2. Preliminaries
There is a general observation [
16] that a block-matrix, forming a semisimple
-ring (Artinian ring with binary addition and
k-ary multiplication) has the shape:
In other words, it is given by the cyclic shift matrix, in which identities are replaced by blocks of suitable sizes and with arbitrary entries.
The set
is closed with respect to the product of
k matrices, while the product of two (or less than
k of such matrices) has not the same form as (
1). In this sense, the
k-ary multiplication is not reducible (or derived) to the binary multiplication; therefore, we will call them
k-ary matrices. This “non-reducibility” is the key property of
k-ary matrices, which will be used in the below constructions. The matrices of the shape (
1) form a
k-ary semigroup (which cannot be reduced to binary semigroups), and, when the blocks are over an associative binary ring, then the total
k-ary associativity follows from the ordinary associativity of the binary matrix multiplication of the blocks.
Our proposal is to use single arbitrary elements (from rings with associative multiplication) in place of the blocks , 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 , 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 polyadic-binary correspondence).
In the lowest arity case
, the ternary case, the
matrices
are anti-triangle. From
and
(where
is the ternary identity; see below), we obtain the correspondences of the above conditions on
with the ordinary regular semigroups and braid groups, respectively. In this way, we extend the polyadic-binary correspondence on -arities
to get the higher relations
where
is the
k-ary identity (see below), and
q is a fixed element of the braid group.
4. Polyadic Matrix Semigroup Corresponding to
the Higher Regular Semigroup
Next we extend the ternary-binary correspondence (
9) to the
k-ary matrix case (
1) and thereby obtain higher
k-regular binary semigroups. We use the following notation:
Round brackets: is size of matrix , as well as the sequential number of a matrix element.
Square brackets: is number of multipliers in the regularity and braid conditions.
Angle brackets: is the polyadic power (number of k-ary multiplications).
Let us introduce the
matrix over a binary group
of the form (
1)
where
.
Definition 4. The set of k-ary matrices (10) over is a k-ary matrix semigroup , where the multiplicationis the ordinary product of k matrices ; see (10). Recall that the polyadic power
ℓ of an element
M from a
k-ary semigroup
is defined by (e.g., Reference [
19])
such that
ℓ coincides with the
number of k-ary multiplications. In the binary case
the polyadic power is connected with the ordinary power
p (
number of elements in the product) as
, i.e.,
. In the ternary case
, we have
, and so the l.h.s. of (
6) is of polyadic power
.
Definition 5. An element of a k-ary semigroup is called idempotent, if its first polyadic power coincides with itselfand -idempotent, if Definition 6. A k-ary semigroup is called idempotent (ℓ-idempotent), if each of its elements is idempotent (-idempotent).
Assertion 1. From it follows that , but not vice-versa; therefore, all -idempotent elements are -idempotent, but an -idempotent element need not be -idempotent.
Therefore, the definition given in(
14) makes sense.
Proposition 2. If a k-ary matrix is idempotent (13), then its elements satisfy the relations Proof. This follows from (
10), (
11) and (
13). □
Definition 7. The relations (15)–(17) are called (higher) -regularity (or higher k-degree regularity). The case is the standard regularity (-regularity in our notation) (7) and (8). Proposition 3. If a k-ary matrix is -idempotent (14), then its elements satisfy the following relations Proof. This also follows from (
10), (
11), and (
14). □
Definition 8. The relations (15)–(17) are called (higher) --regularity. The case (7) and (8) is the standard regularity (--regularity in this notation). Definition 9. A binary semigroup , in which any elements are -regular (--regular), is called a higher -regular (--regular) semigroup ().
Similarly to Assertion 1, it is seen that
-
-regularity (
18)–(20) follows from
-regularity (
15)–(17), but not the other way around; therefore, we have:
Assertion 2. If a binary semigroup is -regular, then it is --regular as well, but not vice-versa.
Proposition 4. The set of idempotent (-idempotent) k-ary matrices form a k-ary semigroup , if and only if () is abelian.
Proof. It follows from (
15)–(20) that the idempotence (
-idempotence) and the following
-regularity (
-
-regularity) are preserved with respect the
k-ary multiplication (
11) only in the case, when all
mutually commute. □
By analogy with (
9), we have:
Definition 10. We will say that the set of k-ary matrices (10) over the underlying set is in polyadic-binary correspondence with the binary -regular semigroup and write this as Thus, using the idempotence condition for
k-ary matrices in components (being simultaneously elements of a binary semigroup
) and the polyadic-binary correspondence (
21) we obtain the higher regularity conditions (
15)–(20) generalizing the ordinary regularity (
7) and (8), which allows us to define the higher
-regular binary semigroups
(
).
