Higher braid groups and regular semigroups from polyadic-binary correspondence

In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and $R$-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, $n$-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.


INTRODUCTION
We begin by observing that the defining relations of the von Neumann regular semigroups (e.g. GRILLET [1995], HOWIE [1976], PETRICH [1984]) and the Artin braid group KASSEL AND TURAEV [2008], KAUFFMAN [1991] correspond to such properties of ternary matrices (over the same set) as idempotence and the orders of elements (period). We then generalize the correspondence thus introduced to the polyadic case and thereby obtain higher degree (in our definition) analogs of the former. The higher (degree) regular semigroups obtained in this way have appeared previously in semisupermanifold theory DUPLIJ [2000], and higher regular categories in TQFT DUPLIJ AND MARCINEK [2002]. The representations of the higher braid relations in vector spaces coincide with the higher braid equation and corresponding generalized R-matrix obtained in DUPLIJ [2018b], as do the ordinary braid group and the Yang-Baxter equation TURAEV [1988]. The proposed constructions use polyadic group methods and differ from the tetrahedron equation ZAMOLODCHIKOV [1981] and n-simplex equations HIETARINTA [1997] connected with the braid group representations LI AND HU [1995], HU [1997], also from higher braid groups of MANIN AND SCHECHTMAN [1989]. Finally, we define higher degree (in our sense) versions of the Coxeter group and the symmetric group and show that they are connected in the classical (i.e. non-higher) case only.

PRELIMINARIES
There is a general observation NIKITIN [1984], that a block-matrix, forming a semisimple p2, kqring (Artinian ring with binary addition and k-ary multiplication) has the shape In other words, it is given by the cyclic shift pk´1qˆpk´1q matrix, in which identities are replaced by blocks of suitable sizes and with arbitrary entries. The set tM pk´1qu is closed with respect to the product of k matrices, and we will therefore call them k-ary matrices. They form a k-ary semigroup, and when the blocks are over an associative binary ring, then total associativity follows from the associativity of matrix multiplication.
Our proposal is to use single arbitrary elements (from rings with associative multiplication) in place of the blocks m piˆjq , supposing that the elements of the multiplicative part G of the rings form binary (semi)groups having some special properties. Then we investigate the similar correspondence between the (multiplicative) properties of the matrices M pk´1q , related to idempotence and order, and the appearance of the relations in G leading to regular semigroups and braid groups, respectively. We call this connection a polyadic matrix-binary (semi)group correspondence (or in short the polyadicbinary correspondence).

PRELIMINARIES
In the lowest -arity case k " 3, the ternary case, the 2ˆ2 matrices M p2q are anti-triangle. From pM p2qq 3 " M p2q and pM p2qq 3 " E p2q (where E p2q is the ternary identity, see below), we obtain the correspondences of the above conditions on M p2q with the ordinary regular semigroups and braid groups, respectively. In this way we extend the polyadic-binary correspondence on -arities k ě 4 to get the higher relations pM pk´1qq k " " " M pk´1q corresponds to higher k-degree regular semigroups, " qE pk´1q corresponds to higher k-degree braid groups, where E pk´1q is the k-ary identity (see below) and q is a fixed element of the braid group.

TERNARY MATRIX GROUP CORRESPONDING TO THE REGULAR SEMIGROUP
Let G f ree " tG | µ g 2 u be a free semigroup with the underlying set G " g piq ( and the binary multiplication. The anti-diagonal matrices over G f ree where M g p2q " tM g p2qu is the set of ternary matrices (3.1) closed under the ternary multiplication being the ordinary matrix product. Recall that an element M g p2q P M g which in the matrix form (3.2) leads to We denote the set of idempotent ternary matrices by M g id p2q " tM g id p2qu. Definition 3.1. A ternary matrix semigroup in which every element is idempotent (3.4) is called an idempotent ternary semigroup.
Using (3.1) and (3.4) the idempotence expressed in components gives the regularity conditions g p1q g p2q g p1q " g p1q , (3.5) Definition 3.2. A binary semigroup G f ree in which any two elements are mutually regular (3.5)-(3.6) is called a regular semigroup G reg .
Proposition 3.3. The set of idempotent ternary matrices (3.4) form a ternary semigroup M g 3,id " tM id p2q | µ 3 u, if G reg is abelian.
Proof. It follows from (3.5)-(3.6), that idempotence (and following from it regularity) is preserved with respect to the ternary multiplication (3.2), only when any g p1q , g p2q P G f ree commute.
Definition 3.4. We say that the set of idempotent ternary matrices M g id p2q (3.4) is in ternary-binary correspondence with the regular (binary) semigroup G reg and write this as This means that such property of the ternary matrices as their idempotence (3.4) leads to the regularity conditions (3.5)-(3.6) in the correspondent binary group G f ree .

