Polyadic Hopf algebras and quantum groups

This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, 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, and 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. As a possible application to quantum group theory, 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). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and $M$-matrix, for which the compatibility equations are found.

From another perspective, the concepts of polyadic vector space, polyadic algebras and polyadic tensor product over general polyadic fields were introduced in DUPLIJ [2019]. They differ from the standard definitions of n-ary algebras DE AZCARRAGA AND IZQUIERDO [2010], MICHOR AND VINOGRADOV [1996], GOZE AND RAUSCH DE TRAUBENBERG [2009] in considering an arbitrary arity shape for all operations, and not the algebra multiplication alone. This means that the arities of addition in the algebra, the multiplication and addition in the underlying field can all be different from binary and the number of places in the multiaction (polyadic module) can be more than one DUPLIJ [2018a]. The connection between arities is determined by their arity shapes DUPLIJ [2019] ("arity freedom principle"). Note that our approach is somewhat different from the operad approach (see, e.g., MARKL ET AL. [2002], LODAY AND VALLETTE [2012]).
Here we propose a similar and consequent polyadic generalization of Hopf algebras. First, we define polyadic coalgebras and study their homomorphisms and tensor products. In the construction of the polyadic convolution product and bialgebras we propose considering different arities for the algebra and coalgebra, which is a crucial difference from the binary case. Instead of the antipode, we introduce its polyadic version, the querantipode, by analogy with the querelement in n-ary groups DÖRNTE [1929]. We then consider polyadic analogs of braidings, almost co-commutativity and the R-matrix, together with the quasitriangularity equations. This description is not unique, as with the polyadic analog of the twist map, while the medial map is unique for all arities. Therefore, a new (unique) concept of deformation is proposed: almost co-mediality with the corresponding Mmatrix. The medial analogs of braidings and quasitriangularity are introduced, and the equations for M-matrix are obtained.

POLYADIC FIELDS AND VECTOR SPACES
Let k = k (m k ,n k ) = K | ν (m k ) k , µ (n k ) k be a polyadic or (m k , n k )-ary field with n k -ary multiplication µ (n k ) k : K n k → K and m k -ary addition ν (m k ) k : K m k → K which are (polyadically) associative and distributive, such that K | µ (n k ) and K | ν (m k ) are both commutative polyadic groups CROMBEZ [1972], LEESON AND BUTSON [1980]. This means that µ • τ m k , where τ n k ∈ S n k , τ m k ∈ S m k , and S n k , S m k are the symmetry permutation groups. A polyadic field k (m k ,n k ) is derived, if µ (n k ) k and ν (m k ) k are iterations of the corresponding binary operations: ordinary multiplication and addition. The polyadic fields considered in LEESON AND BUTSON [1980] were derived. The simplest example of a nonderived (2, 3)-ary field is k (2,3) = {iR}, and of a nonderived (3, 3)-ary field is k (3,3) = {ip/q}, where p, q ∈ Z odd (i 2 = −1, and the operations are in C). Polyadic analogs of prime Galois fields including nonderived ones were presented in DUPLIJ [2017].
Recall that a polyadic zero z in any X | ν (m) (with ν (m) being an addition-like operation) is defined (if it exists) by where z can be on any place, and x is any polyad of length m − 1 (as a sequence of elements) in X. A polyadic unit in any X | µ (n) (with µ being a multiplication-like operation) is an e ∈ X (if it exists) such that µ (n) e n−1 , x = x, ∀x ∈ X, (2.2) where x can be on any place, and the repeated entries in a polyad are denoted by a power n x, . . . , x ≡ x n . It follows from (2.2), that for n ≥ 3 the polyad e can play the role of a unit, and is called a neutral sequenceǓSAN [2003] µ (n) [ê, x] = x, ∀x ∈ X,ê ∈ X n−1 . (

2.3)
This is a crucial difference from the binary case, as the neutral sequenceê can (possibly) be nonunique. The nonderived polyadic fields obey unusual properties: they can have several (polyadic) units or no units at all (nonunital, as in k (2,3) and k (3,3) above), no (polyadic) zeros (zeroless, as k (3,3) above), or they can consist of units only (for some examples, see DUPLIJ AND WERNER [2015], DUPLIJ [2017]). This may lead, in general, to new features of the algebraic structures using the polyadic fields as the underlying fields (e.g. scalars for vector spaces, etc.) DUPLIJ [2019].
Moreover, polyadic invertibility is not connected with units, but is governed by the special element, analogous to an inverse, the so called querelementx, which for any X | µ (n) is defined by DÖRNTE [1929] µ (n) x n−1 ,x = x, ∀x ∈ X, (2.4) wherex can be on any place (instead of the binary inverse "xx −1 = e"). An element x ∈ X for which (2.4) has a solution underx is called querable or "polyadically invertible". If all elements in X are querable, and the operation µ (n) is polyadically associative, then X | µ (n) is a n-ary group. Polyadic associativity in X | µ (n) can be defined as a kind of invariance relationship wherex,ŷ,ẑ are polyads of the needed size in X, and µ (n) [ŷ] can be on any place, and we therefore will not use additional brackets. Using polyadic associativity (2.5) we introduce ℓ-iterated multiplication by ],x ∈ X ℓ(n−1)+1 , (2.6) where ℓ is "number of multiplications". Therefore, the admissible length of any n-ary word is not arbitrary, as in the binary n = 2 case, but fixed ("quantized") to ℓ (n − 1) + 1.
The polyadic analogs of vector spaces and tensor products were introduced in DUPLIJ [2019]. Briefly, consider a set V of "polyadic vectors" with the addition-like m v -ary operation ν is a commutative m v -ary group. The key differences from the binary case are: 1) The zero vector z v does not necessarily exist (see the above example for k (3,3) field); 2) The role of a negative vector is played by the additive querelementṽ in V | ν (2.7) If the unit e k exists in k (m k ,n k ) , then the multiaction can be normalized (analog of "1v = v") by (2.8) Under the composition • nρ (given by the arity changing formula DUPLIJ [2018a]), the set of multiactions form a n ρ -ary semigroup S (nρ) ρ = ρ (rv) V | • nρ . Its arity is less or equal than n k and depends on one integer parameter (the number of intact elements in the composition), which is less than (r v − 1) (for details see DUPLIJ [2019]).
A polyadic vector space over the polyadic field k (m k ,n k ) is is a polyadic field, ρ (rv) V | • nρ is a n ρ -ary semigroup, the multiaction ρ (rρ) is distributive with respect to the polyadic additions ν ). If instead of the underlying field, we consider a ring, then (2.9) define a polyadic module together with (2.7). The dimension d v of a polyadic vector space is the number of elements in its polyadic basis, and we denote it V dv = V (mv ;m k ,n k ;rv) dv . The polyadic direct sum and polyadic tensor product of polyadic vector spaces were constructed in DUPLIJ [2019] (see (3.25) and (3.39) there). They have an unusual peculiarity (which is not possible in the binary case): the polyadic vector spaces of different arities can be added and multiplied. The polyadic tensor product is "k-linear" in the usual sense, only instead of "multiplication by scalar" one uses the multiaction ρ (rv) V (see DUPLIJ [2019] for details). Because of associativity, we will use the binary-like notation for polyadic tensor products (implying ⊗ = ⊗ k ) and powers of them (for instance, n x ⊗ x ⊗ . . . ⊗ x = x ⊗n ) to be clearer in computations and as customary in diagrams.

