´ 1q p4 ´ 1q by (6.23), dots denote zeroes and
stars denote nonzero entries. The dimension of this polyadic quaternion algebra represented by (6.24) is
– 18 –

19. Complex number ternary division algebra POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION
22
¨ 24 “ 96, because of two simple field extensions (6.15)–(6.18). This is 13-ary quaternion non-division
algebra. If there would be no higher stages of the Cayley-Dickson process, and the 24 ˆ 24 monomial
matrix would not be composed, having the form (3.6), then it will give 25-ary algebra.
Thus, we present the polyadic Cayley-Dickson process in the component form (6.12), but instead of
elements zpℓq of ℓ-th stage, we place Z-matrices (3.6) of suitable sizes. The resulting matrix is still
monomial, and so its determinant is proportional to the product of elements. The subalgebra of invertible
matrices corresponds to the division n-ary algebra.
Theorem 6.9. The invertible elements (which are described by Z-matrices with nonzero determinant) of
the polyadic Cayley-Dickson construction Drns
CD (for first stages ℓ “ 0, 1, 2, 3) (6.23) form the subalgebra
Drns
CD,div Ă Drns
CD which is the polyadic division n-ary algebra Drns
CD,div corresponding to the binary division
algebras D “ R, C, H, O.
To conclude, the matrix polyadization procedure applied for division algebras leads, in general, to non-
division algebras, because it is presented by the monomial matrices (representing nonzero noninvertible
elements) which become noninvertible when at least one entry vanishes. However, the subalgebras of
invertible elements can be considered as new n-ary division algebras.
7. POLYADIC IMAGINARY DIVISION ALGEBRAS
Here we introduce non-matrix polyadization procedure, which allows to obtain ternary division al-
gebras from the ordinary binary normed division algebras. Let us exploit the notations of the concrete
hyperembedding approach from Subsection 6.2. First, we show that some version of ternary division
algebra structure can be obtained by a new iterative process without introducing additional variables.
Definition 7.1. We define the ternary “imaginary tower” of algebras by
A
r3s
CD,ℓ`1 “ ACD piℓq ¨ iℓ`1 “
A
zpℓq piℓq ¨ iℓ`1
(
| p`q , µ
r3s
ℓ`1
E
, i2
ℓ “ ´1, iℓ`1iℓ “ ´iℓiℓ`1, ℓ ě 0.
(7.1)
The multiplication µ
r3s
ℓ`1 in (7.1) is nonderived ternary, because i3
ℓ “ ´iℓ for ℓ ě 1.
Theorem 7.2. If the initial algebra ACD piℓq is a normed division algebra, then its iterated imaginary
version A
r3s
CD,ℓ`1 (7.1) is the ternary division algebra of the same (initial) dimension Dpℓq and norm.
Assertion 7.3. The ternary algebras A
r3s
CD,ℓ`1 are not subalgebras of the initial algebras ACD piℓq.
Proof. This is because multiplications in the above algebras have different arities, despite the underlying
sets of imaginary algebras are subsets of the corresponding initial algebras.
Let us present the concrete expressions for the initial division algebras.
7.1. Complex number ternary division algebra. The first algebra (with ℓ “ 0) is the ternary division
algebra of pure imaginary complex numbers having the dimension D p0q “ D pRq “ 1
A
r3s
CD,1 ” C
r3s
“ R ¨ i1, zp1q “ bi1 P C, i2
1 “ ´1, b P R, (7.2)
which is unitless. The multiplication in C
r3s
is ternary nonderived and commutative
µ
r3s
1
“
z1
p1q, z2
p1q, z3
p1q
‰
“ ´b1
b2
b3
i1 P C
r3s
. (7.3)
– 19 –

