Summary: We prove that a set of v-2 symmetric idempotent latin squares of order v, such that no two of them agree in an off-diagonal position, exists for all sufficiently large odd v. We describe how the techniques used in the proof relate to techniques used in [IMA Vol. Math. Appl. 20, 368-378 (1990; Zbl 0727.05010)] to construct generalized idempotent ternary quasigroups whose conjugate invariant group contains some specified subgroup. We also show how these techniques fit into the more general context of trying to extend group divisible design methods to combinatorial structures with \(t>3\), using closure spaces.