Steven Duplij, "Polyadic Algebraic Structures", book Front matter

Polyadic Algebraic Structures
Steven Duplij
Center for Information Processing WWU IT University of Münster,
48149 Münster, Germany
IOP Publishing, Bristol, UK

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ISBN 978-0-7503-2648-3 (ebook)
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DOI 10.1088/978-0-7503-2648-3
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To my parents Lidia and Anatoliy Duplij who made many vital efforts in my formation
and development for the book to be born, and to my wife Pierpaola whose everyday
assistance, inspiration and general discussions allowed me to crystallize ideas and not
be afraid of introducing new concepts.

Contents
Preface xiii
Acknowledgements xiv
Author biography xv
Symbols xvi
Introduction xvii
New constructions and ideas xx
Part I One-set polyadic algebraic structures
1 One-set algebraic structures and Hosszú–Gluskin theorem 1-1
1.1 General properties of one-set one-operation polyadic structures 1-1
1.1.1 Changing arity of one-set polyadic algebraic structures 1-2
1.1.2 Special elements in one-set polyadic structures 1-5
1.2 Polyadic semigroups, quasigroups and groups 1-7
1.3 Polyadic direct products and changed arity powers 1-13
1.3.1 Polyadic products of semigroups and groups 1-13
1.3.2 Full polyadic external product 1-14
1.3.3 Mixed arity iterated product 1-15
1.3.4 Polyadic hetero product 1-17
1.4 The deformed Hosszú–Gluskin theorem 1-22
1.4.1 The Hosszú–Gluskin theorem 1-22
1.4.2 ‘Deformation’ of Hosszú–Gluskin chain formula 1-29
1.4.3 Generalized ‘deformed’ version of the homomorphism theorem 1-39
1.5 Polyadic analog of Grothendieck group 1-40
1.5.1 Grothendieck group of commutative monoid 1-40
1.5.2 The n-ary group completion of m-ary semigroup 1-44
References 1-60
2 Representations and heteromorphisms 2-1
2.1 Homomorphisms of one-set polyadic algebraic structures 2-1
2.2 Heteromorphisms of one-set polyadic algebraic structures 2-3
2.2.1 Multiplace mappings and heteromorphisms 2-3
2.2.2 Heteromorphisms and associativity quivers 2-9
vii

2.3 Hetero-covering of algebraic structures 2-13
2.4 Multiplace representations of polyadic algebraic structures 2-16
2.5 k-Place actions 2-21
2.5.1 k-Place actions and G-spaces 2-21
2.5.2 Regular k-place actions 2-24
References 2-27
3 Polyadic semigroups and higher regularity 3-1
3.1 Generalized q-regular elements in semigroups 3-1
3.1.1 Binary q-regular single elements 3-2
3.1.2 Polyadic q-regular single elements 3-3
3.2 Higher q-inverse semigroups 3-7
3.2.1 Higher q-regular semigroups 3-8
3.2.2 Idempotents and higher q-inverse semigroups 3-13
3.3 Higher q-inverse polyadic semigroups 3-14
3.3.1 Higher q-regular polyadic semigroups 3-14
3.3.2 Sandwich polyadic q-regularity 3-18
3.3.3 Sandwich regularity with generalized idempotents 3-23
3.4 Polyadic-binary correspondence, regular semigroups, braid groups 3-26
3.4.1 Polyadic-binary correspondence 3-26
3.4.2 Ternary matrix group corresponding to the regular
semigroup
3-27
3.4.3 Polyadic matrix semigroup corresponding to the higher
regular semigroup
3-29
3.4.4 Ternary matrix group corresponding to the braid group 3-33
3.4.5 Ternary matrix generators 3-35
3.4.6 Generated n-ary matrix group corresponding the higher
braid group
3-37
References 3-47
4 Polyadic rings, fields and integer numbers 4-1
4.1 One-set polyadic ‘linear’ structures 4-2
4.1.1 Polyadic distributivity 4-2
4.1.2 Polyadic rings and fields 4-3
4.2 Polyadic direct products of rings and fields 4-9
4.2.1 External direct product of binary rings 4-10
4.2.2 Full polyadic external direct product of m n
( , )-rings 4-11
viii

