- Cwatset Examples Properties
- Generalized cwatset Definitions Examples Properties
- References
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
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 
. 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
- All subgroups of — that is, nonempty subsets of 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
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
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
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 
, 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
- 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
- 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
- {{citation | doi=10.2307/2690684 | first1=Gary J. | last1=Sherman | first2=Martin | last2=Wattenberg | year=1994 | title=Introducing … cwatsets! | jstor=2690684 | journal=Mathematics Magazine | volume=67 | pages=109–117 | issue=2 }}.
- 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