Locally convex topological MV-algebras and uniformization of MV-algebras

MV-algebras are equivalent to MV-semirings that are neither rings nor groups; consequently, the theory of topological MV-algebras does not follow directly from the results on topological rings or topological groups. In this paper, we prove that every locally convex topological MV-algebra is uniformizable, which implies that it is completely regular. Furthermore, we show that for any locally convex topological MV-algebra, the separation axioms T₀, T₁, T₂, T₃, and T₃₁₂ are equivalent. Moreover, building upon the work of [H. Weber, Topology and its Applications (2012), Proposition 2.3], we provide a more precise formulation of the conclusion. We also investigate the uniformity induced by a family of ideals in an MV-algebra A , and demonstrate that the uniform topology generated by this uniformity makes A a zero-dimensional topological MV-algebra.