TERNARY MATRIX GROUP CORRESPONDING TO THE REGULAR SEMIGROUP
Remark 3.5. The correspondence (3.7) is not a homomorphism and not a bi-element mapping BOROWIEC ET AL. [2006], and also not a heteromorphism in sense of DUPLIJ [2018a], because we do not demand that the set of idempotent matrices M g id p2q form a ternary semigroup (which is possible in commutative case of G f ree only, see Proposition 3.3).

POLYADIC MATRIX SEMIGROUP CORRESPONDING TO THE HIGHER REGULAR SEMIGROUP
We next extend the ternary-binary correspondence (3.7) to the k-ary matrix case (2.1) and thereby obtain higher k-regular binary semigroups 1 .
Let us introduce the pk´1qˆpk´1q matrix over a binary group G f ree of the form (2.1) Definition 4.1. The set of k-ary matrices M g pk´1q (4.1) over G f ree is a k-ary matrix semigroup is the ordinary product of k matrices M g i pk´1q " M ppk´1qˆpk´1qq´g p1q i , g p2q i , . . . , g pk´1q i¯, see (4.1).
Recall that the polyadic power ℓ of an element M from a k-ary semigroup M k is defined by (e.g. POST [1940]) such that ℓ coincides with the number of k-ary multiplications. In the binary case k " 2 the polyadic power is connected with the ordinary power p (number of elements in the product) as p " ℓ`1, i.e. M xℓy 2 " M ℓ`1 " M p . In the ternary case k " 3 we have xℓy 3 " 2ℓ`1, and so the l.h.s. of (3.4) is of polyadic power ℓ " 1. Assertion 4.4. From M x1y k " M it follows that M xℓy k " M, but not vice-versa, therefore all x1yidempotent elements are xℓy-idempotent, but an xℓy-idempotent element need not be x1y-idempotent.
Therefore, the definition given in(4.5) makes sense.
Similarly to Assertion 4.4, it is seen that rks-xℓy-regularity (4.9)-(4.11) follows from rksregularity (4.6)-(4.8), but not the other way around, and therefore we have Assertion 4.10. If a binary semigroup G reg rks is rks-regular, then it is rks-xℓy-regular as well, but not vice-versa.
By analogy with (3.7), we have Definition 4.12. We will say that the set of k-ary pk´1qˆpk´1q matrices M g id pk´1q (4.1) over the underlying set G is in polyadic-binary correspondence with the binary rks-regular semigroup G reg rks and write this as Thus, using the idempotence condition for k-ary matrices in components (being simultaneously elements of a binary semigroup G f ree ) and the polyadic-binary correspondence (4.12) we obtain the higher regularity conditions (4.6)-(4.11) generalizing the ordinary regularity (3.5)-(3.6), which allows us to define the higher rks-regular binary semigroups G reg rks (G xℓy-reg rks).
The higher regularity conditions (4.14)-(4.16) obtained above from the idempotence of polyadic matrices using the polyadic-binary correspondence, appeared first in DUPLIJ [1998] and were then used for transition functions in the investigation of semisupermanifolds DUPLIJ [2000] and higher regular categories in TQFT DUPLIJ AND MARCINEK [2001,2002]. Now we turn to the second line of (2.2), and in the same way as above introduce higher degree braid groups.

