Polyadization of algebraic structures

A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in block diagonal matrix form (resulting in the Wedderburn decomposition), a general form of polyadic structures is given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures ("double"decomposition of two kinds). We then introduce the polyadization concept (a"polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The"deformation"by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.


INTRODUCTION
I am no poet, but if you think for yourselves, as I proceed, the facts will form a poem in your minds.
Here we first propose a generalization of this concept for polyadic algebraic structures DUPLIJ [2022a], which can also be important, e.g. in the operad theory MARKL ET AL. [2002], LODAY AND VALLETTE [2012] and nonassociative structures ZHEVLAKOV ET AL. [1982], SABININ ET AL. [2006]. If semisimple structures can be presented in the block-diagonal matrix form (resulting to the Wedderburn decomposition WEDDERBURN [1908], HERSTEIN [1996], LAM [1991]), a corresponding general form for polyadic rings can be decomposed to a kind of block-shift matrices NIKITIN [1984]. We combine these forms and introduce a general shape of semisimple polyadic structures, which are nonderived in the sense that they cannot be obtained as a successive composition of binary operations, which can be treated as a polyadic ("double") decomposition.
Second, going in the opposite direction, we define the polyadization concept ("polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. Then we briefly describe supersymmetric structure polyadization.
Third, we propose operations "deformed" by shifts to obtain a nonderived n-ary multiplication on the direct power of binary algebraic structures.
For these new constructions some illustrative concrete examples are given.

PRELIMINARIES
We use notation from DUPLIJ [2022a,b]. In brief, a (one-set) polyadic algebraic structure A is a set A closed with respect to polyadic operations (or n-ary multiplication) µ rns : A n Ñ A (n-ary magma). We denote polyads POST [1940] by bold letters a " a pnq " pa 1 , . . . , a n q, a i P A. A polyadic zero is defined by µ rns " a pn´1q , z ‰ " z, z P A, a pn´1q P A n´1 , where z can be on any place. A (positive) polyadic power ℓ µ P N is a xℓµy "`µ rns˘˝ℓ µ " a ℓµpn´1q`1 ‰ , a P A. An element of a polyadic algebraic structure a is called ℓ µ -nilpotent (or simply nilpotent for ℓ µ " 1), if there exist ℓ µ such that a xℓµy " z. A polyadic (or n-ary) identity (or neutral element) is defined by µ rns ra, e n´1 s " a, @a P A, where a can be on any place -2 -Simple polyadic structures PRELIMINARIES in the l.h.s. A one-set polyadic algebraic structure @ A | µ rns D is totally associative, if`µ rns˘˝2 ra, b, cs " µ rns " a, µ rns rbs , c ‰ " invariant, with respect to placement of the internal multiplication on any of the n places, and a, b, c are polyads of the necessary sizes DUPLIJ [2018,2019]. A polyadic semigroup S pnq is a one-set and one-operation structure in which µ rns is totally associative. A polyadic structure is commutative, if µ rns " µ rns˝σ , or µ rns ras " µ rns rσ˝as, a P A n , for all σ P S n .
A polyadic structure is solvable, if for all polyads b, c and an element x, one can (uniquely) resolve the equation (with respect to h) for µ rns rb, x, cs " a, where x can be on any place, and b, c are polyads of the needed lengths. A solvable polyadic structure is called a polyadic quasigroup BELOUSOV [1972]. An associative polyadic quasigroup is called a n-ary (or polyadic) group GAL'MAK [2003]. In an n-ary group the only solution of µ rns rb,ās " a, a,ā P A, b P A n´1 (2.1) is called a querelement of a and denoted byā DÖRNTE [1929], whereā can be on any place. Any idempotent a coincides with its querelementā " a. The relation (2.1) can be considered as a definition of the unary queroperationμ p1q ras "ā GLEICHGEWICHT AND GŁAZEK [1967]. For further details and definitions, see DUPLIJ [2022a].
