Polyadic Algebraic Structures

PRELIMS.pdf
Preface
Acknowledgements
Author biography
Steven Duplij
Symbols
Introduction
New constructions and ideas
References
CH001.pdf
Chapter 1 One-set algebraic structures and Hosszú–Gluskin theorem
1.1 General properties of one-set one-operation polyadic structures
1.1.1 Changing arity of one-set polyadic algebraic structures
1.1.2 Special elements in one-set polyadic structures
1.2 Polyadic semigroups, quasigroups and groups
1.3 Polyadic direct products and changed arity powers
1.3.1 Polyadic products of semigroups and groups
1.3.2 Full polyadic external product
1.3.3 Mixed arity iterated product
1.3.4 Polyadic hetero product
1.4 The deformed Hosszú–Gluskin theorem
1.4.1 The Hosszú–Gluskin theorem
1.4.2 ‘Deformation’ of Hosszú–Gluskin chain formula
1.4.3 Generalized ‘deformed’ version of the homomorphism theorem
1.5 Polyadic analog of Grothendieck group
1.5.1 Grothendieck group of commutative monoid
1.5.2 The n-ary group completion of m-ary semigroup
Example 1.5.24. Negative numbers (continued)
References
CH002.pdf
Chapter 2 Representations and heteromorphisms
2.1 Homomorphisms of one-set polyadic algebraic structures
2.2 Heteromorphisms of one-set polyadic algebraic structures
2.2.1 Multiplace mappings and heteromorphisms
2.2.2 Heteromorphisms and associativity quivers
2.3 Hetero-covering of algebraic structures
2.4 Multiplace representations of polyadic algebraic structures
2.5 k-Place actions
2.5.1 k-Place actions and G-spaces
2.5.2 Regular k-place actions
References
CH003.pdf
Chapter 3 Polyadic semigroups and higher regularity
3.1 Generalized q-regular elements in semigroups
3.1.1 Binary q-regular single elements
3.1.2 Polyadic q-regular single elements
3.2 Higher q-inverse semigroups
3.2.1 Higher q-regular semigroups
3.2.2 Idempotents and higher q-inverse semigroups
3.3 Higher q-inverse polyadic semigroups
3.3.1 Higher q-regular polyadic semigroups
3.3.2 Sandwich polyadic q-regularity
3.3.3 Sandwich regularity with generalized idempotents
3.4 Polyadic-binary correspondence, regular semigroups, braid groups
3.4.1 Polyadic-binary correspondence
3.4.2 Ternary matrix group corresponding to the regular semigroup
3.4.3 Polyadic matrix semigroup corresponding to the higher regular semigroup
3.4.4 Ternary matrix group corresponding to the braid group
3.4.5 Ternary matrix generators
3.4.6 Generated n-ary matrix group corresponding the higher braid group
References
CH004.pdf
Chapter 4 Polyadic rings, fields and integer numbers
4.1 One-set polyadic ‘linear’ structures
4.1.1 Polyadic distributivity
4.1.2 Polyadic rings and fields
Definition 4.1.14 Diagrammatic definition of (m,n)-field.
4.2 Polyadic direct products of rings and fields
4.2.1 External direct product of binary rings
4.2.2 Full polyadic external direct product of (m,n)-rings
4.2.3 Mixed arity iterated product of (m,n)-rings
4.2.4 Polyadic hetero product of (m,n)-fields
4.3 Polyadic integer numbers
4.3.1 Congruence classes and operations
4.3.2 Polyadic rings in limiting cases
4.3.3 Prime polyadic integers
4.3.4 The mapping of parameters to arity
4.4 Finite polyadic rings of integers
4.4.1 Secondary congruence classes
4.4.2 Finite polyadic rings of secondary classes
4.5 Finite polyadic fields of integer numbers
4.5.1 Abstract finite polyadic fields
4.5.2 Multiplicative structure of finite polyadic fields
4.5.3 Examples of exotic finite polyadic fields
4.6 Diophantine equations over polyadic integers and Fermat’s theorem
4.6.1 Polyadic analog of the Lander–Parkin–Selfridge conjecture
Conjecture 4.6.3. Lander–Parkin–Selfridge (Lander et al 1967)
Conjecture 4.6.6. Polyadic analog of Fermat’s Last Theorem
4.6.2 Frolov’s theorem and the Tarry–Escott problem
References
CH005.pdf
Chapter 5 Polyadic algebras and deformations
5.1 Two-set polyadic structures
5.1.1 Polyadic vector spaces
5.1.2 One-set polyadic vector space
5.2 Mappings between polyadic vector spaces
5.2.1 Polyadic functionals and dual polyadic vector spaces
5.2.2 Polyadic direct sum and tensor product
5.3 Polyadic associative algebras
5.3.1 ‘Elementwise’ description
Theorem 5.3.12. The arity partial freedom principle
5.3.2 Polyadic analog of the functions on group
5.3.3 ‘Diagrammatic’ description
Definition 5.3.20. Algebra associativity axiom
Definition 5.3.25. Algebra unit axiom