20. POLYADIC IMAGINARY DIVISION ALGEBRAS Half-quaternion ternary division algebra
The norm is the absolute value in C (module)
›
›zp1q
›
›
p1q
“ |b|, and it is ternary multiplicative (from (7.3))
›
›
›µ
r3s
1
“
z1
p1q, z2
p1q, z3
p1q
‰›
›
›
p1q
“
›
›z1
p1q
›
›
p1q
›
›z2
p1q
›
›
p1q
›
›z3
p1q
›
›
p1q
P R, (7.4)
such that the corresponding map C
r3s
Ñ R is a ternary homomorphism. The querelement of zp1q (2.5) is
now defined by
µ
r3s
1
“
zp1q, zp1q, r
zp1q
‰
“ zp1q, (7.5)
which gives
r
zp1q “
1
zp1q
“ ´
i1
b
, b P Rz t0u . (7.6)
Thus, C
r3s
is a ternary commutative algebra, which is indeed a division algebra, because each nonzero
element has a querelement, i.e. it is invertible, so all equation of type (7.5) with different z1
s have solution.
7.2. Half-quaternion ternary division algebra. The next iteration case (ℓ “ 1) of the “imaginary
tower” (7.1) is more complicated and leads to pure imaginary ternary quaternions of the dimension
D p1q “ D pCq “ 2
A
r3s
CD,2 ” H
r3s
“ C pi1q ¨ i2, zp2q “ pc ` di1q i2 “ ci2 ` di1i2 P H
r3s
,
i2
1 “ i2
2 “ ´1, i1i2 “ ´i2i1, i1 P C, i2 P HzC, c, d P R. (7.7)
We informally can call H
r3s
the ternary algebra of imaginary “half-quaternions”, because in the standard
notation (see Example 6.1) zp2q from (7.7) is q “ a`bi`cj`dk
a“0,b“0
ÝÑ qhalf “ cj`dk. The nonderived
ternary algebra H
r3s
is obviously unitless. The multiplication of the half-quaternions H
r3s
µ
r3s
2
“
z1
p2q, z2
p2q, z3
p2q
‰
“ z1
p2q ¨ z2
p2q ¨ z3
p2q “ pd1
c2
d3
´ c1
d2
d3
´ d1
d2
c3
´ c1
c1
c3
q i2
` pc1
d2
c3
´ d1
c2
c3
´ c1
c2
d3
´ d1
d2
d3
q i1i2 P H
r3s
, (7.8)
is ternary nonderived (the algebra is closed with respect of product of 3 elements, but not less), noncom-
mutative and totally ternary associative (2.4)
µ
r3s
2
”
µ
r3s
2
“
z1
p2q, z2
p2q, z3
p2q
‰
z4
p2q, z41
p2q
ı
“ µ
r3s
2
”
z1
p2q, µ
r3s
2
“
z2
p2q, z3
p2qz4
p2q
‰
, z41
p2q
ı
“ µ
r3s
2
”
z1
p2q, z2
p2q, µ
r3s
2
“
z3
p2qz4
p2q, z41
p2q
‰ı
. (7.9)
The norm is defined by the absolute value in the quaternion algebra H in the standard way
›
›zp2q
›
›
p2q
“
ˇ
ˇzp2q
ˇ
ˇ “
?
c2 ` d2. (7.10)
The norm (7.10) is ternary multiplicative
›
›
›µ
r3s
2
“
z1
p2q, z2
p2q, z3
p2q
‰›
›
›
p2q
“
›
›z1
p2q
›
›
p2q
›
›z2
p2q
›
›
p2q
›
›z3
p2q
›
›
p2q
P R, (7.11)
such that the corresponding map H
r3s
Ñ R is a ternary homomorphism.
Proposition 7.4. Ternary analog of the sum of two squares identity is
pd1
c2
d3
´ c1
d2
d3
´ c1
c1
c3
´ d1
d2
c3
q
2
` pc1
d2
c3
´ d1
c2
c3
´ c1
c2
d3
´ d1
d2
d3
q
2
“
`
pc1
q2
` pd1
q2
˘ `
pc2
q2
` pd2
q2
˘ `
pc3
q2
` pd3
q2
˘
. (7.12)
– 20 –

21. Half-octonion ternary division algebra POLYADIC IMAGINARY DIVISION ALGEBRAS
Proof. It follows from the ternary multiplication (7.8) and the multiplicativity (7.11) of the norm (7.10).
Remark 7.5. The ternary sum of two squares identity (7.12) cannot be derived from the binary sum of
two squares (Diophantus, Fibonacci) identity or from Euler’s sum of four squares identity, while it can be
treated as the intermediate (triple) identity.
The querelement of zp2q (2.5) is now defined by
µ
r3s
2
“
zp2q, zp2q, r
zp2q
‰
“ zp2q, (7.13)
which gives
r
zp2q “ ´
ci2 ` di1i2
c2 ` d2
P H
r3s
, c2
` d2
‰ 0, c, d P R. (7.14)
It follows from (7.14) that H
r3s
(7.7) is a noncommutative nonderived ternary algebra, which is indeed
a division algebra, because each nonzero element zp2q has a querelement r
zp2q, i.e. it is invertible, and
equation of type (7.5) have solutions.
7.3. Half-octonion ternary division algebra. The next case (ℓ “ 2) gives pure imaginary ternary octo-
nions of the dimension D p2q “ D pHq “ 4
A
r3s
CD,3 ” O
r3s
“ H pi1, i2q ¨ i3, zp3q “ pa ` bi1 ` ci2 ` di1i2q i3 “ ai3 ` bi1i3 ` ci2i3 ` di1i2i3,
i2
1 “ i2
2 “ i2
3 “ ´1, i1i2 “ ´i2i1, i1i3 “ ´i3i1, i2i3 “ ´i3i2,
i1 P C, i2 P HzC, i3 P OzH, a, b, c, d P R. (7.15)
We informally can call O
r3s
the nonderived ternary algebra of imaginary “half-octonions” which is ob-
viously unitless. In the standard notation (with seven imaginary units e1 . . . e7) the “half-octonion” is
ohalf “ zp3q “ ae4 ` be5 ` ce6 ` de7.
It is well-known (see, e.g. BAEZ [2002], DRAY AND MANOGUE [2015]), that the algebra of octonions
O is not a field, but a special nonassociative loop (quasigroup with an identity), the Moufang loop (see,
e.g. BRUCK AND KLEINFELD [1951], PUMPLÜN [2014]). Because the ternary algebra O
r3s
is unitless, it
cannot be a ternary loop.
In this way, we introduce the ternary multiplication for “half-octonions” not by a triple product in O
(as in (7.8)) which is not unique due to nonassociativity, but in the following (unique) way
µ
r3s
3
“
z1
p3q, z2
p3q, z3
p3q
‰
“
´
z1
p3q ‚ z2
p3q
¯
‚ z3
p3q ` z1
p3q ‚
´
z2
p3q ‚ z3
p3q
¯
2
, (7.16)
where p‚q is the binary product in O.
Proposition 7.6. The nonderived nonunital ternary algebra of imaginary “half-octonions” (7.15)
O
r3s
“
A
zp3q
(
| p`q , µ
r3s
3
E
(7.17)
is closed under the multiplication (7.16) and ternary nonassociative.
– 21 –