3.1. Simple polyadic structures. According to the Wedderburn-Artin theorem (see, e.g., HERSTEIN [1996], LAM [1991], HAZEWINKEL AND GUBARENI [2016]), a ring which is simple (having no twosided ideals, except zero and the ring itself) and Artinian (having minimal right ideals) R simple is isomorphic to a full dˆd matrix ring R simple -Mat f ull dˆd pDq (3.1) over a division ring D. As a corollary, R simple -Hom D pV pd | Dq , V pd | Dqq " End D pV pd | Dqq , where V pd | Dq is a d-finite-dimensional vector space (left module) over D. In the same way, a finitedimensional simple associative algebra A over an algebraically closed field F is In the polyadic case, the structure of a simple Artinian r2, ns-ring R r2,ns simple (with binary addition and n-ary multiplication µ rns ) was obtained in NIKITIN [1984], where the Wedderburn-Artin theorem for r2, ns-rings was proved. So instead of one vector space V pd | Dq, one should consider a direct sum of pn´1q vector spaces (over the same division ring D), that is (3.5) This means that after choosing a suitable basis in terms of matrices (when the ring multiplication µ rns coincides with the product of n matrices) we have -3 - where ν r2s and µ rns are binary addition and ordinary product of n matrices, M shif t dˆd is the block-shift (traceless) matrix over D of the form (which follows from (3.5)) M shif tpnq pdˆdq "¨0 (3.7) where pn´1q blocks are nonsquare matrices B i pd 1ˆd2 q P Mat f ull d 1ˆd2 pDq over the division ring D, and Remark 3.2. The set of the fixed size blocks tB i pd 1ˆd2 qu does not form a binary ring, because d 1 ‰ d 2 .
Assertion 3.3. The block-shift matrices of the form (3.7) are closed with respect to n-ary multiplication and binary addition, and we call them n-ary matrices.
Taking distributivity into account we arrive at the polyadic ring structure (3.6).
Assertion 3.5. A finite-dimensional simple associative n-ary algebra A pnq over an algebraically closed field F CARLSSON [1980] is isomorphic to the block-shift n-ary matrix (3.7) over F A pnq -Mat shif tpnq dˆd pFq .
(3.9) 3.2. Semisimple polyadic structures. The Wedderburn-Artin theorem for semisimple Artinian rings R semispl states that R semispl is a finite direct product of k simple rings, each of which has the form (3.1).
where d " q p1q`qp2q`. . .`q pkq . Then, instead of (3.2) we have the isomorphism 1 (3.11) In a suitable basis the Wedderburn-Artin theorem follows Theorem 3.6. A semisimple Artinian (binary) ring R semispl is isomorphic to the dˆd matrix ring R semispl -Mat diagpkq q pjqˆqpjq pDq " M diagpkq pdˆdq | ν r2s , µ r2s ( , (3.12) 1 We enumerate simple components by an upper index in round brackets pkq, block-shift components by lower index without brackets, and the arity is an upper index in square brackets rns.
where ν r2s and µ r2s are binary addition and binary product of matrices, M diagpkq pdˆdq are blockdiagonal matrices of the form (which follows from (3.11)) where k square blocks are full matrix rings over division rings D pjq (3.14) The same matrix structure has a finite-dimensional semisimple associative algebra A over an algebraically closed field F (see (3.3)). For further details, see, e.g., HERSTEIN [1996], LAM [1991], HAZEWINKEL AND GUBARENI [2016].
Thus, our task is to decompose each of the V i pd i q, in (3.4) into components as in (3.10) In matrix language this means that each block B d 1ˆd2 from the polyadic ring (3.7) should have the semisimple decomposition (3.13), i.e. be a block-diagonal square matrix of the same size pˆp, where p " d 1 " d 2 " . . . " d n´1 and the total matrix size becomes d " pn´1q p. Moreover, all the blocks B's should have diagonal blocks A's of the same size, and therefore q pjq " q pjq 1 " q pjq 2 " . . . " q pjq n´1 for all j " 1, . . . , k and p " q p1q`qp2q`. . .`q pkq , where k is the number of semisimple components. In this way the cyclic direct sum of homomorphisms for the semisimple polyadic rings becomes (we use different division rings for each semisimple component as in (3.14)) R r2,ns After choosing a suitable basis we obtain a polyadic analog of the Wedderburn-Artin theorem for semisimple Artinian r2, ns-rings R r2,ns semispl , which can be called as the double decomposition (of the first kind or shift-diagonal ).
