Polyadic Algebraic Structures

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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