Fig. 1: Illustration of how symmetry applies to four different structures.

Each structure consists of a space along with a set of transformations of that space. The transformation for the linear and bisected linear structures inverts the one-dimensional space. The transformations for the bisected circular structure are a reflection in the vertical axis and a rotation through 180^∘, and the transformations for the binary feature structure exchange the values of the two features. A category system is symmetric if invariant under all transformations shown, and three symmetric systems are shown for each structure. The top row shows examples of domains that can be modeled using each structure, and the bottom row indicates whether semantic systems in those domains are expected to have both parities or to show a preference towards odd or even sizes.