Closure with a twist

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

${\displaystyle y_{1},y_{2}\in Y}$

there exists an automorphism ${\displaystyle \phi }$ of ${\displaystyle X}$ and an element ${\displaystyle y_{3}\in Y}$ such that

${\displaystyle y_{1}\cdot y_{2}=\phi (y_{3})}$

where "${\displaystyle \cdot }$" is notation for an operation on ${\displaystyle X}$ preserved by ${\displaystyle \phi }$.

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