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 which exhibit closure with a twist are the cwatset and the generalized cwatset, or 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 ${\displaystyle \mathbb {Z} _{2}^{n))$. 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, ${\displaystyle {\text{Sym))(n)}$, acts on ${\displaystyle \mathbb {Z} _{2}^{n))$ by bit permutation:

${\displaystyle p((c_{1},\ldots ,c_{n}))=(c_{p(1)},\ldots ,c_{p(n)}),}$

where ${\displaystyle c=(c_{1},\ldots ,c_{n})}$ is an element of ${\displaystyle \mathbb {Z} _{2}^{n))$ and p is an element of ${\displaystyle {\text{Sym))(n)}$. Closure with a twist now means that for each element c in C, there exists some permutation ${\displaystyle p_{c))$ 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 ${\displaystyle +}$, 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 ${\displaystyle \mathbb {Z} _{2}^{n))$ — that is, nonempty subsets of ${\displaystyle \mathbb {Z} _{2}^{n))$ 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

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

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 ${\displaystyle C\subset \mathbb {Z} _{2}^{n))$ 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 ${\displaystyle 2^{n}!}$.

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 ${\displaystyle h\in H}$, there exists a ${\displaystyle \phi _{h}\in {\text{Aut))(G)}$ such that ${\displaystyle \phi _{h}(h)\cdot H=\phi _{h}(H)}$.

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

Examples

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

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 ${\displaystyle p^{q))$ 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.