Two Spaces of Minimal Primes

This paper studies algebraic frames L and the set Min(L) of minimal prime elements of L. We will endow the set Min(L) with two well-known topologies, known as the Hullkernel (or Zariski) topology and the inverse topology, and discuss several properties of these two spaces. It will be shown that Min(L) endowed with the Hull-kernel topology is a zero-dimensional, Hausdorff space; whereas, Min(L) endowed with the inverse topology is a T1, compact space. The main goal will be to find conditions on L for the spaces Min(L) and Min(L)−1 to have various topological properties; for example, compact, locally compact, Hausdorff, zero-dimensional, and extremally disconnected. We will also discuss when the two topological spaces are Boolean and Stone spaces.