4.2.3 Mixed arity iterated product of m n
( , )-rings 4-12
4.2.4 Polyadic hetero product of m n
( , )-fields 4-15
4.3 Polyadic integer numbers 4-19
4.3.1 Congruence classes and operations 4-19
4.3.2 Polyadic rings in limiting cases 4-24
4.3.3 Prime polyadic integers 4-27
4.3.4 The mapping of parameters to arity 4-33
4.4 Finite polyadic rings of integers 4-35
4.4.1 Secondary congruence classes 4-35
4.4.2 Finite polyadic rings of secondary classes 4-36
4.5 Finite polyadic fields of integer numbers 4-39
4.5.1 Abstract finite polyadic fields 4-39
4.5.2 Multiplicative structure of finite polyadic fields 4-41
4.5.3 Examples of exotic finite polyadic fields 4-50
4.6 Diophantine equations over polyadic integers and Fermat’s theorem 4-52
4.6.1 Polyadic analog of the Lander–Parkin–Selfridge conjecture 4-53
4.6.2 Frolov’s theorem and the Tarry–Escott problem 4-59
References 4-61
Part II Two-sets polyadic algebraic structures
5 Polyadic algebras and deformations 5-1
5.1 Two-set polyadic structures 5-1
5.1.1 Polyadic vector spaces 5-1
5.1.2 One-set polyadic vector space 5-7
5.2 Mappings between polyadic vector spaces 5-8
5.2.1 Polyadic functionals and dual polyadic vector spaces 5-10
5.2.2 Polyadic direct sum and tensor product 5-13
5.3 Polyadic associative algebras 5-17
5.3.1 ‘Elementwise’ description 5-17
5.3.2 Polyadic analog of the functions on group 5-24
5.3.3 ‘Diagrammatic’ description 5-26
5.3.4 Medial map and polyadic permutations 5-30
5.3.5 Tensor product of polyadic algebras 5-33
5.3.6 Heteromorphisms of polyadic associative algebras 5-34
5.3.7 Structure constants 5-36
References 5-38
ix

6 Polyadic inner spaces and operators 6-1
6.1 Polyadic inner pairing spaces and norms 6-1
6.2 Elements of polyadic operator theory 6-6
6.2.1 Multistars and polyadic adjoints 6-8
6.2.2 Polyadic isometry and projection 6-14
6.2.3 Towards a polyadic analog of *
C -algebras 6-16
References 6-19
7 Medial deformation of n-ary algebras 7-1
7.1 Almost commutative graded algebra 7-2
7.1.1 Almost commutativity 7-3
7.1.2 Tower of higher level commutation brackets 7-5
7.2 Almost medial graded algebras 7-7
7.2.1 Medial binary magmas and quasigroups 7-7
7.2.2 Almost mediality 7-8
7.2.3 Tower of higher binary mediality brackets 7-11
7.3 Medial n-ary algebras 7-13
7.4 Almost medial n-ary graded algebras 7-14
7.4.1 Higher level mediality n2-ary brackets 7-18
7.5 Toyoda’s theorem for almost medial algebras 7-20
References 7-21
8 Membership deformations and obscure n-ary algebras 8-1
8.1 Graded algebras and Shur factors 8-1
8.2 Membership function and obscure algebras 8-3
8.3 Membership deformation of commutativity 8-4
8.3.1 Deformation of commutative algebras 8-4
8.3.2 Deformation of almost-commutative algebras 8-7
8.3.3 Double almost-Lie algebras 8-9
8.4 Projective representations 8-10
8.4.1 Binary projective representations 8-10
8.4.2 n-Ary projective representations 8-12
8.5 n-ary double commutative algebras 8-14
8.5.1 n-ary almost-commutative algebras 8-14
8.5.2 Membership deformed n-ary algebras 8-18
8.6 Conclusions 8-21
References 8-21
x