POLYADIC ASSOCIATIVE ALGEBRAS
Here we introduce operations on elements of a polyadic vector space, which leads to the notion of a polyadic algebra.
3.1. "Elementwise" description. Here we formulate the polyadic algebras in terms of sets and operations written in a manifest form. The arities will be initially taken as arbitrary, but then relations between them will follow from compatibility conditions (as in DUPLIJ [2019]).
Definition 3.1. A polyadic (associative) algebra (or k-algebra) is a tuple consisting of 2 sets and 5 operations (3.1) where: is a polyadic field with the m k -ary field (scalar) addition ν (m k ) k : K m k → K and n k -ary field (scalar) multiplication µ is a polyadic vector space with the m a -ary vector addition ν (ma) A : A na → A and the r a -place multiaction ρ where the second product µ can be on any place in brackets andâ,b,ĉ are polyads; 4) The multiacton ρ (ra) A is compatible with vector and field operations ν Definition 3.2. We call the tuple (m a , n a ; m k , n k ; r a ) an arity shape of the polyadic algebra A.
Proposition 3.6. If the multiaction ρ (ra) A is an ordinary action K × A → A, then all ℓ-arities are minimal ℓ = ℓ ′ = ℓ ′′ = 1, and the arity shape of A is determined by two integers (m, n), such that the arities of the algebra and underlying field are equal, and the arity n ρ of the action semigroup S ρ is equal to the arity of multiplication in the underlying field (3.20) As it was shown in DUPLIJ [2017], there exist zeroless and nonunital polyadic fields and rings. Therefore, the main difference with the binary algebras is the possible absence of a zero and/or unit in the polyadic field k (m k ,n k ) and/or in the polyadic ring (3.21) and so the additional axioms are needed iff such elements exist. This was the reason we have started from Definition 3.1, where no existence of zeroes and units in k (m k ,n k ) and A ring is implied. If they exist, denote possible units and zeroes by e k ∈ k (m k ,n k ) , z k ∈ k (m k ,n k ) and e A ∈ A (ma,na) , z A ∈ A (ma,na) . In this way we have 4 choices for each k (m k ,n k ) and A (ma,na) , and these 16 possible kinds of polyadic algebras are presented in TABLE 1. The most exotic case is at the bottom right, where both k (m k ,n k ) and A (ma,na) are zeroless nonunital, which cannot exist in either binary algebras or n-ary algebras DE AZCARRAGA AND IZQUIERDO [2010].  The standard case is that in the upper left corner, when both k (m k ,n k ) and A (ma,na) have a zero and unit.

3.2.
Polyadic analog of the functions on group. In the search for a polyadic version of the algebra of k-valued functions (which is isomorphic and dual to the corresponding group algebra) we can not only have more complicated arity shapes than in the binary case, but also the exotic possibility that the arities of the field and group are different as can be possible for multiplace functions.
Let us consider a n g -ary group G = G (ng) = G | µ (ng) g , which does not necessarily contain the identity e g , and where each element is querable (see (2.4)). Now we introduce the set A f of multiplace (s-place) functions f i (g 1 , . . . , g s ) (of finite support) which take value in the polyadic field k (m k ,n k ) such that f i : G s → K. To endow A f with the structure of a polyadic associative algebra (3.1), we should consistently define the m k -ary addition ν ). Thus we write for the algebra of k-valued functions The simplest operation here is the addition of the k-valued functions which, obviously, coincides with the field addition ν Construction 3.8. Because all arguments of the multiacton ρ (rf ) f are in the field, the only possibility for the r.h.s. is its multiplication (similar to the regular representation) (3.25) and in addition we have the arity shape relation (3.26) which is satisfied "automatically" in the binary case.
The polyadic analog of k-valued function convolution (" , where ℓ ν is the "number of additions", can be constructed according to the arity rules from DUPLIJ [2018a, 2019]. Definition 3.9. The polyadic convolution of s-place k-valued functions is defined as the admissible polyadic sum of ℓ ν (m k − 1) + 1 products where ℓ id is the number of intact elements in the determining equations ("h 1 h 2 = g") under the field sum ν k . The arity shape is determined by (3.28) which gives the connection between the field and the group arities.
Definition 3.15. A polyadic algebra A (n) is called totally commutative, if where τ n ∈ S n , and S n is the symmetry permutation group on n elements.
Remark 3.16. Initially, there are no other axioms in the definition of a polyadic algebra, because polyadic fields and vector spaces do not necessarily contain zeroes and units (see TABLE 1).
A special kind of polyadic algebra can appear, when the multiplication is "iterated" from lower arity ones, which is one of 3 kinds of arity changing for polyadic systems DUPLIJ [2018a].
The normalization of the multiaction (2.8) gives the corresponding normalization of the map η (r,n) (instead of "η (e k ) = e a ") (3.41) Assertion 3.20. In the "elementwise" description (see Subsection 3.1) the polyadic unit η (r,n) of A (n) is a (n − 1)-valued function of r arguments.
Introduce a "derived" version of the polyadic unit by analogy with the neutral sequence (2.3).
Definition 3.22. The k-linear derived polyadic unit (neutral unit sequence) of n-ary algebra A (n) is the setη (r) = η where id A can be on any place. If η (3.44) The normalization of the maps η (r) i is given by . . ⊗ e k = e a , i = 1, . . . , n − 1, e a ∈ A, e k ∈ K, (3.45) and in the "elementwise" description η (r) i is a function of r arguments, satisfying η (r) where ρ (r) is the multiaction (2.7).
Definition 3.23. A polyadic associative algebra A 0 (formally, because id A in (3.43) can be on any place). The particular case n = 3 and r = 1 was considered in DUPLIJ [2001, 2018b] (with examples). Invertibility in a polyadic algebra is not connected with the unit or zero (as in n-ary groups DÖRNTE [1929]), but is determined by the querelement (2.4). Introduce the corresponding mappings for the subsets of the additively querable elements A   (3.48) and the multiplicative quermap q mult : A (mult) quer is defined by where D (n) a : A → A ⊗n is the diagonal map given by a → n a ⊗ . . . ⊗ a, while q add and q mult can be on any place. They send an element to the additive querelement a q add →ã, a ∈ A (add) Example 3.25. For the polyadic algebra A (3,3;3,3;2) from Example 3.7 all elements are additively and multiplicatively querable, and so the sets of querable elements coincide A (add) The additive quermap q add and multiplicative quermap q mult act as follows (the operations are in C) (3.52) Example 3.26. The polyadic field k (m k ,n k ) is a polyadic algebra over itself. We identify A = K, µ , and the multiplication is defined by the multiaction as (3.53) Therefore, we have the additional arity conditions which are trivially satisfied in the binary case. Now the polyadic unit map η (r,n) (3.41) is the identity in each tensor component.

