Dimension and partial groups

Source abstract

A partial group with n + 1 n+1 elements is, when regarded as a symmetric simplicial set, of dimension at most n n . This dimension is n n if and only if the partial group is a group. As a consequence of the first statement, finite partial groups are genuinely finite, despite being seemingly specified by infinitely much data. In particular, finite partial groups have only finitely many im-partial subgroups. We also consider dimension of partial groupoids.