Part III Polyadic quantum groups
9 Polyadic Hopf algebras 9-1
9.1 Polyadic coalgebras 9-2
9.1.1 Polyadic comultiplication 9-3
9.1.2 Homomorphisms of polyadic coalgebras 9-8
9.1.3 Tensor product of polyadic coalgebras 9-8
9.1.4 Polyadic coalgebras in the Sweedler notation 9-9
9.1.5 Polyadic group-like and primitive elements 9-11
9.1.6 Polyadic analog of duality 9-13
9.1.7 Polyadic convolution product 9-15
9.2 Polyadic bialgebras 9-22
9.3 Polyadic Hopf algebras 9-25
9.4 Ternary examples 9-31
9.4.1 Ternary Sweedler Hopf algebra 9-31
9.4.2 Ternary quantum group examples 9-32
9.5 Polyadic almost co-commutativity and co-mediality 9-34
9.5.1 Quantum Yang–Baxter equation 9-34
9.5.2 ′
n -Ary braid equation 9-37
9.5.3 Polyadic almost co-commutativity 9-39
9.5.4 Equations for the ′
n -ary R-matrix 9-40
9.5.5 Polyadic triangularity 9-42
9.5.6 Almost co-medial polyadic bialgebras 9-43
9.5.7 Equations for the M-matrix 9-46
9.5.8 Medial analog of triangularity 9-50
References 9-52
10 Solutions to higher braid equations 10-1
10.1 Yang–Baxter operators 10-1
10.1.1 Yang–Baxter maps and braid group 10-1
10.1.2 Constant matrix solutions to the Yang–Baxter equation 10-2
10.1.3 Partial identity and unitarity 10-4
10.1.4 Permutation and parameter-permutation 4-vertex
Yang–Baxter maps
10-6
10.1.5 Group structure of 4-vertex and 8-vertex matrices 10-8
10.1.6 Star 8-vertex and circle 8-vertex Yang–Baxter maps 10-16
10.1.7 Triangle invertible 9- and 10-vertex solutions 10-19
xi

10.2 Polyadic braid operators and higher braid equations 10-22
10.3 Solutions to the ternary braid equations 10-24
10.3.1 Constant matrix solutions 10-24
10.3.2 Permutation and parameter-permutation 8-vertex solutions 10-27
10.3.3 Group structure of the star and circle 8-vertex matrices 10-30
10.3.4 Group structure of the star and circle 16-vertex matrices 10-35
10.3.5 Pauli matrix presentation of the star and circle
16-vertex constant matrices
10-38
10.3.6 Invertible and non-invertible 16-vertex solutions to the
ternary braid equations
10-39
10.3.7 Higher 2n
-vertex constant solutions to n-ary braid equations 10-42
References 10-43
Part IV Polyadic categories
11 Polyadic tensor categories 11-1
11.1 Binary tensor categories 11-1
11.2 Polyadic tensor categories 11-6
11.2.1 Polyadic semigroupal categories 11-7
11.2.2 n-Ary coherence 11-8
11.3 Polyadic units, unitors and quertors 11-9
11.3.1 Polyadic monoidal categories 11-10
11.3.2 Polyadic nonunital groupal categories 11-11
11.4 Braided tensor categories 11-13
11.4.1 Braided binary tensor categories 11-13
11.4.2 Braided polyadic tensor categories 11-16
11.5 Medialed polyadic tensor categories 11-20
11.5.1 Medialed binary and ternary categories 11-22
11.6 Conclusions 11-25
References 11-26
xii

Preface
Nowadays, the arity of algebraic structures tends to be investigated from either a too
general (e.g., in abstract algebra, as sets with operations of arbitrary arity satisfying
axioms), or a too narrow (e.g., in physics, as group-like, ring-like and algebra-like
concrete structures with mostly binary operations) point of view.
This book can be treated as a ‘bridge’ between these approaches. We propose
using a full ‘polyadization’ procedure: take a concrete structure with binary
operations and ‘free-up’ all arities while saving the ‘interaction’ between them.
The latter gives equations relating arities, and since these are integers, we obtain
‘quantization’ rules for algebraic structures, restricting their arity shape. Another
feature of this thorough ‘polyadization’ is the appearance of exotic properties which
are not possible for binary arities, e.g., polyadic groups without identity and
polyadic ﬁelds without zero. Another unusual possibility is mappings between
structures of different arities, or heteromorphisms. This leads to a new kind of
representation theory, in which the arity of an algebraic structure need not coincide
with the arity of a representation. Further, new quantum groups, higher braidings,
M-matrices and medialed categories are introduced.
The book contains original constructions and ideas developed by the author,
while known material is given in brief for notational purposes only. It should attract
the attention of researchers in modern theoretical physics and mathematics who are
interested in new directions and developments.
Steven Duplij
Münster, Germany
December 2021
xiii

