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.
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
To demonstrate that F is a cwatset, observe that
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.
Note that for , .
Let be a cwatset.
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.
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.
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 .
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.
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
To demonstrate that F is a cwatset, observe that
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.
Note that for , .
Let be a cwatset.
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.
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.
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 .