Medial map and polyadic permutations.
Recall that the binary medial map for the tensor product of algebras (as vector spaces) (3.56) It is obvious that 57) where τ op : A 1 ⊗ A 2 → A 1 ⊗ A 2 is the permutation of 2 elements (twist/flip) of the tensor product, such that a (1) ⊗ a (2) τop → a (2) ⊗ a (1) , a (1) ∈ A 1 , a (2) ∈ A 2 , τ op ∈ S 2 . This may be presented (3.56) in the matrix form where T is the ordinary matrix transposition. Let us apply (3.55) to arbitrary tensor products. By analogy, if we have a tensor product of mn elements (of any nature) grouped by n elements (e.g. m elements from n different vector spaces), as in (3.56), (3.58), we can write the tensor product in the Definition 3.27. The polyadic medial map τ (n,m) ⊗n is defined as the transposition of the tensor product matrix (3.59) by the evaluation (cf. the binary case (3.56)) We can extend the mediality concept EVANS [1963], BELOUSOV [1972] to polyadic algebras using the medial map. If we have an algebra with n-ary multiplication (3.31), then the mediality relation follows from (3.59) with m = n and contains (n + 1) multiplications acting on n 2 elements.
Definition 3.28. A k-linear polyadic algebra A (n) (3.32) is called medial, if its n-ary multiplication map satisfies the relation where τ (n,n) medial is given by (3.60), or in the manifest elementwise form (evaluation) Let us "polyadize" the binary twist map τ op from (3.57), which can be suitable for operations with polyadic tensor products. Informally, we can interpret (3.57), as "omitting the fixed points" of the binary medial map τ medial , and denote this procedure by "τ op = τ medial \ id".
where ℓ τ = mn − k f ixed , and k f ixed is the number of fixed points of the medial map τ (n,m) medial . Assertion 3.30. If m = n, then ℓ τ = mn − 2. If m = n, then the polyadic twist map τ Proof. This follows from the matrix form (3.59) and (3.60).
Therefore the number of places ℓ τ is "quantized" and for lowest m, n is presented in TABLE 2. This generalizes the binary twist in a more unique way, which gives polyadic commutativity.
Remark 3.31. The polyadic twist map τ op is one element of the symmetry permutation group S ℓτ which is fixed by the medial map τ (n,m) medial and the special condition (3.63), and it therefore respects polyadic tensor product operations.
Tensor product of polyadic algebras 3. POLYADIC ASSOCIATIVE ALGEBRAS Example 3.32. In the matrix representation we have (3.66) The introduction of the polyadic twist gives us the possibility to generalize (in a way consistent with the medial map) the notion of the opposite algebra.
Definition 3.33. For a polyadic algebra A (n) = A | µ (n) , an opposite polyadic algebra exists if the number of places for the polyadic twist map (which coincides in (3.67) with the arity of algebra multiplication ℓ τ = n) is allowed (see TABLE 2).
op is the medially allowed polyadic twist map.
3.5. Tensor product of polyadic algebras. Let us consider a polyadic tensor product n i=1 A (n) i of n polyadic associative n-ary algebras A with the structure of an algebra, we will use the medial map τ (n,m) medial (3.60).
Proposition 3.35. The tensor product of n associative n-ary algebras A (n) i has the structure of the polyadic algebra A (3.71) Proof. We act by the multiplication map µ ⊗ on the element's tensor product matrix (3.59) and obtain which proves that µ ⊗ is indeed a polyadic algebra multiplication. To prove the associativity (3.70) we repeat the same derivation (3.72) twice and show that the result is independent of i, j.
have their polyadic unit map η (r,n) i defined by (3.39) and acting as (3.41), then we have ⊗ is a (n 2 − n)-valued function of nr arguments. Note that concepts of tensor product and derived polyadic algebras are different.
3.6. Heteromorphisms of polyadic associative algebras. The standard homomorphism between binary associative algebras is defined as a linear map ϕ which "commutes" with the algebra multiplications ("ϕ • µ 1 = µ 2 • (ϕ ⊗ ϕ)"). In the polyadic case there exists the possibility to change arity of the algebras, and for that, one needs to use the heteromorphisms (or multiplace maps) introduced in DUPLIJ [2018a]. Let us consider two polyadic k-algebras A (over the same polyadic field k).
Definition 3.38. A heteromorphism between two polyadic k-algebras A (n 1 ) 1 and A (n 2 ) 2 (of different arities n 1 and n 2 ) is a s-place k-linear map Φ (n 1 ,n 2 ) s (3.74) and the diagram commutes. The arities satisfy Assertion 3.39. If ℓ id = 0 (there are no "intact elements"), then the (s-place) heteromorphism does not change the arity of the polyadic algebra. commutes.
The above definitions do not include the behavior of the polyadic unit under heteromorphism, because a polyadic associative algebra need not contain a unit. However, if both units exist, this will lead to strong arity restrictions. commutes.