Acknowledgements
The author is deeply grateful to Vladimir Akulov, Bernd Billhardt, Andrzej
Borowiec, Andrew Bruce, Vesna Celakoska, Miroslav Ćirić Joachim Cuntz,
Costantino Delizia, Christopher Deninger, Branko Dragovich, Wieslaw Dudek,
Siegfried Echterhoff, Murray Geshtenhaber, Nick Gilbert, Gerald Goldin, Mike
Hewitt, Lacrimioara Iancu, Vladimir Kalashnikov, Richard Kerner, Maurice
Kibler, Mikhail Krivoruchenko, Grigorij Kurinnoj, Michel Lapidus, Dimitrij
Leites, Daniel Lenz, Jerzy Lukierski, Yuri Manin, Riivo Must, Chi-Keung Ng,
Peter Nonnenmann, Thomas Nordahl, Wolfram Von Oertzen, Valentin Ovsienko,
Walther Paravicini, Christopher Phillips, Norbert Ponchin, Maria Pop, Vasile Pop,
Sergey Prokushkin, Anatoliy Prykarpatsky, Ioan Purdea, Mathias Pütz, Dominik
Rudolph, Jim Stasheff, Vladimir Tkach, Ron Umble, Raimund Vogl, Thomas
Timmermann, Alexander Voronov, Raimar Wulkenhaar, and Wend Werner for
fruitful discussions, important help and valuable support.
xiv

Author biography
Steven Duplij
Steven Duplij (Stepan Douplii) is a theoretical and mathematical
physicist from the University of Münster, Germany. He was born
in Chernyshevsk–Zabaikalsky, Russia, and studied at Kharkiv
University, Ukraine where he gained his PhD in 1983. While
working at Kharkiv he received the title Doctor of Physical and
Mathematical Sciences by Habilitation in 1999.
Dr Duplij is the editor-compiler of ‘Concise Encyclopedia of
Supersymmetry’ (2005, Springer), and is the author of more than a hundred scientiﬁc
publications and several books. He is listed in the World Directory Of
Mathematicians, Marques Who Is Who In America, the Encyclopedia of Modern
Ukraine, the Academic Genealogy of Theoretical Physicists and the Mathematics
Genealogy Project. His scientiﬁc directions include supersymmetry and quantum
groups, advanced algebraic structures, gravity and nonlinear electrodynamics,
constrained systems and quantum computing.
xv

Symbols
A list of symbols and notations used in the text.
S T G
, , — sets
μ ν ρ
, ,
n n n
( ) ( ) ( ) — n-ary multiplications, additions, n-place actions (‘elementwise’
products)
μ ν ρ
, ,
n n n
( ) ( ) ( ) — n-ary multiplications, additions, n-place actions (‘diagrammatic’
linear maps)
Δ n
( ) — n-ary comultiplications (‘elementwise’)
σ
Δ ,
n n
( ) ( ) — n-ary comultiplications, coactions (‘diagrammatic’ linear maps)
μ ◦
( )
n ℓ
( ) — composition of ℓ n-ary multiplications (additions, actions)
s g h E
, , , — …
s s
( , , )
n
1 , n-tuples, polyads, sequences
, , ,
n n n n
( ) ( ) ( ) ( )
S G R A —semigroups, groups, rings, algebras of arity n
  
, , — ground fields
,
Alg Grp — category names
Φ Ψ Π
, , — homomorphisms, heteromorphisms, representations
z
, ,
K
0 1 — zero and unity of ground fields
z z
, V — zero of semigroups and algebras, vector spaces
e —identity of groups, semigroups and algebras
B, A — braidings, associators
,
U T — unitors, natural transformations
F C
, — functors, categories
L M
, — Lie brackets, medial brackets
×
A n — direct product of n copies of A
⊗
A n — tensor product of n copies of A
′ ′
a
,
G — grading groups, elements
f — membership functions
( )
A f — obscure (fuzzy) set
H — cohomology classes
Italic — definitions
San serif — stressing
Quotes — figurative meaning
xvi

