In analogy with effect algebras, we introduce the test spaces and $MV$-test spaces. A test corresponds to a hypothesis on the propositional system, or, equivalently, to a partition of unity. We show that there is a close correspondence between $MV$-algebras and $MV$-test spaces.
We give two variations of the Holland representation theorem for $\ell $-groups and of its generalization of Glass for directed interpolation po-groups as groups of automorphisms of a linearly ordered set or of an antilattice, respectively. We show that every pseudo-effect algebra with some kind of the Riesz decomposition property as well as any pseudo $MV$-algebra can be represented as a pseudo-effect algebra or as a pseudo $MV$-algebra of automorphisms of some antilattice or of some linearly ordered set.