Spectral resolutions in effect algebras.

Compressions on an effect algebra $E$, analogous to compressions on operator algebras, order unit spaces or unital abelian groups, are studied. A special family of compressions on $E$ is called a compression base. Elements of a compression base are in one-to-one correspondence with certain elements of $E$, called projections. A compression base is spectral if it has two special properties: the projection cover property (i.e., for every element $a$ in $E$ there is a smallest projection majorizing $a$), and the so-called b-comparability property, which is an analogue of general comparability in operator algebras or unital abelian groups. An effect algebra is called spectral if it has a distinguished spectral compression base. It is shown that in a spectral effect algebra $E$, every $a\in E$ admits a unique rational spectral resolution and its properties are studied. If in addition $E$ possesses a separating set of states, then every element $a\in E$ is determined by its spectral resolution. It is also proved that for some types of interval effect algebras (with RDP, Archimedean divisible), spectrality of $E$ is equivalent to spectrality of its universal group and the corresponding rational spectral resolutions are the same. In particular, for convex Archimedean effect algebras, spectral resolutions in $E$ are in agreement with spectral resolutions in the corresponding order unit space.