Introduction
The study of polyadic (or higher arity) algebraic structures has a two-century long history,
commencing with works by Cayley, Sylvester, Kasner, Prüfer, Dörnte, Lehmer, Post, etc.
They took a single set, closed under one (main) binary operation having special properties
(the so-called ’group-like’ structure), and ‘generalized’ it by increasing the arity of that
operation, which can then be called a polyadic operation and the corresponding algebraic
structure polyadic. We will use the term ‘polyadic’ in this sense only, while other uses are
extantintheliterature(see,e.g.,Halmos1962).An‘abstractway’tostudypolyadicalgebraic
structures is via theuse of universalalgebras definedas sets with various axioms(equational
laws) for the polyadic operations (Cohn 1965, Grätser 1968, Bergman 2012). However,
some important algebraic structures cannot be described in this language, e.g., ordered
groups, fields, etc (Denecke and Wismath 2009). It is therefore also worthwhile to pursue a
‘concrete approach’ by studying examples of binary algebraic structures and then to
‘polyadize’ them in the proper way. This method has initiated the development of a
corresponding theory of n-ary quasigroups (Belousov 1972), n-ary semigroups (Monk and
Sioson 1966, Zupnik 1967) and n-ary groups (Gal’mak 2003, Rusakov 1998) (for a more
recent review, see, e.g., Duplij 2012 and a comprehensive list of references therein). Binary
algebraic structures with two operations (addition and multiplication) on one set (the so-
called’ring-like’structures)werelatergeneralizedto m n
( , )-rings(Celakoski1977,Crombez
1972, Leeson and Butson 1980) and m n
( , )-fields (Iancu and Pop 1996) (for a recent study,
see Duplij 2017), while these were mostly investigated in a more restrictive manner by
considering particular cases: ternary rings (or (2, 3)-rings) (Lister 1971), m
( , 2)-rings
(Boccioni 1965, Pop and Pop 2002), as well as (3, 2)-fields (Duplij and Werner 2021).
In chapter 1 one-set polyadic algebraic structures are reviewed and their
classification is given. The general ways of changing arity and special elements are
considered. Polyadic direct products of (semi)groups which can change arity are
then introduced. A new form of the Hosszú–Gluskin theorem is presented in terms
of polyadic powers and using the language of diagrams. It is shown that the Hosszú–
Gluskin chain formula is not unique and can be ‘deformed’ using a parameter q
which takes special integer values, which leads to the concept of a sandwich
nonunital endomorphism. The ‘q-deformed’ analog of the Hosszú–Gluskin theorem
is then formulated, and a ‘q-deformed’ homomorphism theorem is given. Then we
generalize the Grothendieck construction of the completion group for a monoid (being
the starting point of the algebraic K-theory) to the polyadic case, when an initial
semigroup in m-ary and the corresponding final class group K m n
0
( , )
can be n-ary.
In chapter 2 a new multiplace generalization of associativity preserving homo-
morphisms, a ‘heteromorphism’ which connects polyadic systems having unequal arities,
is introduced via an explicit formula, together with related definitions for multiplace
representations and multiactions. The classification of heteromorphisms is given, and it
is seen that their arity are not arbitrary, but ‘quantized’. Then regular multiactions are
introduced and the corresponding regular multiplace representations are defined.
In chapter 3 the regularity concept for semigroups is generalized in two ways
simultaneously: to higher regularity and to higher arity. We show that in this
approach the one-relational and multi-relational formulations of higher regularity
xvii

