Two kinds of enriched topological representations of Q-algebras

Abstract

Abstract 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.