Theorem 3.7. The semisimple polyadic Artinian ring R r2,ns semispl (of the first kind) is isomorphic to the dˆd matrix ring where ν r2s , µ rns are binary addition and ordinary product of n matrices, N shif t-diagpn,kq dˆd (n is arity of N's and k is number of simple components of N's) are the block-shift n-ary matrices with block-diagonal square blocks (which follows from (3.16)) and the k square blocks A's are full matrix rings over the division rings D pjq Remark 3.8. By analogy with (3.8), in the limiting case n " 2, we have in (3.18) one block B pkq 1 ppˆpq only and (3.19) gives its standard (binary) semisimple ring decomposition.
This allows us to introduce another possible double decomposition in the opposite sequence to (3.18)-(3.19), we call it of second kind or reverse, or diagonal-shift. Indeed, in a suitable basis we first provide the standard block-diagonal decomposition (3.13), and then each block obeys the block-shift decomposition (3.7). Here we do not write the "reverse" analog of (3.16) and arrive directly to Theorem 3.9. The semisimple polyadic Artinian ringR r2,ns where ν r2s , µ rns are binary addition and ordinary product of n matrices,N diag-shif tpn,kq dˆd (n is arity ofN's and k is number of simple components ofN's) are the block-diagonal n-ary matrices with block-shift nonsquare blockŝ  .25)) is called polyadic ring of the first kind (resp. of the second kind ).
Proposition 3.11. The polyadic rings of the first and second kind are not isomorphic.
Thus the two double decompositions introduced above can lead to a new classification for polyadic analogs of semisimple rings.
Example 3.12. Let us consider the double decomposition of two kinds for ternary (n " 3) rings with two semisimple components (k " 2) and blocks as full qˆq matrix rings over C. Indeed, we have for the ternary nonderived rings R r2,3s semispl andR r2,3s semispl of the first and second kind, respectively, the following block structures In terms of component blocks, the ternary multiplications in the rings R r2,3s semispl andR r2,3s semispl are kind I: It follows from (3.30)-(3.31) and (3.32)-(3.33) that R r2,3s semispl andR r2,3s semispl are not ternary isomorphic. Note that the sum of the block structures (3.29) obeys nontrivial properties.
Remark 3.13. Consider a binary sum of the block matrices of the first and second kind (3.29) The set of matrices (3.34) forms the nonderived r2, 3s-ring P r2,3s over C P r2,3s " where ν r2s , µ r3s are binary addition and ordinary product of 3 matrices (3.34).
Notice that the P-matrices (3.34) are the block-matrix version of the circle matrices M circ which were studied in DUPLIJ AND VOGL [2021] in connection with 8-vertex solutions to the constant Yang-Baxter equation LAMBE AND RADFORD [1997] and the corresponding braiding quantum gates KAUFFMAN AND LOMONACO [2004], MELNIKOV ET AL. [2018].
3.2.1. Supersymmetric double decomposition. Let us generalize the above double decomposition (of the first kind) to superrings and superalgebras. For that we first assume that the constituent vector spaces (entering in (3.16)) are super vector spaces (Z 2 -graded vector spaces) obeying the standard decomposition into even and odd parts (3.36) where q pjq even and q pjq odd are dimensions of the even and odd spaces,respectively, q pjq " q pjq even`q pjq odd . The parity of a homogeneous element of the vector space see BEREZIN [1987], LEITES [1983]. In the graded case, the k square blocks A's in (3.22) are full supermatrix rings of the size´q pjq even | q pjq odd¯ˆ´q pjq even | q pjq odd¯, while the square B's (3.19) are block-diagonal supermatrices, and the block-shift n-ary supermatrices have a nonstandard form (3.18).