do not coincide, and each element can have several inverses. For polyadic semi-
groups we introduce several types of higher regularity which satisfy the arity
invariance principle: the expressions should not depend of the numerical arity
values. We introduce sandwich higher polyadic regularity with generalized idempo-
tents. We use the polyadic-binary correspondence in which the product of a special
kind of matrices reproduces the higher regular semigroups and higher braid groups,
and the latter are connected with the generalized polyadic braid equation. The
higher degree (in our sense) Coxeter group and symmetry groups are then defined.
In chapter 4 polyadic rings and fields are introduced. Their direct products are
unusual and can have different arity. The polyadic integer numbers, which form a
polyadic ring, are defined. The basics of the new polyadic arithmetic thus introduced
is presented: prime polyadic numbers, the polyadic Euler function, polyadic division
with a remainder, etc. Secondary congruence classes of polyadic integer numbers
and the corresponding finite polyadic rings are defined. A polyadic version of
(prime) finite fields is introduced. These can be zeroless, zeroless and nonunital, or
have several units; it is even possible for all of their elements to be units. None of the
above situations is possible in the binary case. It is conjectured that a finite polyadic
field should contain a certain canonical prime polyadic field, defined here as a
minimal finite subfield, which can be considered as a polyadic analog of GF p
( ). The
Diophantine equations over the polyadic rings are then considered. Polyadic analogs
of the Lander–Parkin–Selfridge conjecture and Fermat’s last theorem are formu-
lated. For polyadic numbers neither of the standard statements holds. Polyadic
versions of the Frolov’s theorem and the Tarry–Escott problem are presented.
In chapter 5 concrete two-set (module-like and algebra-like) algebraic structures are
investigated from the viewpoint that the initial arities of all operations are arbitrary.
Relations between operations arising from the structure definitions, however, lead to
restrictions which determine their possible arity shapes (which are ‘quantized’) and
allow us to formulate the partial arity freedom principle. Polyadic vector spaces and
dual vector spaces, direct sums, associative algebras and tensor products are recon-
sidered, and a classification in terms of the units and zeroes of an algebra and its
underlying field is provided. Medial polyadic associative algebras are introduced, their
‘quantized’ arity shape is given, and polyadic structure constants are defined.
In chapter 6 elements of polyadic operator theory are outlined: inner pairing
spaces, multistars and polyadic analogs of adjoints, operator norms, isometries and
projections are introduced. The arity signature and arity shape of polyadic algebraic
structures is presented as a general table. Then polyadic *
C -algebras, Toeplitz
algebras and Cuntz algebras are defined and represented by polyadic operators.
In chapter 7 algebraic structures in which the property of commutativity is
replaced by the mediality property are introduced. We consider (associative) graded
algebras and in place of almost commutativity, we introduce almost mediality: the
commutativity-to-mediality ansatz. Higher graded twisted products and ‘deforming’
brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem
which connects (universal) medial algebras with abelian algebras is proven for the
almost medial graded algebras introduced here.
In chapter 8 a new general mechanism for ‘breaking’ commutativity in algebras is
proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/
xviii

fuzzy set, the membership function, reflecting the degree of truth that an element
belongs to the set, can be incorporated into the commutation relations. Special
‘deformations’ of commutativity and almost commutativity are introduced in such a
way that equal degrees of truth result in the ‘nondeformed’ case. We also sketch how
to ‘deform’ ε-Lie algebras and Weyl algebras and introduce the double almost-Lie
algebras. Further, the above constructions are extended to n-ary algebras for which
the analogs of projective representations and ε-commutativity are studied.
In chapter 9 generalized polyadic coassociative coalgebras, bialgebras and Hopf
algebras are defined and investigated. They have many unusual features in comparison
with the binary case. For instance, both the algebra and its underlying field can be
zeroless and nonunital, the existence of the unit and counit is not obligatory, while the
dimension of the algebra is not arbitrary, but ‘quantized’. The polyadic convolution
product and bialgebra can be defined, and when the algebra and coalgebra have
unequal arities, the polyadic version of the antipode, the querantipode, has different
properties. We introduce the polyadic version of braidings, almost co-commutativity,
quasitriangularity and the equations for the R-matrix (which can be treated as a
polyadic analog of the Yang–Baxter equation). We propose another concept of
deformation which is governed not by the twist map, but by the medial map. We
present the corresponding braidings, almost co-mediality and M-matrix (the medial
analog of the R-matrix), for which compatibility equations are found.
In chapter 10 constant matrix solutions of the polyadic (higher) braid equations
are introduced and studied. They play a role as various quantum gates in quantum
computation. We define two classes of 8-vertex matrices, star and circle, such that
the magic matrices (connected with the Cartan decomposition) belong to the star
class. We also define partial identity and unitarity. The general algebraic structure of
the classes is described in terms of new semigroups, ternary and 5-ary groups and
modules. We present the invertible and noninvertible 8-, 9- and 10-vertex Yang–
Baxter maps. We then introduce polyadic braid operators for higher braid
equations, find constant matrix solutions for the ternary 8- and 16-vertex cases
and give their Pauli matrix presentation.
In chapter 11 we generalize tensor categories and braided tensor categories using
the commutativity-to-mediality ansatz, as in chapter 7 for algebras. A polyadic
(non-strict) tensor categoryhas an n-ary tensorproduct asan additional multiplication
with −
n 1associators of the arity −
n
2 1satisfying a +
n
( 1)
2 -gon relation, which is a
polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain
several unit objects, and it is also possible that all objects are units. A new kind of
polyadic categories (called groupal) is defined: they are close to monoidal categories,
but may not contain units: instead (by analogy with the querelements in n-ary groups)
the querfunctor and (natural) functorial isomorphisms, the quertors, are considered.
The arity-nonreducible n-ary braiding is introduced and the equation for it is derived,
which for n = 2 coincides with the Yang–Baxter equation. Then, analogously to the
approach in chapter 7, we introduce ‘medialing’ instead of braiding and construct
‘medialed’ polyadic tensor categories.
xix