2) The number of "intact elements" is fixed by its maximum value
such that in the l.h.s. of (3.74) there is only one multiplication µ (n 1 ) 1 .
3.7. Structure constants. Let A (n) be a finite-dimensional polyadic algebra (3.1) having the basis e i ∈ A, i = 1, . . . , N, where N is its dimension as a polyadic vector space (see (2.9) and (3.2), here N = d v ). In the binary case "a = i λ (i) e i ", any element a ∈ A is determined by the number N λ of scalars λ ∈ K, which coincides with the algebra dimension N λ = N, because r = 1. In the polyadic case, it can be that r > 1, and moreover with m ≥ 2 the admissible number of "words" (in the expansion of a by e i ) is "quantized", such Definition 3.42. In N-dimensional n-ary algebra A (n) (with m-ary addition and r-place "scalar" multiplication) the expansion of any element a ∈ A by the basis and is determined by N λ ∈ N "scalars", where (3.86) In the binary case m = 2, the dimension N of an algebra is not restricted and is a natural number, because, N = ℓ N + 1.
Proof. It follows from (3.86) and demanding that the "number of additions" ℓ N is natural or zero.
In a similar way, by considering a product of the basis elements, which can also be expanded in the basis "e i e j = k χ (k) (i,j) e k ", we can define a polyadic analog of the structure constants χ Definition 3.44. The polyadic structure constants χ (j) r,(i 1 ,...in) ∈ K, i 1 , . . . i n , j = 1, . . . , n of the Ndimensional n-ary algebra A (n) (with m-ary addition ν (m) and r-place multiaction ρ (r) ) are defined by the expansion of the n-ary product of the basis elements {e i | i = 1, . . . , N} as where (3.92) As in the binary case, we have Corollary 3.45. The algebra multiplication µ (n) of A (n) is fully determined by the rN n+1 polyadic structure constants χ Contrary to the binary case m = 2, when N χ can be any natural number, we now have Assertion 3.46. The number of the polyadic structure constants N χ of the finite-dimensional nary algebra A (n) with m-ary addition and r-place multiaction is not arbitrary, but "quantized" according to (3.93) Proof. This follows from (3.91) and "quantization" of the algebra dimension N, see Assertion 3.43.

POLYADIC COALGEBRAS
4.1. Motivation. The standard motivation for introducing the comultiplication is from representation theory CURTIS AND REINER [1962], KIRILLOV [1976]. The first examples come from socalled addition formulas for special functions ("anciently" started from sin/cos), which actually arise from representations of groups SHNIDER AND STERNBERG [1993], HAZEWINKEL ET AL. [2010].
In brief (and informally), let π be a finite-dimensional representation of a group G in a vector space V over a field k, such that (4.1) In some basis of V the matrix elements π ij (g) satisfy π ij (gh) = k π ik (g) π kj (h) (from (4.1)) and span a finite dimensional vector space C π of functions with a basis e πm as f π = m α m e πm , f π ∈ C π . Now (4.1) gives f π (gh) = m,n β mn e πm (g) e πn (h), f π ∈ C π . If we omit the evaluation, it can be written in the vector space C π using an additional linear map ∆ π : C π → C π ⊗ C π , in the following way (4.2) Thus, to any finite-dimensional representation π one can define the map ∆ π of vector spaces C π to functions on a group, called a comultiplication.
It is important that all the above operations are binary, and the defining formula for comultiplication (4.2) is fully determined by the definition of a representation (4.1).
The polyadic analog of a representation was introduced and studied in DUPLIJ [2018a]. In the case of multiplace representations, arities of the initial group and its representation can be different.
is a n ′ -ary group of endomorphisms of a polyadic vector space V (2.9).
was considered as a derived one, while here we do not restrict it in this way.
satisfying the associativity preserving heteromorphism equation DUPLIJ [2018a] commutes, and the arity changing formula

Polyadic comultiplication.
Our motivations say that in constructing a polyadic analog of the comultiplication, one should not only "reverse arrows", but also pay thorough attention to arities. Proof. It follows from (4.1), (4.2) and (4.3).
Let us consider a polyadic vector space over the polyadic field k (m k ,n k ) as (see (3.2)) Definition 4.5. A polyadic (coassociative) coalgebra (or k-coalgebra) is the polyadic vector space C vect equipped with the polyadic comultiplication and such that the diagram where τ n ′ ∈ S n ′ , and S n ′ is the permutation symmetry group on n ′ elements.
There are no other axioms in the definition of a polyadic coalgebra, following the same reasoning as for a polyadic algebra: the possible absence of zeroes and units (see Remark 3.16 and TABLE 1). Obviously, in a polyadic coalgebra C (n ′ ) , there is no "unit element", because there is no multiplication, and a polyadic analog of counit can be only defined, when the underlying field k (m k ,n k ) is unital (which is not always the case DUPLIJ [2017]).
By analogy with (2.6), introduce the ℓ ′ -coiterated n ′ -ary comultiplication by (4.13) Therefore, the admissible length of any co-word is fixed ("quantized") as ℓ ′ (n ′ − 1) + 1, but not arbitrary, as in the binary case. Let us introduce a co-analog of the derived n-ary multiplication (3.36) by where and ℓ d ≥ 2 is the "number of coiterations".
Definition 4.11. A polyadic coalgebra C (n ′ ) (4.8) is called co-medial, if its n ′ -ary multiplication map satisfies the relation medial is the polyadic medial map given by (3.59)-(3.60). Introduce a k-linear r ′ -place action mapρ (r ′ ) : Let k (m k ,n k ) be unital with unit e k .
We can provide the definition of counit only in the case where the underlying field k has a unit.
Remark 4.16. We cannot write the "elementwise" normalization action for the counit analogous to (3.41) (and state the Assertion 3.20), because a unit element in a (polyadic) coalgebra is not defined.
By analogy with the derived polyadic unit (see (3.43) and Definition 3.22), consider a "derived" version of the polyadic counit.
Definition 4.17. The k-linear derived polyadic counit (neutral counit sequence) of the polyadic coalgebra C (n ′ ) is the setε (r) = ε where id C can be on any place. If ε 0 , we call it the strong derived polyadic counit. In general, we can define formally, cf. (3.44), (4.25) , if (4.14) and (formally, because id C in (4.24) can be on any place).
Only when the underlying field k is unital, we can also define a morphism for counits.

