Ternary and n-ary f-distributive structures

Definition 2.1

An f-shelf is a triple (X, *, f) in which X is a set, * is a binary operation on X, and f: X → X is a map such that, for any x,y,z ∈ X, the identity

holds. An f-rack is an f-shelf such that, for any x,y ∈ X, there exists a unique z ∈ X such that

An f-quandle is an f-rack such that, for each x ∈ X, the identity

holds.

An f-crossed set is an f-quandle (X, *, f) such that f: X → X satisfies x * y = f(x) whenever y* x = f(y) for any x, y ∈ X.

Definition 2.2

Let (X₁, *₁, f₁) and (X₂, *₂, f₂) be two f-racks (resp. f-quandles). A map ϕ: X₁ → X₂is an f-rack (resp. f-quandle) morphism if it satisfies ϕ (a *₁b) = ϕ(a) *₂ϕ(b) and ϕ ∘ f₁ = f₂ ∘ ϕ.

Remark 2.3

A category of f-quandles is a category whose objects are tuples (A, *, f) which are f-quandles and morphism are f-quandle morphisms.

Examples of f-quandles include the following:

1. Given any set X and map f: X → X, then the operation x * y = f(x) for any x, y ∈ X gives a f-quandle. We call this a trivial f-quandle structure on X.

2. For any group G and any group endomorphism f of G, the operation x * y = y⁻¹x f(y) defines a f-quandle structure on G.

3. Consider the Dihedral quandle R_n, where n ≥ 2, and let f be an automorphism of R_n. Then f is given by f(x) = ax+b, for some invertible element a ∈ ℤ_n and some b ∈ ℤ_n [16]. The binary operation x * y = f(2y − x) = 2ay − ax + b (mod n) gives a f-quandle structure called the Dihedral f-quandle.

4. Any ℤ[T^{± 1}, S]-module M is a f-quandle with x * y = Tx+Sy, x,y ∈ M with TS = ST and f(x) = (S+T)x, called an Alexander f-quandle.

Remark 2.4

Axioms of Definition 2.1 give the following identity,

We note that the two medial terms in this equation are swapped (resembling the mediality condition of a quandle). Note also that the mediality in the general context may not be satisfied for f-quandles. For example one can check that the f-quandle given in item (2) of Examples is not medial.

Definition 2.5

Let Q be a set, α be a morphism and T: Q × Q × Q → Q be a ternary operation on Q. The operation T is said to be right f-distributive with respect to α if it satisfies the following condition for all x,y,z,u,v ∈ Q

The previous condition is called right f-distributivity.

Remark 2.6

Using the diagonal map D: Q → Q × Q × Q = Q^{× 3}such that D(x) = (x,x,x), equation (4) can be written, as a map from Q^{× 5}to Q, in the following form

where id stands for the identity map. In the whole paper we denote by ρ:Q^{× 9} → Q^{× 9}the map defined as ρ = p_6,8∘ p_3,7 ∘ p_2,4where p_i,jis the transposition i^th and j^th elements, i.e.

Definition 2.7

Let T: Q × Q × Q → Q be a ternary operation on a set Q. The triple (Q,T,α) is said to be a ternary f-shelf if the identity (4) holds. If, in addition, for all a,b ∈ Q, the map R_a,b:Q → Qgiven byR_a,b(x) = T(x,a,b) is invertible, then (Q,T,α) is said to be a ternary f-rack. If further T satisfies T(x,x,x) = α(x), for all x ∈ Q, then (Q,T, α) is called a ternary f-quandle.

Remark 2.8

Using the right translation R_a,b : Q → Q defined as R_a,b(x) = T(x, a, b), the identity(4)can be written asRα(u),α(v)∘Ry,z=RRu,v(y),Ru,v(z)∘Ru,v.

Example 2.9

Let (Q,*,f) be a f-quandle and define a ternary operation on Q by

It is straightforward to see that (Q, T, α) is a ternary f-quandle where α = f ∘ f. Note that in this case R_a,b = R_b ∘ R_a. We will say that this ternary f-quandle is induced by a (binary) quandle.

Remark 2.10

Binary f-quandles are related to ternary f-quandles when α = f ∘ f. Also, ternary operations T lead to binary operations by setting for example x*y = T (x,y,y).

Example 2.11

Let (M,*,f) be an Alexander f-quandle, then the ternary f-quandle (M, T,α) coming from M, has the operation T(x,y,z) = Px+Qy+Rz where P, Q, R commutes with each other and α (x) = (P + Q + R)x. It is also called affine ternary f-quandle.

Example 2.12

Any group G with the ternary operation T(x,y,z) = f(xy⁻¹z) gives an example of ternary f-quandle. This is called f heap (sometimes also called a groud) of the group G.

