Topological properties of prime filters in MTL-algebras and fuzzy set representations for MTL-algebras

In the present paper, some basic properties of prime filters in MTL-algebras are studied. By introducing some topological structures on the set of all prime filters and the set of all maximal filters, respectively, and by investigating the topological properties of them, we conclude that the set of all prime filters is a compact T0 topological space and the set of all maximal filters is a compact Hausdorff topological space. By studying the relation between valuations and filters in MTL-algebras, we prove that a MTL-algebra is homomorphic to a subalgebra of a MTL-cube over the set of all prime filters and illustrate that the Stone representation theorem of Boolean algebras is only a special case of fuzzy sets representation theorem of MTL-algebras. Thus the well-known Stone representation theorem of Boolean algebras is extended to MTL-algebras.