New constructions and ideas
Here we list new constructions and ideas presented in the book:
• Polyadization—Exchanging all binary operations in an algebraic structure
with the polyadic ones preserving ‘interrelations’ between them.
• Sandwitch quasi-endomorphism—A nonunital endomorphism of the sand-
witch form (1.4.75).
• Full polyadic direct product—An external direct product of polyadic algebraic
structures of the same arity (1.3.5), and for polyadic rings (4.2.5)–(4.2.18).
• Iterated direct product—A direct product of polyadic algebraic structures of
different arities (1.3.12), see for m n
( , )-rings (4.2.17)–(4.2.18).
• Hetero (‘entangled’) k-power—An external power of polyadic algebraic
structures which changes arity, but preserves associativity (1.3.18).
• Deformed Hosszú–Gluskin chain formula and theorem—see (1.4.98) and
theorem 1.4.12.
• Heteromorphism—A mapping between algebraic structures of different arities
(2.2.4).
• Multiplace representation—A representation (2.4.5) of a on-set polyadic
algebraic structure given by a multiplace function (2.4.4).
• Arity changing formula—Connection between arities of initial and final
algebraic structures in heteromorphism (2.2.16).
• Associative polyadic quiver—A graph preserving associativity with respect to
heteromorphism (2.2.30)–(2.2.31).
• k-place action/multiaction—see (2.5.2).
• Higher regularity—Regularity with multi-element inverses (3.1.3).
• Arity invariance principle—An expression of algebraic structure in term of
operations does not depend on their arities (definition 3.1.4).
• Sandwich polyadic regularity—Polyadic analog of higher regularity in terms
of idempotents (3.3.29).
• n-ary matrices—Set of matrices which is closed with respect to (not less than)
n-ary multiplication (3.4.10).
• Polyadic-binary correspondence—Reproduction of higher regularity by prod-
uct of n-ary matrices (3.4.15)–(3.4.21).
• Higher n-degree braid (binary) group—Generalization of the Artin braid
group for n multipliers (3.4.81)–(3.4.88).
• Higher n-degree Coxeter group—see (3.4.115)–(3.4.116).
• Quer-symmetric polyadic field—A m n
( , )-field in which unary queroperations
commute, see definition 4.1.16.
• Iterated direct product polyadic ring—A direct product of polyadic rings of
different arities (4.2.17)–(4.2.18).
• Polyadic direct product field—An external product of polyadic zeroless fields
of the same arity shape, see definition 4.2.8.
• Polyadic integer number—A m n
( , )-ring of representatives of a congruence
(residue) class (4.3.9).
xx