4.4.
Tensor product of polyadic coalgebras. Let us consider n ′ polyadic equiary coalgebras C Proposition 4.21. The tensor product of the coalgebras has a structure of the polyadic coassociative coalgebra C (n ′ ) medial is defined in (3.60) and ∆ (n ′ ) The proof is in full analogy with that of Proposition 3.35. If all of the coalgebras C Proposition 4.22. The tensor product coalgebra C (n ′ ) ⊗ has a counit which is defined by c i ∈ C i , i = 1, . . . , n ′ (n ′ − 1) , and the arity of the comultiplication coincides with the arity of the underlying field n ′ = n k .
(4.33) 4.5. Polyadic coalgebras in the Sweedler notation. The k-linear coalgebra comultiplication map ∆ (n ′ ) defined in Definition 4.4 is useful for a "diagrammatic" description of polyadic coalgebras, and it corresponds to the algebra multiplication map µ (n) , which both manipulate with sets. However, for concrete computations (with elements) we need an analog of the polyadic algebra multiplication µ (n) ≡ µ (na) A from (3.3). The connection of µ (n) and µ (n) is given by (3.31), which can be treated as a "bridge" between the "diagrammatic" and "elementwise" descriptions. The co-analog of (3.31) was not considered, because the comultiplication has only one argument. To be consistent, we introduce the "elementwise" comultiplication ∆ (n ′ ) as the coanalog of µ (n) by the evaluation (4.34) In general, one does not distinguish ∆ (n ′ ) and ∆ (n ′ ) and may use one symbol in both descriptions.
In real "elementwise" coalgebra computations with many variables and comultiplications acting on them, the indices and various letters reproduce themselves in such a way that it is impossible to observe the structure of the expressions. Therefore, instead of different letters in the binary decomposition ("∆ (c) = i a i ⊗ b i " and (4.2)) it was proposed SWEEDLER [1968]  , c [2] only), because the real indices pullulate in complicated formulas enormously. In simple cases, the sum sign was also omitted ("∆ (c) = c [1] ⊗ c [2] "), which recalls the Einstein index summation rule in physics. This trick abbreviated tedious coalgebra computations and was called the (sumless) Sweedler (sigma) notation (sometimes it is called the Heyneman-Sweedler notation HEYNEMAN AND SWEEDLER [1969]). Now we can write ∆ (n ′ ) as a n ′ -ary decomposition in the manifest "elementwise" form where ℓ ∆ ∈ N 0 is a "number of additions", and N ∆ ∈ N is the "number of summands". In the binary case, the number of summands in the decomposition is not "algebraically" restricted, because N ∆ = ℓ ∆ + 1. In the polyadic case, we have Assertion 4.23. The admissible "number of summands" N ∆ in the polyadic comultiplication is (4.36) Therefore, the "quantization" of N ∆ coincides with that of the N-dimensional polyadic algebra (see Assertion 3.43).
Introduce the polyadic Sweedler notation by exchanging in (4.35) the real m-ary addition ν (m) by the formal addition ν [c] and writing (4.37) Remember here that we can formally add only N ∆ summands, because of the "quantization" (4.36) rule.
The polyadic Sweedler notation power can be seen in the following -25 -

POLYADIC COALGEBRAS
Polyadic group-like and primitive elements Example 4.24. We apply (4.37) to the coassociativity (4.9) with n ′ = 3, to obtain (4.39) After dropping the brackets and applying the Sweedler trick for the second time, we get the same formal expression in all three cases (4.40) Unfortunately, in the polyadic case the Sweedler notation looses too much information to be useful.
Assertion 4.25. The polyadic Sweedler notation can be applied to only the derived polyadic coalgebras (see Definition 4.18).
Nevertheless, if in an expression there are no coiterations, one can formally use it (e.g., in the polyadic analog (4.22) of the counting axiom " ε c [1] c [2] = c").

4.6.
Polyadic group-like and primitive elements. Let us consider some special kinds of elements in a polyadic coalgebra C (n ′ ) . We should take into account that in the polyadic case, as in (4.35), there can only be the admissible "number of summands" N ∆ (4.36).
In this case, we call x a polyadic primitive element.
The most important difference with the binary case is the "intermediate" possibility k p < n ′ − 1, when the r.h.s. is "nonlinear" in x.
Example 4.30. In the case where n ′ = 3 and k p = 1, we have m = 3, and ℓ ∆ = 1 (4.49) (4.50) Now ternary coassociativity cannot be achieved with any values of g i . This is true for any arity n ′ and any "nonlinear" comultiplication. Therefore, we arrive at the general structure Assertion 4.31. In a polyadic coassociative coalgebra C (n ′ ) polyadic primitive elements exist, if and only if the n ′ -ary comultiplication ∆ (n ′ ) is derived (4.14) from the binary comultiplication ∆ (2) .