We assume that in super case a polyadic analog of the Wedderburn-Artin theorem for semisimple Artinian superrings (of the first kind) is also valid, with the form of the double decomposition (3.18)-(3.19) being the same, however now the blocks A's and B's are corresponding supermatrices.

POLYADIZATION CONCEPT
Here we propose a general procedure for how to construct new polyadic algebraic structures from binary (or lower arity) ones, using the "inverse" (informally) to the block-shift matrix decomposition (3.7). It can be considered as a polyadic analog of the inverse problem of the determination of an algebraic structure from the knowledge of its Wedderburn decomposition DIETZEL AND MITTAL [2021].
4.1. Polyadization of binary algebraic structures. Let a binary algebraic structure X be represented by pˆp matrices B y " B y ppˆpq over a ring R (a linear representation), where y is the set of N y parameters corresponding to an element x of X. Because the binary addition in R transfers to the matrix addition without restrictions (as opposed to the polyadic case, see below), we will consider only the multiplicative part of the resulting polyadic matrix ring. In this way, we propose a special block-shift matrix method to obtain n-ary semigroups (n-ary groups) from the binary ones, but the former are not derived from the latter GAL'MAK [2003], DUPLIJ [2022a]. In general, this can lead to new algebraic structures that were not known before.
Definition 4.1. A (block-matrix) polyadization Φ pol of a binary semigroup (or group) X represented by square pˆp matrices B y is an n-ary semigroup (or an n-ary group) represented by the dˆd block-shift matrices (over a ring R) of the form (3.7) as follows Q y 1 ,...,y n´1 " Q Bshif tpnq y 1 ,...,y n´1 pdˆdq "¨0 where d " pn´1q p, and the n-ary multiplication µ rrnss is given by the product of n matrices (4.1).
In terms of the block-matrices B's the multiplication Remark 4.2. The number of parameters N y describing an element x P X increases to pn´1q N y , and the corresponding algebraic structure @ Q y 1 ,...,y n´1 ( | µ rrnss D becomes n-ary, and so (4.1) can be treated as a new algebraic structure, which we denote by the same letter with the arities in double square brackets X rrnss .
We now analyze some of the most general properties of the polyadization map Φ pol which are independent of the concrete form of the block-matrices B's and over which algebraic structure (ring, field, etc...) they are defined. We then present some concrete examples.
Proof. In the case of (4.6) all pn´1q relations (4.3)-(4.5) coincide which means that the ordinary (binary) product of n matrices B y 's is mapped to the n-ary product of matrices Q y 's (4.2) as it should be for an n-ary-binary homomorphism, but not for a homomorphism.
Assertion 4.5. If matrices B y " B y ppˆpq contain the identity matrix E p , then the n-ary identity E pnq d in @ tQ y pdˆdqu | µ rrnss D , d " pn´1q p has the form (4.9) Proof. It follows from (4.1), (4.2) and (4.7).
Let us suppose that on the set of matrices tB y u over a binary ring R one can consider some analog of a multiplicative character χ : tB y u Ñ R, being a (binary) homomorphism, such that χ pB y 1 q χ pB y 2 q " χ pB y 1 B y 2 q . (4.13) For instance, in case B P GL pp, Cq, the determinant can be considered as a (binary) multiplicative character. Similarly, we can introduce Definition 4.7. A polyadized multiplicative character χ : Q y 1 ,...,y n´1 ( Ñ R is proportional to a product of the binary multiplicative characters of the blocks χ pB y i q χ`Q y 1 ,...,y n´1˘" p´1q n χ pB y 1 q χ pB y 2 q . . . χ`B y n´1˘. (4.14) The normalization factor p´1q n in (4.14) is needed to be consistent with the case when R is commutative, and the multiplicative characters are determinants. It can also be consistent in other cases.
Proposition 4.8. If the ring R is commutative, then the polyadized multiplicative character χ is an n-ary-binary homomorphism.

