Test spaces, pseudo-effect algebras, and tensor product of pseudo-effect algebras

Inspiring the ideas of the test spaces of effect algebras, we introduce test spaces of pseudo-effect algebras. We show that there is a one-to-one correspondence among algebraic test spaces and pseudo-effect algebras. This correspondence gives a test on a pseudo-effect algebra as a decomposition of unity, which corresponds to hypothesis in the statistical models. This correspondence allows us to define a tensor product of pseudo-effect algebras as well as bounded Boolean power. In addition, we introduce PMV-test spaces which forces pseudo MV-algebras.