Finiteorthoatomistic effect algebras and regular algebraic E -test spaces

In this article we consider two categories: the category of finiteorthoatomistic effect algebras and the category of regular algebraic E -test spaces. S. Gudder introduced (algebraic) E -test spaces and showed that for each algebraic E -test space ( X; 𝒯 ) there exists an effect algebra Π( X ). Moreover Π is a covariant functor from the category of algebraic E -test spaces to the category of effect algebras. We show that the category of regular and algebraic E -test spaces is equivalent to the category of finiteorthoatomistic effect algebras. We proved that atoms in effect algebras corresponding to regular and algebraic E -test spaces are perspectivity classes of characteristic functions of one-element sets. We found an example of algebraic E -test space which is not regular and in corresponding E -test space some perspectivity class of the characteristic function of a singleton is not an atom.