4.7.
Polyadic analog of duality. The connection between binary associative algebras and coassociative coalgebras (formally named as "reversing arrows") is given in terms of the dual vector space (dual module) concept. Informally, for a binary coalgebra C (2) = C | ∆, ε considered as a vector space over a binary field k (a k-vector space), its dual is C * 2 = Hom k (C, k) with the natural pairing C * × C → k given by f (c), f ∈ C * , c ∈ C. The canonical injection θ : which is an isomorphism in the finite-dimensional case. The transpose of ∆ : C → C ⊗ C is a k-linear map ∆ * : (C ⊗ C) * → C * acting as ∆ * (ξ) (c) = ξ • (∆ (c)), where ξ ∈ (C ⊗ C) * , c ∈ C. The multiplication µ * on the set C * is the map C * ⊗ C * → C * , and therefore we have to use the canonical injection θ as follows (4.53) The associativity of µ * follows from the coassociativity of ∆. Since k * ≃ k, the dual of the counit is the unit η * : k ε * → C * . Therefore, C (2) * = C * | µ * , η * is a binary associative algebra which is called the dual algebra of the binary coalgebra C (2) = C | ∆, ε (see, e.g. RADFORD [2012]).
In the polyadic case, arities of the comultiplication, its dual multiplication and the underlying field can be different, but connected by (4.51). Let us consider a polyadic coassociative coalgebra C (n ′ ) with n ′ -ary comultiplication ∆ (n ′ ) (4.34) over k (m k ,n k ) . In search of the most general polyadic analog of the injection (4.51), we arrive at the possibility of multiplace morphisms.
Definition 4.32. For the polyadic coalgebra C (n ′ ) considered as a polyadic vector space over While constructing a polyadic analog of (4.51), recall that for any n ′ -ary operation the admissible length of a co-word is ℓ ′ (n ′ − 1) + 1, where ℓ ′ is the number of the iterated operation (4.13).
Remark 4.36. If n * = n ′ and s ≥ 2, the word "duality" can only be used conditionally. 4.8. Polyadic convolution product. If A (2) = A | µ, η is a binary algebra and C (2) = C | ∆, ε is a binary coalgebra over a binary field k, then a more general set of k-linear maps Hom k (C, A) can be considered, while its particular case where A (2) = k corresponds to the above duality. The multiplication on Hom k (C, A) is the convolution product (⋆) which can be uniquely constructed in the natural way: by applying first comultiplication ∆ and then multiplication µ ≡ (·) to an element The associativity of the convolution product follows from the associativity of µ and coassociativity of ∆, and the role of the identity (neutral element) in Hom k (C, A) is played by the composition of the unit map η : k → A and the counit map ε : C → k, such that e ⋆ = η • ε ∈ Hom k (C, A), because e ⋆ ⋆ f = f ⋆ e ⋆ = f . Indeed, from the obvious relation id A •f • id C = f and the unit and counit axioms it follows that The polyadic analog of duality and (4.59) offer an idea of how to generalize the binary convolution product to the most exotic case, when the algebra and coalgebra have different arities n = n ′ .
Let A (n) and C (n ′ ) be, respectively, a polyadic associative algebra and a coassociative coalgebra over the same polyadic field k (m k ,n k ) . If they are both unital and counital respectively, then we can consider a polyadic analog of the composition η • ε. The crucial difference from the binary case is that now η (r,n) and ε (n ′ ,r ′ ) are multiplace multivalued maps (3.39) and (4.22). Their composition is where the multiplace multivalued map γ (r ′ ,r) ∈ Hom k K ⊗r ′ , K ⊗r is, obviously, (≃), and the diagram commutes.
The formula (4.61) leads us to propose Conjecture 4.37. A polyadic analog of the convolution should be considered for multiplace multivalued k-linear maps in Hom k C ⊗(n ′ −1) , A ⊗(n−1) .
Definition 4.39. Let A (n) and C (n ′ ) be a n-ary associative algebra and n ′ -ary coassociative coalgebra over a polyadic field k (the existence of the unit and counit here is mandatory), then the set and its arity is given by the following n ⋆ -consistency condition Remark 4.44. If the polyadic tensor product and the underlying polyadic field k are derived (see discussion in SECTION 2 and DUPLIJ [2019]), while all maps coincide f (i) = f, the convolution product (4.68) is called the Sweedler power of f KASHINA ET AL. [2012] or the Adams operator AGUIAR AND LAUVE [2015]. In the binary case they denoted it by (f) n⋆ , but for the n ⋆ -ary product this is the first polyadic power of f (see (2.6)).
Obviously, some interesting algebraic objects are nonderived, and here they are determined by n + n ′ ≥ 5, and also the arities of the algebra and coalgebra can be different n = n ′ , which is a more exotic and exciting possibility. Generally, the arity n ⋆ of the convolution product (4.64) is not arbitrary and is "quantized" by solving (4.65) in integers. The values n ⋆ for minimal arities n, n ′ are presented in TABLE 3.   TABLE 3. Arity values n ⋆ of the polyadic convolution product (4.64), allowed by (4.65). The framed box corresponds to the binary convolution product.
The most unusual possibility is the existence of nondiagonal entries, which correspond to unequal arities of multiplication and comultiplication n = n ′ . The table is symmetric, which means that the arity n ⋆ is invariant under the exchange (n, ℓ) ←→ (n ′ , ℓ ′ ) following from (4.65).
Recall that the associativity of the binary convolution product (⋆) is transparent in the Sweedler Lemma 4.48. The polyadic convolution algebra C (n ′ ,n) ⋆ (4.66) is associative.
Proof. To prove the claimed associativity of polyadic convolution µ (n⋆) ⋆ we express (2.5) in Sweedler notation. Starting from where g is given by (4.73), it follows that h should not depend of place of g in (4.74). Applying h to c ∈ C twice, we obtain for its Sweedler components h [j] , j ∈ 1, . . . , n − 1, (4.75) and here coassociativity and (4.65) gives Since n-ary algebra multiplication µ (n) is associative, the internal µ (n) •ℓ in (4.75) can be on any place, and g in (4.74) can be on any place as well. This means that the polyadic convolution product µ (n⋆) ⋆ is associative.

POLYADIC BIALGEBRAS
The next step is to combine algebras and coalgebras into a common algebraic structure in some "natural" way. Informally, a bialgebra is defined as a vector space which is "simultaneously" an algebra and a coalgebra with some compatibility conditions (e.g., SWEEDLER [1969], ABE [1980]).
In search of a polyadic analog of bialgebras, we observe two structural differences with the binary case: 1) since the unit and counit do not necessarily exist, we obtain 4 different kinds of bialgebras (similar to the unit and zero in TABLE 1); 2) where the most exotic is the possibility of unequal arities of multiplication and comultiplication n = n ′ (see Assertion 4.35). Initially, we take them as arbitrary and then try to find restrictions arising from some "natural" relations.
Let B vect be a polyadic vector space over the polyadic field k (m k ,n k ) as (see (3.2) and (4.7)) where ν (m) : B ×m → B is m-ary addition and ρ (r) : K ×r × B → B is r-place action (see (2.7)).
The equivalence of the compatibility conditions 1b) and 2b) can be expressed in the form (polyadic analog of " medial is the medial map (3.60) acting on B, while the diagram commutes.
Example 5.2 (von Neumann n-regular bialgebra). Let B (n,n) = B | µ (n) , ∆ (n) be a polyadic bialgebra generated by the elements b i ∈ B, i = 1, . . . , n − 1 subject to the nonderived n-ary multiplication (5.11) It is straightforward to check that the compatibility condition (5.4) holds. Many possibilities exist for choosing other operations-algebra addition, field addition and multiplication, action-so to demonstrate the compatibility we have confined ourselves to only the algebra multiplication and comultiplication.
If the n-ary algebra B

(n)
A has unit and/or n ′ -ary coalgebra B (n ′ ) C has counit ε (n ′ ,r ′ ) , we should add the following additional axioms. commutes.
Definition 5.4 (Counit axiom). If B such that the diagram commutes.
If both the polyadic unit and polyadic counit exist, then we include their compatibility condition commutes.
Definition 5.6. A polyadic bialgebra B (n ′ ,n) is called totally co-commutative, if where τ n ∈ S n , τ n ′ ∈ S n ′ , and S n , S n ′ are the symmetry permutation groups on n and n ′ elements respectively.

