Cwatsets were originally defined as subsets of \(\mathbb{Z}_2^d\) that are “closed with a twist.” Attempts have been made to generalize them, but the generalizations have failed to produce notions of subcwatset and quotient cwatset that behave naturally.
We present a new, abstract definition that appears to avoid these problems. The relationship between this new definition and its predecessor is similar to that between the abstract definition of “group” and its original meaning as a set of permutations. To justify the broader definition, we use small cancellation theory to prove a result analogous to the statement that every group is isomorphic to some permutation group. After developing the notion of a quotient cwatset, we prove an analogue of the First Homomorphism Theorem.
1970-2025 CP (Manitoba, Canada) unless otherwise stated.