Closure with a twist

In mathematics, closure with a twist is an algebraic condition describing a subset that maps onto itself ("closure"), not necessarily identically, but under a group automorphism (a "twist") when acted upon by one of its own elements.

Formally, a subset $H$ of a group $G$ is said to exhibit closure with a twist if each of its cosets is an image of $H$ under some automorphism of $G$. That is, for every element $h\in H$, there exists an automorphism $\phi _{h}$ of $G$ such that:

$H\cdot h=\phi _{h}(H)$

where "$\cdot$" denotes the group operation on $G$, and $\phi _{h}(H)$ is the image of the entire subset $H$ under the automorphism $\phi _{h}$. ("Coset" here is used loosely, as this concept is interesting when $H$ is not itself a group.)

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 $\mathbb {Z} _{2}^{n}$. 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, ${\text{Sym}}(n)$, acts on $\mathbb {Z} _{2}^{n}$ by bit permutation:

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

where $c=(c_{1},\ldots ,c_{n})$ is an element of $\mathbb {Z} _{2}^{n}$ and p is an element of ${\text{Sym}}(n)$. 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 $\mathbb {Z} _{2}^{n}$ — that is, nonempty subsets of $\mathbb {Z} _{2}^{n}$ 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

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

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

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 $h\in H$, there exists a $\phi _{h}\in {\text{Aut}}(G)$ 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 $h\in H$ and a $\phi \in {\text{Aut}}(G)$ 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