Closure with a twist

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

 y_1, y_2 \in Y

there exists an automorphism ϕ of X and an element y_3 \in Y such that

 y_1 \cdot y_2 = \phi(y_3)

where "\cdot" is notation for an operation on X preserved by ϕ.

Two examples of algebraic structures with the property of closure with a twist are the cwatset and the GC-set.

Contents

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

\ \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 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
 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 πR and σC 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 Kn, 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, K3 = 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 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 φhAut(G) such that φh(h) \cdot H = φh(H).

Furthermore, a GC-set HG is a cyclic GC-set if there exists an hH and a ϕAut(G) such that H = {h1,h2,...} where h1 = h and hn = h1 \cdot ϕ(hn − 1) for all n > 1.

Examples

  • Any cwatset is a GC-set since C + c = π(C) implies that π − 1(c) + C = π − 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 = Z10.
  • 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 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 pq which divides the order of H, H contains sub-GC-sets of orders p,p2,...,pq. (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 [1]

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