• Secondary congruence class—A double class of polyadic integers which gives
finite polyadic ring (4.4.3).
• Non-extended finite polyadic field—An abstract non-extended prime finite
m n
( , )-field of polyadic integers (4.5.1).
• Diophantine equation over polyadic integers—see (4.6.1).
• Polyadic Lander–Parkin–Selfridge conjecture—see conjecture 4.6.7.
• Polyadic Pythagorean theorem—see definition 4.6.5.
• Polyadic Fermat’s Last Theorem—see conjecture 4.6.6.
• Polyadic vector space—The 2-set 4-operation algebraic structure satisfying 7
axioms, see definition 5.1.6.
• Polyadic associative algebra—The 2-set 5-operation algebraic structure sat-
isfying 4 compatibility conditions (5.3.4)–(5.3.12), see definition 5.3.1.
• Polyadic convolution—see definition 5.3.15.
• Arity partial freedom principle—theorem 5.3.12.
• Polyadic unit map—see (5.3.39).
• Polyadic structure constants—see (5.3.50).
• Arity shape of polyadic algebraic structures—see table 6.1.
• Polyadic multistar adjoint—see (6.2.21).
• Polyadic isometry—A polyadic operator which preserves the polyadic inner
pairing, see definition 6.2.15.
• Polyadic operator *
C -algebra—The operator polyadic Banach algebra sat-
isfying the multi- *
C -relations (6.2.58).
• Membership deformation—Deformation of commutativity using the member-
ship function (8.3.2).
• Higher level Lie bracket—see (7.1.18)–(7.1.20).
• Medial brackets—Deviation from almost mediality (7.2.19).
• Higher level mediality bracket—see (7.4.23)–(7.4.25).
• Ternary Toyoda theorem—A shape of multiplication for almost medial
algebras, see theorem 7.5.2.
• Obscure algebra—An algebra with the multiplication deformed by the
membership function (8.3.1).
• Double almost commutative algebra—An ε-commutative algebra additionally
deformed by the membership function, see definition 8.3.13.
• Double Lie bracket—Deviation from the double commutativity (8.3.23).
• Polyadic projective representation—A multiplace generalization of the pro-
jective representation (8.4.12).
• Polyadic π-commutative algebra—A polyadic graded algebra deformed by n-
ary π-factor (8.5.3).
• Polyadic counit—see (9.1.22).
• Polyadic duality—see definition 9.1.33.
• n-ary convolution product—A product consisting of polyadic multiplication
an comultiplication having different arity (9.1.64).
• Polyadic querantipod—A coquerelement for the identity with respect to the n-
ary convolution product (9.3.10).
• n-ary braid equation—The equation for the morphism of n modules (9.5.23).
xxi

• n-ary R-matrix—An element of polyadic bialgebra which governs ‘interac-
tion’ of product and coproduct (9.5.29).
• Polyadic M-matrix—An element of polyadic almost co-medial bialgebra
which governs ‘interaction’ of product and coproduct (9.5.48).
• Medial quasipolyangularity—see (9.5.70)–(9.5.72).
• Partial unitary matrix—A matrix having a product with its conjugate-trans-
posed as a partial identity (10.1.20).
• Star and circle matrices—Matrices of special form to which belong some
solutions of the Yang–Baxter and higher braid equations (10.1.76)–(10.1.77)
and (10.3.53)–(10.3.54).
• n-ary associator—Polyadic generalization of the associator (11.2.7).
• Polyadic unitor—An n-ary unit morphism (11.3.1).
• Querfunctor, querobject, quermorphism—Analogs of querelement in polyadic
groupal categories, see definition 11.3.4.
• n-ary braiding—Natural transformation which permutes n-ary tensor product
(11.4.9).
• n-ary braid equation—The equation for n-ary braiding (11.4.15).
• n-ary medial braiding—A functorial n2-isomorphism (11.5.1).
• Medialed polyadic monoidal category—see definition 11.5.4.
• Polyadic ‘gauge’ and ‘twisted’ shifts—Polyadic analogs of componentwise and
twisted shifts to define the equaivalence relations on a polyadic semigroup
(1.5.44) and (1.5.45).
• n-ary group completion of m-ary semigroup K m n
0
( , )
—A polyadic group of
equivalence classes for a polyadic semigroup (1.5.70).
References
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Bergman C 2012 Universal Algebra: Fundamentals and Selected Topics (New York: CRC Press)
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Iancu L and Pop M S 1996 A Post type theorem for (m,n) ﬁelds Proc. of the Scientiﬁc
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Leeson J J and Butson A T 1980 On the general theory of m n
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xxiii

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