A morphism of ternary quandles is a map ϕ: (Q, T) → (Q′,T′) such that

A bijective ternary quandle endomorphism is called ternary f-quandle automorphism.

Therefore, we have a category whose objects are ternary f-quandles and morphisms as defined above.

As in the case of the binary quandle there is a notion of medial ternary quandle

Definition 2.13

([28, 29]). A ternary quandle (Q,T, α) is said to be medial if for all a, b, c, d, e, f, g, h, k ∈ Q, the following identity is satisfied

This definition of mediality can be written in term of the following commutative diagram

Example 2.14

Every affine ternary f-quandle, defined in Example 2.11, is medial. That is, condition (2.13) is satisfied. Indeed the left hand side (LHS) of the mediality condition is shown below to equal the right hand side (RHS):

We generalize the notion of ternary f-quandle to n-ary setting.

Definition 2.15

An n-ary distributive set is a triple (Q,T, α) where Q is a set, α: Q → Q is a morphism and T: Q^{× n} → Q is an n-ary operation satisfying the following conditions:

1. T(T(x1,⋯,xn),α(u1),⋯,α(un−1))=T(T(x1,u1,⋯,un−1),T(x2,u1,⋯,un−1),⋯,T(xn,u1,⋯,un−1)),

   ∀ x_i,u_i ∈ Q (f-distributivity).

2. For all a₁,⋯,a_{n − 1} ∈ Q, the map R_{a1,⋯,an−1}:Q → Q given by

   Ra1,⋯,an−1(x)=T(x,a1,⋯,an−1)

   is invertible.

3. For all x ∈ Q,

   T(x,⋯,x)=α(x).

If T satisfies only condition (1), then (Q,T, α) is said to be an n-ary f-shelf. If both conditions (1) and (2) are satisfied then (Q,T, α) is said to be an n-ary f-rack. If all three conditions (1), (2) and (3) are satisfied then (Q,T, α) is said to be an n-ary f-quandle.

Example 2.16

Let (Q,*,f) be an f-quandle and define an n-ary twisted operation on Q by

where i = 1, ⋯, n.

It is straightforward to see that (Q, T, α) is an n-ary f-quandle where α = fⁿ⁻¹.

Example 2.17

Let (M,*,f) be an Alexander f-quandle, then the n-ary f-quandle (M, T,α) coming from M has the operation T(x₁, ⋯, x_n) = S₁x₁+S₂x₂+ ⋯ +S_n x_n, where S_i, i = 1, ⋯, n commutes with each other and α (x) = (S₁ + S₂ + ⋯ + S_n)x.

Definition 2.18

An n-ary quandle (Q,T, α) is said to be medial if for all x_ij ∈ Q, 1 ≤ i,j ≤ n, the following identity is satisfied

Proposition 5.1

Let (X, T, F) be a ternary f-quandle and A be a non-empty set. Let α: X × X × X → Fun(A × A × A, A) be a function and f, g: A → A are maps. Then, X × A is a ternary f-quandle by the operation T((x,a), (y,b), (z,c)) = (T(x,y,z), α_x,y,z(a, b,c)), where T(x, y, z) denotes the ternary f-quandle product in X, if and only if α satisfies the following conditions:

1. α_x,x,x(a, a,a) = g(a) for all x ∈ X and a ∈ A;

2. α_{x, y, z}(−, b, c) : A → A is a bijection for all x, y, z ∈ X and for all b, c ∈ A;

3. α_{T(x, y, z), f(u), f(v)}(α_{x, y, z}(a, b, c), g(d), g(e)) = α_{T(x, u, v), T(y, u, v), T(z, u, v)}(α_{x, u, v}(a, d, e), α_{y, u, v}(b, d, e), α_{z, u, v}(c, d, e)) for all x, y, z, u, v ∈ Xanda, b, c, d, e ∈ A.

Such function α is called a dynamical ternary f-quandle cocycle or dynamical ternary f-rack cocycle (when it satisfies above conditions).

The ternary f-quandle constructed above is denoted by X ×_αA, and it is called extension of X by a dynamical cocycle α. The construction is general, as Andruskiewitch and Graña showed in [11].

Assume (X, T, F) is a ternary f-quandle and α be a dynamical f-cocycle. For x ∈ X, define T_x(a, b, c) := α_x,x,x(a, b, c). Then it is easy to see that (A, T_x, F) is a ternary f-quandle for all x ∈ X.

Remark 5.2

When x = y = z in condition (3) above, we get

for all a, b, c, d, e ∈ A.