Example 1. The lowest nontrivial () case is , where the matrices over are of the shapeand they form the 4-ary matrix semigroup . The idempotence gives three -regularity conditions According to the polyadic-binary correspondence (21), the conditions (23)–(25) are -regularity relations for the binary semigroup , which defines to the higher -regular binary semigroup . In the case , we have , which gives three --regularity conditions (they are different from -regularity)and these define the higher --regular binary semigroup . Obviously, (26)–(28) follow from (23)–(25), but not vice-versa. The higher regularity conditions (
23)–(25) obtained above from the idempotence of polyadic matrices using the polyadic-binary correspondence, appeared first in Reference [
20] and were then used for transition functions in the investigation of semisupermanifolds [
6] and higher regular categories in TQFT [
7,
21].
Now, we turn to the second line of (
2), and in the same way as above introduce higher degree braid groups.
5. Ternary Matrix Group Corresponding to the
Braid Group
Recall the definition of the Artin braid group [
22] in terms of generators and relations [
4] (we follow the algebraic approach; see, e.g., Reference [
23]).
The
Artin braid group (with
n strands and the identity
) has the presentation by
generators
satisfying
relations
where (
29) are called the
braid relations, and (30) are called
far commutativity. A general element of
is a word of the form
where
are (positive or negative) powers of the generators
,
and
.
For instance,
is generated by
and
satisfying one relation
, and is isomorphic to the trefoil knot group. The group
has 3 generators
satisfying
The representation theory of
is well known and well established [
4,
5]. The connections with the Yang-Baxter equation were investigated, e.g., in Reference [
9].
Now, we build a ternary group of matrices over
having generators satisfying relations which are connected with the braid relations (
29) and (30). 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
matrices over
Definition 11. The set of matrices (35) over form a ternary matrix semigroup , where is the arity of the following multiplicationand the associativity is governed by the associativity of both the ordinary matrix product in the r.h.s. of (36) and . Proposition 5. is a ternary matrix group.
Proof. Each element of the ternary matrix semigroup
is invertible (in the ternary sense) and has a
querelement (a polyadic analog of the group inverse [
24]) defined by
It follows from (
36)–(38) that
where
denotes the ordinary matrix inverse (but not the binary group inverse which does not exist in the
k-ary case,
). Non-commutativity of
is provided by (37) and (38). □
The ternary matrix group
has the
ternary identity
where
e is the identity of the binary group
, and
We observe that the ternary product
in components is “naturally braided” (37) and (38). This allows us to ask the question: which generators of the ternary group
can be constructed using the Artin braid group generators
and the relations (
29) and (30)?
7. Generated k-Ary Matrix Group Corresponding the
Higher Braid Group
The above construction of the ternary matrix group
corresponding to the braid group
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
(in our sense: the number of factors in braid relations more than 3). We denote such a braid-like group with
n generators by
, where
k is the number of generator multipliers in the braid relations (as in the regularity relations (
15)–(17)). Simultaneously,
k is the -arity of the matrices (
35); therefore, we call
a higher
k-degree analog of the braid group
. In this notation, the Artin braid group
is
. Now, we build
for any degree
k exploiting the “reverse” procedure, as for
and
in
Section 5. For that, we need a
k-ary generalization of the matrices over
, which, in the ternary case, are the anti-diagonal matrices
(
35), and the generator matrices
(
43). Then, using the
k-ary analog of multiplication (37) and (38) we will obtain the higher degree (than (
29)) 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
, 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 (
21)).
Let us consider a free binary group
and construct over it a
k-ary matrix group along the lines of Reference [
16], similarly to the ternary matrix group
in (
35)–(38).
Definition 15. A set of k-ary matricesform a k-ary matrix semigroup , where is the k-ary multiplicationwhere the r.h.s. of (60) is the ordinary matrix multiplication of k-ary matrices (59) , . Proposition 7. is a k-ary matrix group.
Proof. Because
is a (binary) group with the identity
, each element of the
k-ary matrix semigroup
is invertible (in the
k-ary sense) and has a
querelement (see Reference [
24]) defined by (cf. (
42))
where
can be on any place, and so we have
k conditions (cf. (
39) for
). □
The
k-ary matrix group has the polyadic identity
satisfying
where
can be on any place, and so we have
k conditions (cf. (
42)).
Definition 16. An element of a k-ary group has the polyadic order ℓ, ifwhere is the polyadic identity (65), for ; see (41). Definition 17. An element of the -matrix over that is is of finite q-polyadic order, if there exists a finite ℓ such that Let us assume that the binary group
is presented by generators and relations (cf. the Artin braid group (
29) and (30)), i.e., it is generated by
generators
,
. An element of
is the word of the form (
31). To find the relations between
we construct the corresponding
k-ary matrix generators analogous to the ternary ones (
43). Then, using a
k-ary version of the relations (
45) and (46) for the matrix generators, as the finite order conditions (
68), we will obtain the corresponding higher degree braid relations for the binary generators
and can, therefore, present a higher degree braid group
in the form of generators and relations.
Using
generators
of
, we build
polyadic (or
k-ary)
-matrix generators having
indices
, as follows
For the matrix generator
(
69), its querelement
is defined by (
64).
