The spectrum of the sum of observables on $$\sigma $$ σ -complete MV-effect algebras

Abstract

The natural question about the sum of observables on $$\sigma $$ -complete MV-effect algebras, which was recently defined by A. Dvurecenskij, is how it affects spectra of observables, particularly, their extremal points. We describe boundaries for extremal points of the spectrum of the sum of observables in a general case, and we give necessary and sufficient conditions under which the spectrum attains these boundary values. Moreover, we show that every bounded observable x on a complete MV-effect algebra E can be decomposed into the sum $$x=\tilde{x}+x'$$ , where $$\tilde{x}$$ is the greatest sharp observable less than x and $$x'$$ is a meager and extremally non-invertible observable.