On the use of pairwise balanced designs and closure spaces in the construction of structures of degree at least 3

We prove that a set of v-2 symmetric idempotent latin squares of order v , such that no two of them agree in a off-diagonal position, exists for all odd v>>0 . We describe how the techniques used in the proof relate to techniques used in [17] to construct generalised idempotent ternary quasigroups whose conjugate invariant group contains some specific subgroup. We also showhow these techniques fit into the more general context of trying to extend group divisible design methods to combinatorial structures, using closure spaces.