We now build a
k-ary matrix analog of the braid relations (
29), (
45) and of far commutativity (30), (46). Using (
69), we obtain
conditions that the matrix generators are of finite polyadic order (analog of (
45))
where
are polyadic identities (
65) and
.
We propose a
k-ary version of the far commutativity relation (46) in the following form:
where
is an element the permutation symmetry group
.
In matrix form, we can define
Definition 18. A k-ary (generated) matrix group is presented by the matrix generators (69) and the relations (we use (60))andwhere and . Each element of
is a
k-ary matrix word (analogous to the binary word (
31)) being the
k-ary product of the polyadic powers (
12) of the matrix generators
and their querelements
as in (
50).
Similarly to the ternary case
(
Section 5), we now develop the
k-ary “reverse” procedure and build from
the higher
k-degree braid group
using (
69). Because the presentation of
by generators and relations has already been given in (
77) and (
80), we need to expand them into components and postulate that these new relations between the (binary) generators
present a new higher degree analog of the braid group. This gives:
Definition 19. A higher k-degree braid (binary) group is presented by generators (and the identity ) satisfying the following relations:
higher braid relations -ary far commutativitywhere τ is an element of the permutation symmetry group .
A general element of the higher
k-degree braid group
is a word of the form
where
are (positive or negative) powers of the generators
,
and
.
Remark 4. The ternary case coincides with the Artin braid group (29) and (30). Remark 5. The representation of the higher k-degree braid relations in in the tensor product of vector spaces (similarly to and the Yang-Baxter equation [9]) can be obtained using the -ary braid equation introduced in Reference [8] (Proposition 7.2 and next there). Definition 20. We say that the k-ary matrix group generated by the matrix generators satisfying the relations (77)–(80) is in polyadic-binary correspondence with the higher k-degree braid group , which is denoted as (cf. (47)) Example 4. Let ; then, the 4-ary matrix group is generated by the matrix generators satisfying (77)–(80) 4-ary relations of q-polyadic order (48) Let , ; then, we use the 4-ary matrix presentation for the generators (cf. Example 1): The querelement satisfying Expanding (91)–(92) in components, we obtain the relations for the higher 4-degree braid group as follows. higher 4-degree braid relations ternary far (total) commutativity
In the higher 4-degree braid group, the minimum number of generators is 4, which follows from (
96). In this case, we have a braid relation for
only and no far commutativity relations because of (98). Then:
Example 5. The higher 4-degree braid group is generated by 3 generators , , , which satisfy only the braid relation If , then there will be no far commutativity relations at all, which follows from (98), and so the first higher 4-degree braid group containing far commutativity should have elements.
Example 6. The higher 4-degree braid group is generated by 7 generators , which satisfy the braid relations with together with the ternary far commutativity relation Remark 6. In polyadic group theory, there are several possible modifications of the commutativity property; but, nevertheless, we assume here the total commutativity relations in the k-ary matrix generators and the corresponding far commutativity relations in the higher degree braid groups.
If
is the abelianization defined by
, then
, if and only if
, and
are of infinite order. Moreover, we can prove (as in the ordinary case
[
25]).
Theorem 1. The higher k-degree braid group is torsion-free.
Recall (see, e.g., Reference [
4]) that there exists a surjective homomorphism of the braid group onto the finite symmetry group
by
. The generators
satisfy (
29) and (30), together with the finite order demand
which is called the
Coxeter presentation of the symmetry group
. Indeed, multiplying both sides of (
106) from the right successively by
,
, and
, using (108), we obtain
, and (107) on
and
, we get
. Therefore, a Coxeter group [
26] corresponding to (
106)–(108) is presented by the same generators
and the relations
A general Coxeter group
is presented by
n generators
and the relations [
27]
By analogy with (
106)–(108), we make the following.
Definition 21. A higher analog of , the k-degree symmetry group , is presented by generators , satisfying (81)–(86) together with the additional condition of finite -order , . Example 7. The lowest higher degree case is which is presented by three generators , , satisfying (see (99)) In a similar way, we define a higher degree analog of the Coxeter group (
112).
Definition 22. A higher k-degree Coxeter group is presented by n generators obeying the relations It follows from (116) that all generators are of order . A higher k-degree Coxeter matrix is a hypermatrix having 1 on the main diagonal and other entries .
Example 8. In the lowest higher degree case, and all , we have (instead of commutativity in the ordinary case ) Example 9. A higher 4-degree analog of (109)–(111) is given by It follows from (120) thatwhich cannot be reduced to total commutativity (97). From the first relation (119), we obtainwhich differs from the higher 4-degree braid relations (96). Example 10. In the simplest case, the higher 4-degree Coxeter group has 3 generator , , satisfying Example 11. The minimal case, when the conditions (120) appear is and an analog of commutativity Thus, we arrive at:
Theorem 2. The higher k-degree Coxeter group can present the k-degree symmetry group in the lowest case only, if and only if .
As a further development, it would be interesting to consider the higher degree (in our sense) groups constructed here from a geometric viewpoint (e.g., References [
5,
28]).