Closure with a twist

Closure with a twist is a property of subsets of an algebraic structure. A subset $Y$ of an algebraic structure $X$ is said to exhibit closure with a twist if for every two elements

$y_1, y_2 \in Y$

there exists an automorphism $\phi$ of $X$ and an element $y_3 \in Y$ such that

$y_1 \cdot y_2 = \phi(y_3)$

where "$\cdot$" is notation for an operation on $X$ preserved by $\phi$.

Two examples of algebraic structures with the property of closure with a twist are the cwatset and the GC-set.

Cwatset

In mathematics, a cwatset is a set of bitstrings, all of the same length, which is closed with a twist.

If each string in a cwatset, C, say, is of length n, then C will be a subset of Z₂ⁿ. Thus, two strings in C are added by adding the bits in the strings modulo 2 (that is, addition without carry, or exclusive disjunction). The symmetric group on n letters, Sym(n), acts on Z₂ⁿ by bit permutation:

p((c₁,...,c_n))=(c_p(1),...,c_p(n)),

where c=(c₁,...,c_n) is an element of Z₂ⁿ and p is an element of Sym(n). Closure with a twist now means that for each element c in C, there exists some permutation p_c such that, when you add c to an arbitrary element e in the cwatset and then apply the permutation, the result will also be an element of C. That is, denoting addition without carry by +, C will be a cwatset if and only if

$\ \forall c\in C : \exists p_c\in \text{Sym}(n) : \forall e\in C : p_c(e+c) \in C.$

This condition can also be written as

$\ \forall c\in C : \exists p_c\in \text{Sym}(n) : p_c(C+c)=C.$

Examples

• All subgroups of Z₂ⁿ — that is, nonempty subsets of Z₂ⁿ which are closed under addition-without-carry — are trivially cwatsets, since we can choose each permutation p_c to be the identity permutation.

• An example of a cwatset which is not a group is

F = {000,110,101}.

To demonstrate that F is a cwatset, observe that

F + 000 = F.

F + 110 = {110,000,011}, which is F with the first two bits of each string transposed.

F + 101 = {101,011,000}, which is the same as F after exchanging the first and third bits in each string.

• A matrix representation of a cwatset is formed by writing its words as the rows of a 0-1 matrix. For instance a matrix representation of F is given by

$F = \begin{bmatrix} 0 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}.$

To see that F is a cwatset using this notation, note that

$F + 000 = \begin{bmatrix} 0 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} = F^{id}=F^{(2,3)_R(2,3)_C}.$

$F + 110 = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 1 \end{bmatrix} = F^{(1,2)_R(1,2)_C}=F^{(1,2,3)_R(1,2,3)_C}.$

$F + 101 = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix} = F^{(1,3)_R(1,3)_C}=F^{(1,3,2)_R(1,3,2)_C}.$

where $\pi_R$ and $\sigma_C$ denote permutations of the rows and columns of the matrix, respectively, expressed in cycle notation.

• For any $n \geq 3$ another example of a cwatset is $K_n$, which has $n$-by-$n$ matrix representation

$K_n = \begin{bmatrix} 0 & 0 & 0 & \cdots & 0 & 0 \\ 1 & 1 & 0 & \cdots & 0 & 0 \\ 1 & 0 & 1 & \cdots & 0 & 0 \\ & & & \vdots & & \\ 1 & 0 & 0 & \cdots & 1 & 0 \\ 1 & 0 & 0 & \cdots & 0 & 1 \end{bmatrix}.$

Note that for $n = 3$, $K_3=F$.

• An example of a nongroup cwatset with a rectangular matrix representation is

$W = \begin{bmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1 \end{bmatrix}.$

Properties

Let C $\subset$ Z₂ⁿ be a cwatset.

• The degree of C is equal to the exponent n.

• The order of C, denoted by |C|, is the set cardinality of C.

• There is a necessary condition on the order of a cwatset in terms of its degree, which is

analogous to Lagrange's Theorem in group theory. To wit,

Theorem. If C is a cwatset of degree n and order m, then m divides 2ⁿn!

The divisibility condition is necessary but not sufficient. For example there does not exist a cwatset of degree 5 and order 15.

Generalized cwatset

In mathematics, a generalized cwatset (GC-set) is an algebraic structure generalizing the notion of closure with a twist, the defining characteristic of the cwatset.

Definitions

A subset H of a group G is a GC-set if for each h ∈ H, there exists a $\phi_h$ ∈ Aut(G) such that $\phi_h$(h) $\cdot$ H = $\phi_h$(H).

Furthermore, a GC-set H ⊆ G is a cyclic GC-set if there exists an h ∈ H and a $\phi$ ∈ Aut(G) such that H = {$h_1, h_2, ...$} where $h_1$ = h and $h_n$ = $h_1$ $\cdot$ $\phi$($h_{n-1}$) for all n > 1.

Examples

• Any cwatset is a GC-set since C + c = $\pi$(C) implies that $\pi^{-1}$(c) + C = $\pi^{-1}$(C).
• Any group is a GC-set, satisfying the definition with the identity automorphism.
• A non-trivial example of a GC-set is H = {0, 2} where G = $Z_{10}$.
• A nonexample showing that the definition is not trivial for subsets of $Z_2^n$ is H = {000, 100, 010, 001, 110}.

Properties

• A GC-set H ⊆ G always contains the identity element of G.
• The direct product of GC-sets is again a GC-set.
• A subset H ⊆ G is a GC-set if and only if it is the projection of a subgroup of Aut(G)⋉G, the semi-direct product of Aut(G) and G.
• As a consequence of the previous property, GC-sets have an analogue of Lagrange's Theorem: The order of a GC-set divides the order of Aut(G)⋉G.
• If a GC-set H has the same order as the subgroup of Aut(G)⋉G of which it is the projection then for each prime power $p^{q}$ which divides the order of H, H contains sub-GC-sets of orders p,$p^{2}$,...,$p^{q}$. (Analogue of the first Sylow Theorem)
• A GC-set is cyclic if and only if it is the projection of a cyclic subgroup of Aut(G)⋉G.

References

• Sherman, Gary J.; Wattenberg, Martin (1994), "Introducing … cwatsets!", Mathematics Magazine 67 (2): 109–117, doi:10.2307/2690684, JSTOR 2690684 .
• The Cwatset of a Graph, Nancy-Elizabeth Bush and Paul A. Isihara, Mathematics Magazine 74, #1 (February 2001), pp. 41–47.
• On the symmetry groups of hypergraphs of perfect cwatsets, Daniel K. Biss, Ars Combinatorica 56 (2000), pp. 271–288.
• Automorphic Subsets of the n-dimensional Cube, Gareth Jones, Mikhail Klin, and Felix Lazebnik, Beiträge zur Algebra und Geometrie 41 (2000), #2, pp. 303–323.
• Daniel C. Smith (2003)RHIT-UMJ, RHIT [1]