# Closure with a twist

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

${\displaystyle y_{1},y_{2}\in Y}$

there exists an automorphism ${\displaystyle \phi }$ of ${\displaystyle X}$ and an element ${\displaystyle y_{3}\in Y}$ such that

${\displaystyle y_{1}\cdot y_{2}=\phi (y_{3})}$

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

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,...,cn))=(cp(1),...,cp(n)),

where c=(c1,...,cn) 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 pc 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

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

This condition can also be written as

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

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

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

${\displaystyle F+000={\begin{bmatrix}0&0&0\\1&1&0\\1&0&1\end{bmatrix}}=F^{id}=F^{(2,3)_{R}(2,3)_{C}}.}$
${\displaystyle F+110={\begin{bmatrix}1&1&0\\0&0&0\\0&1&1\end{bmatrix}}=F^{(1,2)_{R}(1,2)_{C}}=F^{(1,2,3)_{R}(1,2,3)_{C}}.}$
${\displaystyle F+101={\begin{bmatrix}1&0&1\\0&1&1\\0&0&0\end{bmatrix}}=F^{(1,3)_{R}(1,3)_{C}}=F^{(1,3,2)_{R}(1,3,2)_{C}}.}$

where ${\displaystyle \pi _{R}}$ and ${\displaystyle \sigma _{C}}$ denote permutations of the rows and columns of the matrix, respectively, expressed in cycle notation.

• For any ${\displaystyle n\geq 3}$ another example of a cwatset is ${\displaystyle K_{n}}$, which has ${\displaystyle n}$-by-${\displaystyle n}$ matrix representation
${\displaystyle K_{n}={\begin{bmatrix}0&0&0&\cdots &0&0\\1&1&0&\cdots &0&0\\1&0&1&\cdots &0&0\\&&&\vdots &&\\1&0&0&\cdots &1&0\\1&0&0&\cdots &0&1\end{bmatrix}}.}$

Note that for ${\displaystyle n=3}$, ${\displaystyle K_{3}=F}$.

• An example of a nongroup cwatset with a rectangular matrix representation is
${\displaystyle W={\begin{bmatrix}0&0&0\\1&0&0\\1&1&0\\1&1&1\\0&1&1\\0&0&1\end{bmatrix}}.}$

### Properties

Let C ${\displaystyle \subset }$ 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 hH, there exists a ${\displaystyle \phi _{h}}$Aut(G) such that ${\displaystyle \phi _{h}}$(h) ${\displaystyle \cdot }$ H = ${\displaystyle \phi _{h}}$(H).

Furthermore, a GC-set HG is a cyclic GC-set if there exists an hH and a ${\displaystyle \phi }$Aut(G) such that H = {${\displaystyle h_{1},h_{2},...}$} where ${\displaystyle h_{1}}$ = h and ${\displaystyle h_{n}}$ = ${\displaystyle h_{1}}$ ${\displaystyle \cdot }$ ${\displaystyle \phi }$(${\displaystyle h_{n-1}}$) for all n > 1.

### Examples

• Any cwatset is a GC-set since C + c = ${\displaystyle \pi }$(C) implies that ${\displaystyle \pi ^{-1}}$(c) + C = ${\displaystyle \pi ^{-1}}$(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 = ${\displaystyle Z_{10}}$.
• A nonexample showing that the definition is not trivial for subsets of ${\displaystyle Z_{2}^{n}}$ is H = {000, 100, 010, 001, 110}.

### Properties

• A GC-set HG always contains the identity element of G.
• The direct product of GC-sets is again a GC-set.
• A subset HG 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 ${\displaystyle p^{q}}$ which divides the order of H, H contains sub-GC-sets of orders p,${\displaystyle p^{2}}$,...,${\displaystyle p^{q}}$. (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, JSTOR 2690684, doi:10.2307/2690684.
• 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]