22. POLYADIC IMAGINARY DIVISION ALGEBRAS Half-octonion ternary division algebra
In components the multiplication (7.16) becomes
µ
r3s
3
“
z1
p3q, z2
p3q, z3
p3q
‰
“ pa2
pb1
b3
` c1
c3
` d1
d3
q ´ a3
pb1
b2
` c1
c2
` d1
d2
q ` a1
pb2
b3
` c2
c3
` d2
d3
q ´ a1
a2
a3
q i3
` pb2
pa1
a3
` c1
c3
` d1
d3
q ` b3
pa1
a2
` c1
c2
` d1
d2
q ´ b1
pa2
a3
` c2
c3
` d2
d3
q ´ b1
b2
b3
q i1i3
` pc2
pa1
a3
` b1
b3
` d1
d3
q ´ c3
pa1
a2
` b1
b2
` d1
d2
q ´ c1
pa2
a3
` b2
b3
` d2
d3
q ´ c1
c2
c3
q i2i3
` pd2
pa1
a3
` b1
b3
` c1
c3
q ´ d3
pa1
a2
` b1
b2
` c1
c2
q ´ d1
pa2
a3
` b2
b3
` c2
c3
q ´ d1
d2
d3
q i1i2i3.
(7.18)
The querelement r
zp3q (2.5) of zp3q from O
r3s
is defined by
µ
r3s
3
“
zp3q, zp3q, r
zp3q
‰
“ zp3q, (7.19)
where r
zp3q can be on any place. In components we obtain (cf. half-quaternions (7.14))
r
zp3q “ ´
ai3 ` bi1i3 ` ci2i3 ` di1i2i3
a2 ` b2 ` c2 ` d2
P O
r3s
, a2
` b2
` c2
` d2
‰ 0, a, b, c, d P R. (7.20)
It follows from (7.20) that O
r3s
is a nonderived noncommutative nonassociative ternary algebra, which is
indeed a division algebra, because each nonzero element zp3q has a querelement r
zp3q, i.e. it is invertible.
The algebra of (binary) octonions O is nonassociative, but has a very close property, it is alternative.
Recall that a (binary) algebra A is left and right alternative if two relations hold valid
pxxq y “ x pxyq , (7.21)
x pyyq “ pxyq y, x, y P A. (7.22)
The ternary analog of alternativity requires three set of two relations, in general for n-ary algebra we
will need n sets by pn ´ 1q relations, i.e. in total n pn ´ 1q relations.
Definition 7.7. For a nonassociative ternary algebra Ar3s
`
p`q , µr3s
˘
we define the ternary alternativity
by 3 sets (left, middle and right alternativity) of 2 relations
µr3s
“
µr3s
rx, x, xs , y, t
‰
“ µr3s
“
x, µr3s
rx, x, ys , t
‰
“ µr3s
“
x, x, µr3s
rx, y, ts
‰
, (7.23)
µr3s
“
x, µr3s
ry, y, ys , t
‰
“ µr3s
“
x, y, µr3s
ry, y, ts
‰
“ µr3s
“
µr3s
rx, y, ys , y, t
‰
, (7.24)
µr3s
“
x, y, µr3s
rt, t, ts
‰
“ µr3s
“
x, µr3s
ry, t, ts , t
‰
“ µr3s
“
µr3s
rx, y, ts , t, t
‰
, x, y, t P Ar3s
. (7.25)
Obviously, that a ternary associative algebra (satisfying (7.9)) is ternary alternative, but not vise versa.
Proposition 7.8. The ternary algebra of imaginary “half-octonions” O
r3s
with the product (7.18) is
ternary alternative, i.e. it satisfies (7.23)–(7.25).
Thus, we have constructed the noncommutative nonassociative nonderived ternary division algebra of
half-octonions O
r3s
which is nonunital and ternary alternative.
– 22 –

23. REFERENCES
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