POLYADIC HOPF ALGEBRAS
Here we introduce the most general approach to "polyadization" of the Hopf algebra concept ABE [1980], SWEEDLER [1969], RADFORD [2012]. Informally, the transition from bialgebra to Hopf algebra is, in some sense, "dualizing" the passage from semigroup (containing noninvertible elements) to group (in which all elements are invertible). Schematically, if multiplication µ = (·) in a semigroup G is binary, the invertibility of all elements demands two extra and necessary set-ups: 1) An additional element (identity e ∈ G or the corresponding map from a one point set to group ǫ); 2) An additional map (inverse ι : G → G), such that g · ι (g) = e in diagrammatic form is µ • (id G ×ι) • D 2 = ǫ (D 2 : G → G × G is the diagonal map). When "dualizing", in a (binary) bialgebra B (with multiplication µ and comultiplication ∆) again two set-ups should be considered in order to get a (binary) Hopf algebra: 1) An analog of identity e ⋆ = ηε (where η : k → B is unit and ε : B → k is counit); 2) An analog of inverse S : B → B called the antipode, such that µ • (id B ⊗S) • ∆ = e ⋆ or in terms of the (binary) convolution product id B ⋆ S = e ⋆ . By multiplying both sides by S from the left and by id B from the right, we obtain weaker (von Neumann regularity) conditions S ⋆ id B ⋆ S = S, id B ⋆ S ⋆ id B = id B , which do not contain an identity e ⋆ and lead to the concept of weak Hopf algebras DUPLIJ AND LI [2001], LI AND DUPLIJ [2002], SZLACHÁNYI [1996].
The crucial peculiarity of the polyadic generalization is the possible absence of an identity or 1) in both cases. The role and necessity of the polyadic identity (2.2) is not so important: there polyadic groups without identity exist (see, e.g. GAL'MAK [2003], and the discussion after (2.3)). Invertibility is determined by the querelement (2.4) in n-ary group or the quermap (3.48) in polyadic algebra. So there are two ways forward: "dualize" the quermap (3.48) directly (as in the binary case) or use the most general version of the polyadic convolution product (4.64) and apply possible restrictions, if any. We will choose the second method, because the first one is a particular case of it. Thus, if the standard (binary) antipode is the convolution inverse (coinverse) to the identity in a bialgebra, then its polyadic counterpart should be a coquerelement (4.78) of some polyadic analog for the identity map in the polyadic bialgebra. We consider two possibilities to define a polyadic analog of identity: 1) Singular case. The comultiplication is binary n ′ = 2; 2) Symmetric case. The arities of multiplication and comultiplication need not be binary, but should coincide n = n ′ .
In the singular case a polyadic multivalued map in End k B, B ⊗(n−1) is a reminder of how an identity can be defined: its components are to be functions of one variable. That is, with more than one argument it is not possible to determine its value when these are unequal.
Definition 6.1. We take for a singular polyadic identity Id 0 the diagonal map Id 0 = D ∈ End k B, B ⊗(n−1) , such that b → b ⊗(n−1) , for any b ∈ B.
In Sweedler notation Definition 6.5. A polyadic bialgebra B (2,n) equipped with the reduced n ⋆ -ary convolution product µ (n⋆) ⋆ and the singular querantipode Q 0 (6.4) is called a singular polyadic Hopf algebra and is denoted by H (n) Due to their exotic properties we will not consider singular polyadic Hopf algebras H (n) sing in detail. In the symmetric case a polyadic identity-like map in End k B ⊗(n−1) , B ⊗(n−1) can be defined in a more natural way.
Definition 6.6. A symmetric polyadic identity Id : B ⊗(n−1) → B ⊗(n−1) is a polyadic tensor product of ordinary identities in B (n,n) The numbers of iterations are now equal ℓ = ℓ ′ , and the consistency condition (4.65) becomes Definition 6.7. The set of the multiplace multivalued maps f (i) ∈ End k B ⊗(n−1) , B ⊗(n−1) (together with the polyadic identity Id) endowed with the symmetric convolution productμ (6.9) For a polyadic analog of antipode in the symmetric case we have Definition 6.8. A multiplace multivalued map Q id : B ⊗(n−1) → B ⊗(n−1) in the polyadic bialgebra B (n,n) is called a symmetric querantipode, if it is the coquerelement (see (4.78)) of the polyadic identity Q id = q ⋆ (Id) in the symmetric n ⋆ -ary convolution algebrâ where Q id can be on any place, such that the diagram Id ⊗(n⋆−1) ⊗Q id ❄ (6.11) commutes.
Recall that the main property of the antipode S of a binary bialgebra B is its "anticommutation" with the multiplication µ and comultiplication ∆ (e.g., SWEEDLER [1969]) where τ op is the binary twist (see (3.57)). The first relation means that S is an algebra antiendomorphism, because in the elementwise description S (a · b) = S (b) · S (a), a, b ∈ B, (·) ≡ µ. We propose the polyadic analogs of (6.14)-(6.15) without proofs, which are too cumbersome, but their derivations almost coincide with those for the binary case.
Proposition 6.14. If in a symmetric Hopf algebra H (n) sym either multiplication or comultiplication is invariant under polyadic twist map τ (ℓτ ) op (3.63), then the querantipode Q id (6.10) satisfieŝ (6.22) where Q •2 id can be on any place, or the convolution querelement (4.78) of the querantipode Q id is Proof. The proposition follows from applying either (6.16) or (6.17) to the l.h.s. of (6.22), to use (4.78).