TERNARY MATRIX GROUP CORRESPONDING TO THE BRAID GROUP
Recall the definition of the Artin braid group ARTIN [1947] in terms of generators and relations KASSEL AND TURAEV [2008] (we follow the algebraic approach, see, e.g. MARKOV [1945]).
The Artin braid group B n (with n strands and the identity e P B n ) has the presentation by n´1 generators σ 1 , . . . , σ n´1 satisfying n pn´1q {2 relations where (5.1) are called the braid relations, and (5.2) are called far commutativity. A general element of B n is a word of the form where p r P Z are (positive or negative) powers of the generators σ ir , r " 1, . . . , m and m P N.
For instance, B 3 is generated by σ 1 and σ 2 satisfying one relation σ 1 σ 2 σ 1 " σ 2 σ 1 σ 2 , and is isomorphic to the trefoil knot group. The group B 4 has 3 generators σ 1 , σ 2 , σ 3 satisfying The representation theory of B n is well known and well established KASSEL AND TURAEV [2008], KAUFFMAN [1991]. The connections with the Yang-Baxter equation were investigated, e.g. in TURAEV [1988]. Now we build a ternary group of matrices over B n having generators satisfying relations which are connected with the braid relations (5.1)-(5.2). We then generalize our construction to a k-ary matrix group, which gives us the possibility to "go back" and define some special higher analogs of the Artin braid group.
Let us consider the set of anti-diagonal 2ˆ2 matrices over B n Definition 5.1. The set of matrices M p2q " tM p2qu (5.7) over B n form a ternary matrix semigroup M k"3 " M 3 " tM p2q | µ 3 u, where k " 3 is the -arity of the following multiplication and the associativity is governed by the associativity of both the ordinary matrix product in the r.h.s. of (5.8) and B n .
Proposition 5.2. M p3q is a ternary matrix group.
Proof. Each element of the ternary matrix semigroup M p2q P M 3 is invertible (in the ternary sense) and has a querelementM p2q (a polyadic analog of the group inverse DÖRNTE [1929]) defined by It follows from (5.8)-(5.10), that (5.12) where`M p2q˘´1 denotes the ordinary matrix inverse (but not the binary group inverse which does not exist in the k-ary case, k ě 3). Non-commutativity of µ 3 is provided by (5.9)-(5.10).
The ternary matrix group M 3 has the ternary identity where e is the identity of the binary group B n , and (5.14) We observe that the ternary product µ 3 in components is "naturally braided" (5.9)-(5.10). This allows us to ask the question: which generators of the ternary group M 3 can be constructed using the Artin braid group generators σ i P B n and the relations (5.1)-(5.2)?

TERNARY MATRIX GENERATORS
Let us introduce pn´1q 2 ternary 2ˆ2 matrix generators where σ i P B n , i " 1 . . . , n´1 are generators of the Artin braid group. The querelement of Σ ij p2q is defined by analogy with (5.12) as Now we are in a position to present a ternary matrix group with multiplication µ 3 in terms of generators and relations in such a way that the braid group relations (5.1)-(5.2) will be reproduced.
Definition 6.2. We say that the ternary matrix group M gen-Σ 3 generated by the matrix generators Σ ij p2q satisfying the relations (6.3)-(6.4) is in ternary-binary correspondence with the braid (binary) group B n , which is denoted as (cf. (3.7)) M gen-Σ 3 ≎ B n .
Remark 6.3. Note that the above construction is totally different from the bi-element representations of ternary groups considered in BOROWIEC ET AL. [2006] (for k-ary groups see DUPLIJ [2018a]).
Definition 6.5. An element M p2q P M 3 is of finite q-polyadic (q-ternary) order, if there exists a finite ℓ such that M p2q xℓy 3 " M p2q 2ℓ`1 " qE p2q , q P B n . (6.7) The relations (6.3) therefore say that the ternary matrix generators Σ i,j`1 p2q are of finite q-ternary order. Each element of M gen-Σ 3 is a ternary matrix word (analogous to the binary word (5.3)), being the ternary product of the polyadic powers (4.3) of the 2ˆ2 matrix generators Σ p2q ij and their querelementsΣ p2q ij (on choosing the first or second row) where r " 1, . . . , m, i r , j r " 1, . . . , n (from B n ), ℓ r , m P N. In the ternary case the total number of multipliers in (6.8) should be compatible with (4.3), i.e. p2ℓ 1`1 q`. . .`p2ℓ r`1 q`. . .p 2ℓ m`1 q " 2ℓ W`1 , ℓ W P N, and m is therefore odd. Thus, we have Remark 6.6. The ternary words (6.8) in components give only a subset of the binary words (5.3), and so M gen-Σ 3 corresponds to B n , but does not present it.

