Pseudocompact Topological \(MV\)-Algebras

Abstract

Recently, topological MV-algebras have been investigated by several mathematicians. In this paper, we find that every topological \(MV\)-algebra is a Mal'tsev space introduced by Mal'tsev in 1954. Hence, applying the theorem of Reznichenko and Uspenskij on pseudocompact Mal'tsev spaces, we show that the product of arbitrary family of pseudocompact topological \(MV\)-algebras are pseudocompact. We also prove that every $σ$-compact topological \(MV\)-algebra is ccc. Secondly, we obtain that the Stone-Čech compactification of a pseudocompact topological \(MV\)-algebra carries a natural compact topological \(MV\)-algebra structure extending the original one. Finally, we prove that: let \(I\) be a closed ideal in a pseudocompact topological \(MV\)-algebra \(A\) and \(ι_1:A\hookrightarrow\bet A\) is the naturally injective; then \(\cl_{βA}ι_1(I)\) is a closed ideal of \(βA\) and \( βA/\cl_{βA}ι_1(I)\cong β(A/I)\).