Now, we discuss Extensions with constant cocycles. Let (X, T, F) be a ternary f-rack and λ: X × X × X → S_A where S_A is group of permutations of X.

If λ_{T(x, y, z), F(u), F(v)}λ_{x, y, z} = λ_{T(x, u, v), T(y, u, v), T(z, u, v)}λ_{x, u, v} we say λ is a constant ternary f-rack cocycle.

If (X, T, F) is a ternary f-quandle and further satisfies λ_x,x,x = id for all x ∈ X, then we say λ is a constant ternary f-quandle cocycle.

Definition 5.3

Let (X, T, F) be a ternary f-rack, A be an abelian group and f, g: X → X be homomorphisms. A structure of X-module on A consists of a family of automorphisms (η_ijk)_{i, j, k ∈ X}and a family of endomorphisms (τ_ijk)_{i, j, k ∈ X}of A satisfying the following conditions:

In the n-ary case, we generalized the above definition as follows.

Definition 5.4

Let (X, T, F) be an n-ary f-rack, A be an abelian group and f, g: X → X be homomorphisms. A structure of X-module on A consists of a family of automorphisms (η_ijk)_{i, j, k ∈ X}and a family of endomorphisms(τijki)i,j,k∈Xof A satisfying the following conditions:

Remark 5.5

If X is a ternary f-quandle, a ternary f-quandle structure of X-module on A is a structure of an X-module further satisfies

and

Furthermore, if f, g = id maps, then it satisfies

Remark 5.6

When x = y = z in(8), we get

Example 5.7

Let A be a non-empty set and (X, T, F) be a ternary f-quandle, and κ be a generalized 2-cocycle. For a, b, c ∈ A, let

Then, it can be verified directly that α is a dynamical cocycle and the following relations hold:

Definition 5.8

When κ further satisfies κ_x,x,x = 0 in (23) for any x ∈ X, we call it a generalized ternary f-quandle 2-cocycle.

Recall that the quandle algebra of an f-quandle (X, ⊲, f) is a ℤ-algebra ℤ(X) presented by generators as in [11] with relations (8), (9), (10), (11), (12), (13), (14).

Example 5.9

Let (X,T, F) be a ternary f-quandle and A be an abelian group. Set η_{x, y, z} = τ_{x, y, z}, μ_{x, y, z} = 0, κ_{x, y, z} = ϕ(x, y, z). Then ϕ is a 2-cocycle. That is,

Example 5.10

Let Γ = ℤ[P,Q,R] denote the ring of Laurent polynomials. Then any Γ-module M is a ℤ(X)-module for any ternary f-quandle (X, T, F) by η_{x, y, z}(a) = Pa,τ_{x, y,z}(b) = Qb and μ_{x, y, z}(c) = Rc for anyx, y, z ∈ X.

Definition 5.11

A set G equipped with a ternary operator T: G × G × G → G is said to be a ternary group (G, T) if it satisfies the following condition:

1. T(T(x, y, z), u, v) = T(x, T(y, z, u), v) = T(x, y, T(z, u, v)) (associativity),

2. T(e, x, e) = T(x, e, e) = T(e, e, x) = x (existence of identity element),

3. T(x, y, y) = T(y, x, y) = T(y, y, x) = e (existence of inverse element).

Example 5.12

Here we provide an example of a ternary f-quandle module and explicit formula of the ternary f-quandle 2-cocycle obtained from a group 2-cocycle. Let G be a group and let 0 → A → E → G → 1 be a short exact sequence of groups where E = A ⋊_θG by a group 2-cocycle θ and A is an Abelian group.

The multiplication rule in E is given by (a, x) ⋅ (b, y) ⋅ (c, z) = (a + x ⋅ b + y ⋅ c + θ(x, y, z), T(x, y, z)), where x ⋅ b means the action of A on G. Recall that the group 3-cocycle condition is

Now, let X = G be a ternary f-quandle with the operation T(x, y, z) = f(xy⁻¹z) and let g: A → A be a map on A so that we have a map F: E → E given by F(a,x) = (g(a), f(x)). Therefore the group E becomes a ternary f-quandle with the operation

Explicit computations give that η_{x, y, z}(a) = g(a), τ_{x, y, z}(b) = −2xy⁻¹g(b), μ_{x, y, z} = y⁻¹g(c) and κ_{x, y, z} = g[θ(xy, y, y⁻¹) − θ(xy⁻¹, y, y) + θ(xy, y⁻¹, y) + 2θ(x, y, e) − θ(x, y², y⁻¹) − θ(x, y⁻¹, y²) + θ(x, y⁻¹, y) − θ(x, y,y) − θ(x, y, y⁻¹) + θ(x, y⁻¹, z)].