GENERATED k-ARY MATRIX GROUP CORRESPONDING THE HIGHER BRAID GROUP
The above construction of the ternary matrix group M gen-Σ 3 corresponding to the braid group B n can be naturally extended to the k-ary case, which will allow us to "go in the opposite way" and build so called higher degree analogs of B n (in our sense: the number of factors in braid relations more than 3). We denote such a braid-like group with n generators by B n rks, where k is the number of generator multipliers in the braid relations (as in the regularity relations (4.6)-(4.8)). Simultaneously k is the -arity of the matrices (5.7), we therefore call B n rks a higher k-degree analog of the braid group B n . In this notation the Artin braid group B n is B n r3s. Now we build B n rks for any degree k exploiting the "reverse" procedure, as for k " 3 and B n in SECTION 5. For that we need a kary generalization of the matrices over B n , which in the ternary case are the anti-diagonal matrices M p2q (5.7), and the generator matrices Σ ij p2q (6.1). Then, using the k-ary analog of multiplication (5.9)-(5.10) we will obtain the higher degree (than (5.1)) braid relations which generate the so called higher k-degree braid group. In distinction to the higher degree regular semigroup construction from SECTION 4, where the k-ary matrices form a semigroup for the Abelian group G f ree , using the generator matrices, we construct a k-ary matrix semigroup (presented by generators and relations) for any (even non-commutative) matrix entries. In this way the polyadic-binary correspondence will connect k-ary matrix groups of finite order with higher binary braid groups (cf. idempotent k-ary matrices and higher regular semigroups (4.12)).
Let us consider a free binary group B f ree and construct over it a k-ary matrix group along the lines of NIKITIN [1984], similarly to the ternary matrix group M 3 in (5.7)-(5.10).
Definition 7.1. A set M pk´1q " tM pk´1qu of k-ary pk´1qˆpk´1q matrices form a k-ary matrix semigroup M k " tM pk´1q | µ k u, where µ k is the k-ary multiplication (7.5) where the r.h.s. of (7.2) is the ordinary matrix multiplication of k-ary matrices (7.1) Proposition 7.2. M k is a k-ary matrix group.
Proof. Because B f ree is a (binary) group with the identity e P B f ree , each element of the kary matrix semigroup M pk´1q P M k is invertible (in the k-ary sense) and has a querelement M pk´1q (see DÖRNTE [1929]) defined by (cf. (5.14)) whereM pk´1q can be on any place, and so we have k conditions (cf. (5.11) for k " 3).
Definition 7.3. An element of a k-ary group M pk´1q P M k has the polyadic order ℓ, if pM pk´1qq xℓy k " pM pk´1qq ℓpk´1q`1 " E pk´1q , (7.9) where E pk´1q P M k is the polyadic identity (7.7), for k " 3 see (5.13).
Definition 7.4. An element (pk´1qˆpk´1q-matrix over B f ree ) M pk´1q P M tku is of finite q-polyadic order, if there exists a finite ℓ such that pM pk´1qq xℓy k " pM pk´1qq ℓpk´1q`1 " qE pk´1q , q P B f ree . (7.10) Let us assume that the binary group B f ree is presented by generators and relations (cf. the Artin braid group (5.1)-(5.2)), i.e. it is generated by n´1 generators σ i , i " 1, . . . , n´1. An element of B gen-σ n " B f ree pe, σ i q is the word of the form (5.3). To find the relations between σ i we construct the corresponding k-ary matrix generators analogous to the ternary ones (6.1). Then using a k-ary version of the relations (6.3)-(6.4) for the matrix generators, as the finite order conditions (7.10), we will obtain the corresponding higher degree braid relations for the binary generators σ i , and can therefore present a higher degree braid group B n rks in the form of generators and relations. Using n´1 generators σ i of B gen-σ n we build pn´1q k polyadic (or k-ary) pk´1qˆpk´1qmatrix generators having k´1 indices i 1 , . . . , i k´1 " 1, . . . , n´1, as follows For the matrix generator Σ i 1 ,...,i k´1 pk´1q (7.11) its querelementΣ i 1 ,...,i k´1 pk´1q is defined by (7.6).
