# Closure with a twist

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

there exists an automorphism of and an element such that

where "" is notation for an operation on preserved by .

Two examples of algebraic structures which exhibit closure with a twist are the **cwatset** and the **generalized cwatset**, or **GC-set**.

## Cwatset

[edit]In mathematics, a **cwatset** is a set of bitstrings, all of the same length, which is **c**losed **w**ith **a** **t**wist.

If each string in a cwatset, *C*, say, is of length *n*, then *C* will be a subset of . 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, , acts on by bit permutation:

where is an element of and *p* is an element of . Closure *with a twist* now means that for each element *c* in *C*, there exists some permutation 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

This condition can also be written as

### Examples

[edit]- All subgroups of — that is, nonempty subsets of 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

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

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

- For any another example of a cwatset is , which has -by- matrix representation

Note that for , .

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

### Properties

[edit]Let 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 .

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

## Generalized cwatset

[edit]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

[edit]A subset *H* of a group *G* is a *GC-set* if for each , there exists a such that .

Furthermore, a GC-set *H* ⊆ *G* is a *cyclic GC-set* if there exists an and a such that where and for all .

### Examples

[edit]- Any cwatset is a GC-set, since implies that .
- Any group is a GC-set, satisfying the definition with the identity automorphism.
- A non-trivial example of a GC-set is where .
- A nonexample showing that the definition is not trivial for subsets of is .

### Properties

[edit]- 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 which divides the order of*H*,*H*contains sub-GC-sets of orders*p*,,...,. (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

[edit]- 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]