Module Theory in $${\mathtt {}}$$

This chapter is an introduction to module theory in the category \({\mathtt {Sup}}\) of complete lattices and join preserving maps — these are modules on unital quantales. In this context, an interesting class of modules on involutive and unital quantales is induced by irreducible representations of \(C^*\)-algebras. Since \({\mathtt {Sup}}\) is \(*\)-autonomous, there exist constructions in the connection with modules on unital quantales which are not possible in the traditional theory of modules in the category of abelian groups. As an important example, this chapter explains the details of the equivalence between right \({\mathfrak {Q}}\)-modules and join-complete \({\mathfrak {Q}}\)-valued lattices where \({\mathfrak {Q}}\) is a unital quantale. Finally, a sketch of automata in \({\mathtt {Sup}}\) is given showing that every automaton in \({\mathtt {Sup}}\) gives rise to a right module on the free unital quantale generated by the corresponding input alphabet.