Two kinds of enriched topological representations of Q-algebras

In this paper, we continue the study of the enriched topological representation of Q-algebras where Q is a unital quantale and give two kinds of enriched topological representations of Q-algebras. The first one is based on strong M-3-valued Q-algebra homomorphisms, and the second way is based on strong M-6-valued Q-algebra homomorphisms. For the first way, we first construct a spatial and semiunital Q-algebra M-3 containing three elements and show that prime elements of semiunital Q-algebras are identified with strong Q-algebra homomorphisms taking their values in M-3. Then we prove that a semiunital Q-algebra is spatial iff strong Q-algebra homomorphisms with values in M-3 separate elements. Based on this, we obtain that every spatial and semiunital Q-algebra can be identified with an M-3-enriched sober space. For the second enriched topological representation, we construct a spatial and semiunital Q-algebra M-6 containing exactly six elements and prove that every spatial and semiunital Q-algebra can be identified with an M-6-enriched sober space.