Each element of M gen-Σ k is a k-ary matrix word (analogous to the binary word (5.3)) being the kary product of the polyadic powers (4.3) of the matrix generators Σ i 1 ,...,i k´1 pk´1q and their querele-mentsΣ i 1 ,...,i k´1 pk´1q as in (6.8).
Similarly to the ternary case k " 3 (SECTION 5) we now develop the k-ary "reverse" procedure and build from B gen-σ n the higher k-degree braid group B n rks using (7.11). Because the presentation of M gen-Σ k by generators and relations has already been given in (7.19)-(7.22), we need to expand them into components and postulate that these new relations between the (binary) generators σ i present a new higher degree analog of the braid group. This gives Definition 7.6. A higher k-degree braid (binary) group B n rks is presented by pn´1q generators σ i " σ rks i (and the identity e) satisfying the following relations ‚ pk´1q higher braid relations (7.28) if all |i p´is | ě k´1, p, s " 1, . . . , k´1, (7.29) I f ar " tn´k, . . . , n´1u , (7.30) where τ is an element of the permutation symmetry group τ P S k´1 .
A general element of the higher k-degree braid group B n rks is a word of the form w " σ p 1 i 1 . . . σ pr ir . . . σ pm im , i m " 1, . . . , n, (7.31) where p r P Z are (positive or negative) powers of the generators σ ir , r " 1, . . . , m and m P N.
Remark 7.8. The representation of the higher k-degree braid relations in B n rks in the tensor product of vector spaces (similarly to B n and the Yang-Baxter equation TURAEV [1988]) can be obtained using the n 1 -ary braid equation introduced in DUPLIJ [2018b] (Proposition 7.2 and next there).
If B n rks Ñ Z is the abelianization defined by σȋ Ñ˘1, then σ p i " e, if and only if p " 0, and σ i are of infinite order. Moreover, we can prove (as in the ordinary case k " 3 DYER [1980]) Theorem 7.14. The higher k-degree braid group B n rks is torsion-free.
Recall (see, e.g. KASSEL AND TURAEV [2008]) that there exists a surjective homomorphism of the braid group onto the finite symmetry group B n Ñ S n by σ i Ñ s i " pi, i`1q P S n . The generators s i satisfy (5.1)-(5.2) together with the finite order demand which is called the Coxeter presentation of the symmetry group S n . Indeed, multiplying both sides of (7.48) from the right successively by s i`1 , s i , and s i`1 , using (7.50), we obtain ps i s i`1 q 3 " 1, and (7.49) on s i and s j , we get ps i s j q 2 " 1. Therefore, a Coxeter group BRIESKORN AND SAITO [1972] corresponding (7.48)-(7.50) is presented by the same generators s i and the relations ps i s i`1 q 3 " 1, 1 ď i ď n´2, (7.51) ps i s j q 2 " 1, |i´j| ě 2, (7.52) s 2 i " e, i " 1, . . . , n´1. (7.53) A general Coxeter group W n " W n pe, r i q is presented by n generators r i and the relations BJÖRNER AND BRENTI [2005] pr i r j q m ij " e, m ij " " 1, i " j, ě 2, i ‰ j. (7.54) By analogy with (7.48)-(7.50), we make the following Definition 7.15. A higher analog of S n , the k-degree symmetry group S n rks " S rks n pe, s i q, is presented by generators s i , i " 1, . . . , n´1 satisfying (7.23)-(7.28) together with the additional condition of finite pk´1q-order s pk´1q i " e, i " 1, . . . , n.
(7.72) Thus, we arrive at Theorem 7.22. The higher k-degree Coxeter group can present the k-degree symmetry group in the lowest case only, if and only if k " 3.