4.2.
Concrete examples of the polyadization procedure.
Remark 4.11. The above example shows, how "far" polyadic groups can be formed from ordinary (binary) groups: the former can contain infinite number of 4-ary idempotents determined by (4.33)-(4.36), in addition to the standard idempotent in any group, the 4-ary identity (4.29).

4.2.2.
Polyadization of SO p2, Rq. Here we provide a polyadization for the simplest subgroup of GL p2, Cq, the special orthogonal group SO p2, Rq. In the matrix form SO p2, Rq is represented by the one-parameter rotation matrix B pαq "ˆc os α´sin α sin α cos α˙P SO p2, Rq , α P R ä 2πZ, (4.37) satisfying the commutative multiplication (4.38) and the (binary) identity E 2 is B p0q. Therefore, the inverse element for B pαq is B p´αq.
The determinants of B pαq and Q pα, β, γq are 1, and therefore the corresponding multiplicative characters and polyadized multiplicative characters (4.14) are also equal to 1.
Thus, we conclude that just as the binary product of B-matrices corresponds to the ordinary angle addition (4.38), the 4-ary multiplication of polyadic rotation Q-matrices (4.39) corresponds to the 4-ary cyclic shift addition (4.51) through (4.52). 4.3. "Deformation" of binary operations by shifts. The concrete example from the previous subsection shows the strong connection (4.52) between the polyadization procedure and the shifted operations (4.51). Here we generalize it to an n-ary case for any semigroup.
(4.56) Now using shifts, instead of (4.56) we construct a nonderived n-ary operation on the direct power.
To obtain a nonderived n-ary operation, by analogy with (4.50), we deform by shifts the derived n-ary operation (4.56).
Definition 4.14. The shift deformation by (4.57) of the derived operation ν rns on the direct power A m is defined noncomponentwise by ν rns s ra 1 , a 2 , . . . , a n s " n ÿ i"1 s i´1 a i " a 1ˆs a 2ˆ. . .ˆs n´1 a n , (4.58) where a P A m (4.54) and s 0 " id.
Proof. The definition of polyadic identity in terms of the deformed n-ary product in the direct power is ν rns s « n´1 hkkkkikkkkj e, e, . . . , e, a ff " a, @a P A n´1 . (4.69) Using (4.58) we get the equation eˆseˆs 2 eˆ. . .ˆs n´2 eˆa " a. (4.70) After cancellation by a we obtain (4.68).
Thus, starting from a binary semigroup A, using our polyadization procedure we have obtained a nonderived n-ary group on pn´1qth direct power A n´1 with the shift deformed multiplication. This construction reminds the Post-like associative quiver from DUPLIJ [2018, 2022a], and allows us to construct a nonderived n-ary group from any semigroup in the unified way presented here.

Polyadization of binary supergroups.
Here we consider a more exotic possibility, when the Bmatrices are defined over the Grassmann algebra, and therefore can represent supergroups (see (3.36) and below). In this case B's can be supermatrices of two kinds, even and odd, which have different properties BEREZIN [1987], LEITES [1983]. The general polyadization procedure remains the same, as for ordinary matrices considered before (see Definition 4.1), and therefore we confine ourselves to examples only.
In the same way one can polyadize any supergroup that can be presented by supermatrices.

CONCLUSIONS
In this paper we have given answers to the following important questions: how to obtain nonderived polyadic structures from binary ones, and what would be a matrix form of their semisimple versions? First, we introduced a general matrix form for polyadic structures in terms of block-shift matrices. If the blocks correspond to a binary structure (a ring, semigroup, group or supergroup), this can be treated as a polyadization procedure for them. Second, the semisimple blocks which further have a block-diagonal form give rise to semisimple nonderived polyadic structures. For a deeper and expanded understanding of the new constructions introduced, we have given clarifying examples. The polyadic structures presented can be used, e.g. for the further development of differential geometry and operad theory, as well as in other directions which use higher arity and nontrivial properties of the constituent universal objects.