TOWARDS POLYADIC QUANTUM GROUPS
Bialgebras with a special relaxation of co-commutativity, almost co-commutativity, are the ground objects in the construction of quantum groups identified with the non-commutative and non-cocommutative quasitriangular Hopf algebras DRINFELD [1987, 1989a]. 7.1. Quantum Yang-Baxter equation. Here we recall the binary case (informally) in a notation that will allow us to provide the "polyadization" in a clearer way.
Let us consider a (binary) bialgebra B (2,2) = B | µ, ∆ , where µ = µ (2) is the binary multiplication, ∆ = ∆ (2) (see Definition 5.1), and the opposite comultiplication is given by ∆ cop ≡ τ op • ∆, where τ op is the binary twist (3.57). To relax the co-commutativity condition (∆ cop = ∆), the following construction inspired by conjugation in groups was proposed DRINFELD [1987, 1989a]. A bialgebra B (2,2) is almost co-commutative, if there exists R ∈ B ⊗ B such that (in the elementwise notation) (7.1) A fixed element R of a bialgebra satisfying (7.1) is called a universal R-matrix. For a cocommutative bialgebra we have R = e B ⊗ e B , where e B ∈ B is the unit (element) of the algebra If we demand that B | ∆ cop is the opposite coalgebra of B | ∆ , and therefore ∆ cop be coassociative, then R cannot be arbitrary, but has to satisfy some additional conditions, which we will call the almost co-commutativity equations for the R-matrix. Indeed, using (7.1) we can write Therefore, the coassociativity of ∆ cop leads to the first almost co-commutativity equation On the other hand, directly from (7.1), we have relations which can be treated as the next two almost co-commutativity equations (unconnected to the coassociativity of ∆ cop ) The equations (7.4)-(7.6) for the components of In components the almost co-commutativity (7.1) can be expressed as follows α ≡ e B ⊗ R. (7.11) Obviously, one can try to solve (7.4)-(7.6) with respect to the r (1) α , r (2) α directly, but then we are confronted with a difficulty arising from the Sweedler components, because now (see (4.35)-(4.37)) (7.13) To avoid computations in the Sweedler components, one can substitute them by the components of R directly as r (i) [j] −→ r (i) (schematically). This allows us to express (7.12)-(7.13) solely through elements of the "extended" R-matrix R ij by (2) β , (7.14) which do not contain Sweedler components of R at all. The equations (7.14)-(7.15) define a quasitriangular R-matrix DRINFELD [1987]. The corresponding almost co-commutative (binary) bialgebra B (2) braid = B (2,2) , R is called a quasitriangular almost co-commutative bialgebra (or braided bialgebra KASSEL [1995]). Only for them can the almost co-commutativity equations (7.4)-(7.6) be n ′ -ary braid equation 7. TOWARDS POLYADIC QUANTUM GROUPS expressed solely in terms of R-matrix components or through the "extended" R-matrix R ij , using (7.14)-(7.15).
Theorem 7.1. In the binary case, three almost co-commutativity equations for the R-matrix coincide with (7.16) Conversely, any quasitriangular R-matrix is a solution of (7.16) by the above construction. The equation for the "extended" R-matrix R ij (7.16) is called the quantum Yang-Baxter equation RADFORD [1997], MAJID [1995] (or the triangle relation DRINFELD [1989a]). In terms of the R-matrix components (7.7) the quantum Yang-Baxter equation (7.16) takes the form β ′ .
Proof. Use the associative quiver technique from DUPLIJ [2018a] (The Post-like quiver in Section 6).
Remark 7.3. There can be additional equations depending on the concrete values of n ′ which can contain a different number of brackets determined by the corresponding diagram commutation.
Example 7.4. In case n ′ = 3 we have the ternary braided equation The ternary compatibility conditions for c V 1 V 2 V 3 (corresponding to (7.19)-(7.20)) are Now we follow the opposite (to the standard DRINFELD [1989b]), but consistent way: using the equations (7.24)-(7.26) we find polyadic analogs of the corresponding equations for the R-matrix and the quasitriangularity conditions (7.14)-(7.15), which will fix the comultiplication structure of a polyadic bialgebra B (n ′ ,n) . 7.3. Polyadic almost co-commutativity. We will see that the almost co-commutativity equations for the R-matrix are more complicated in the polyadic case, because the main condition (7.1) will have a different form coming from n-ary group theory GAL'MAK [2003]. Indeed, let G (n) = G | µ (n) be an n-ary group and H ′ = H ′ | µ (n) , H ′′ = H ′′ | µ (n) are its n-ary subgroups. Recall GAL' MAK [2003] that H ′ and H ′′ are semiconjugated in G (n) , if there exist g ∈ G, such that µ (n) g, h ′ 1 , . . . , h ′ n−1 = µ (n) h ′′ 1 , . . . , h ′′ n−1 , g , h ′ i ∈ H ′ , h ′′ i ∈ H ′′ , and if g can be on any place, then H ′ and H ′′ are conjugated in G (n) . Based on this notion and on analogy with (2.3), we can "polyadize" the almost co-commutativity condition (7.1) in the following way.
Remark 7.17. The twist of the modules τ V 1 ...V n ′ should be compatible with the polyadic twist τ (n ′ ) op in (7.27). In the binary case they are both the same flip 12 21 , but in the n ′ -ary case they can be different. Example 7.18. Consider the ternary braided equation (7.24), but now for the braiding c V 1 V 2 V 3 , instead of (7.37), where we have where ρ (2) : B ⊗ B ⊗ V → V is a 2-place action (2.7). In this way (7.40) is consistent with (7.31).
In each place of the 2-place action ρ (2) we then obtain the relation (7.38).
7.5. Polyadic triangularity. A polyadic analog of triangularity DRINFELD [1987] can be defined, if we rewrite (7.15) as where τ op is the binary twist. Instead of the R-matrix formulation (the left equality in (7.41)), we use the component approach by RADFORD [2012], and propose the following . . .
(7.45) Remark 7.20. As opposed to the binary case (7.14)-(7.15), the right hand sides here can be expressed in terms of the extended R-matrix in the first equation (7.42) and the last one (7.44) only, because in the intermediate equations the sequences of R-matrix elements are permuted. For instance, it is clear that the binary product α,β r (1) β ⊗ e B cannot be expressed in terms of the extended binary R-matrix (7.9). 7.6. Almost co-medial polyadic bialgebras. The previous considerations showed that cocommutativity and almost co-commutativity in the polyadic case are not unique and do not describe the bialgebras to the fullest extent. This happens because mediality is a more general and consequent property of polyadic algebraic structures, while commutativity can be treated as a particular case of it (see Subsection 3.4 and (3.57)). Therefore, we propose here to deform co-mediality (rather than co-commutativity as in DRINFELD [1987DRINFELD [ , 1989a).
Corollary 7.27. Polyadic almost co-commutativity is a particular case of polyadic co-mediality with the special "medial-like" twist map τ (n ′ ,n ′ ) R (7.53) and the composite M-matrix (7.54) consisting of n ′ copies of the R-matrix (7.30).
We observe that even in the binary case the medial braid equations are cumbersome and nontrivial.
Remark 7.38. Two other compatibility equations corresponding to the intermediate quasipolyangularity equations (7.76)-(7.77) can be written in component form only (see Remark 7.35).
The solutions to (7.81)-(7.82) can be found in matrix form by choosing an appropriate basis and using the standard methods (see, e.g., KASSEL [1995], LAMBE AND RADFORD [1997]).

CONCLUSIONS
We have presented the "polyadization" procedure of the following algebra-like structures: algebras, coalgebras, bialgebras and Hopf algebras (see DUPLIJ [2017DUPLIJ [ , 2019 for ring-like structures). In our concrete constructions the initial arities of operations are taken as arbitrary, and we then try to restrict them only by means of natural relations which bring to mind the binary case. This leads to many exotic properties and unexpected connections between arities and a fixing of their values called "quantization". For instance, the unit and counit (which do not always exist) can be multivalued many place maps, polyadic algebras can be zeroless, the qeurelements should be considered instead of inverse elements under addition and multiplication, a polyadic bialgebra can consist of an algebra and coalgebra of different arities, and a polyadic analog of Hopf algebras contains (instead of the ordinary antipode) the querantipode, which has different properties.
The formulas and constructions introduced for concrete algebra-like structures can have many applications, e.g., in combinatorics, quantum logic,or representation theory. As an example, we have introduced possible polyadic analogs of braidings, almost co-commutativity and a version of the R-matrix. A new concept of deformation (using the medial map) is proposed: this is unique and therefore can be more consequential and suitable in the polyadic case.