The Loomis--Sikorski Theorem for $EMV$-algebras

Abstract

Recently, in [DvZa], we have introduced $EMV$-algebras which resemble $MV$-algebras but the top element is not guaranteed for them. For $\sigma$-complete $EMV$-algebras, we prove an analogue of the Loomis--Sikorski Theorem showing that every $\sigma$-complete $EMV$-algebra is a $\sigma$-homomorphic image of an $EMV$-tribe of fuzzy sets where all algebraic operations are defined by points. To prove it, some topological properties of the state-morphism space and the space of maximal ideals are established.