5.3.4 Medial map and polyadic permutations
5.3.5 Tensor product of polyadic algebras
5.3.6 Heteromorphisms of polyadic associative algebras
5.3.7 Structure constants
References
CH006.pdf
Chapter 6 Polyadic inner spaces and operators
6.1 Polyadic inner pairing spaces and norms
6.2 Elements of polyadic operator theory
6.2.1 Multistars and polyadic adjoints
6.2.2 Polyadic isometry and projection
6.2.3 Towards a polyadic analog of C*-algebras
References
CH007.pdf
Chapter 7 Medial deformation of n-ary algebras
7.1 Almost commutative graded algebra
7.1.1 Almost commutativity
7.1.2 Tower of higher level commutation brackets
7.2 Almost medial graded algebras
7.2.1 Medial binary magmas and quasigroups
Theorem 7.2.3. Toyoda theorem
7.2.2 Almost mediality
7.2.3 Tower of higher binary mediality brackets
7.3 Medial n-ary algebras
7.4 Almost medial n-ary graded algebras
7.4.1 Higher level mediality n2-ary brackets
7.5 Toyoda’s theorem for almost medial algebras
References
CH008.pdf
Chapter 8 Membership deformations and obscure n-ary algebras
8.1 Graded algebras and Shur factors
8.2 Membership function and obscure algebras
8.3 Membership deformation of commutativity
8.3.1 Deformation of commutative algebras
8.3.2 Deformation of almost-commutative algebras
8.3.3 Double almost-Lie algebras
8.4 Projective representations
8.4.1 Binary projective representations
8.4.2 n-Ary projective representations
8.5 n-ary double commutative algebras
8.5.1 n-ary almost-commutative algebras
8.5.2 Membership deformed n-ary algebras
8.6 Conclusions
References
CH009.pdf
Chapter 9 Polyadic Hopf algebras
9.1 Polyadic coalgebras
9.1.1 Polyadic comultiplication
Definition 9.1.14. Counit axiom
9.1.2 Homomorphisms of polyadic coalgebras
9.1.3 Tensor product of polyadic coalgebras
9.1.4 Polyadic coalgebras in the Sweedler notation
9.1.5 Polyadic group-like and primitive elements
9.1.6 Polyadic analog of duality
9.1.7 Polyadic convolution product
Example 9.1.44. HomkC,A
Example 9.1.45. HomkC,A⊗2, HomkC⊗2,A .
Example 9.1.46. HomkC⊗2,A⊗2.
9.2 Polyadic bialgebras
Example 9.2.2. von Neumann n-regular bialgebra
Definition 9.2.3. Unit axiom
Definition 9.2.4. Counit axiom
9.3 Polyadic Hopf algebras
9.4 Ternary examples
9.4.1 Ternary Sweedler Hopf algebra
9.4.2 Ternary quantum group examples
9.5 Polyadic almost co-commutativity and co-mediality
9.5.1 Quantum Yang–Baxter equation
9.5.2 n′-Ary braid equation
9.5.3 Polyadic almost co-commutativity
9.5.4 Equations for the n′-ary R-matrix
9.5.5 Polyadic triangularity
9.5.6 Almost co-medial polyadic bialgebras
9.5.7 Equations for the M-matrix
9.5.8 Medial analog of triangularity
References
CH010.pdf
Chapter 10 Solutions to higher braid equations
10.1 Yang–Baxter operators
10.1.1 Yang–Baxter maps and braid group
10.1.2 Constant matrix solutions to the Yang–Baxter equation
10.1.3 Partial identity and unitarity
10.1.4 Permutation and parameter-permutation 4-vertex Yang–Baxter maps
10.1.5 Group structure of 4-vertex and 8-vertex matrices
10.1.6 Star 8-vertex and circle 8-vertex Yang–Baxter maps
10.1.7 Triangle invertible 9- and 10-vertex solutions
10.2 Polyadic braid operators and higher braid equations
10.3 Solutions to the ternary braid equations
10.3.1 Constant matrix solutions
10.3.2 Permutation and parameter-permutation 8-vertex solutions
10.3.3 Group structure of the star and circle 8-vertex matrices
10.3.4 Group structure of the star and circle 16-vertex matrices
10.3.5 Pauli matrix presentation of the star and circle 16-vertex constant matrices
10.3.6 Invertible and non-invertible 16-vertex solutions to the ternary braid equations
10.3.7 Higher 2n-vertex constant solutions to n-ary braid equations
References
CH011.pdf
Chapter 11 Polyadic tensor categories
11.1 Binary tensor categories
11.2 Polyadic tensor categories
11.2.1 Polyadic semigroupal categories
11.2.2 n-Ary coherence
Conjecture 11.2.8. N-ary coherence
11.3 Polyadic units, unitors and quertors
11.3.1 Polyadic monoidal categories
11.3.2 Polyadic nonunital groupal categories
11.4 Braided tensor categories
11.4.1 Braided binary tensor categories
11.4.2 Braided polyadic tensor categories
Conjecture 11.4.10. Braided n-ary coherence
11.5 Medialed polyadic tensor categories
11.5.1 Medialed binary and ternary categories
11.6 Conclusions
References