The Category of $$\omega $$ -Effect Algebras: Tensor Product and $$\omega $$ -Completion

Abstract

Effect algebras are certain ordered structures that serve as a general framework for studying the algebraic semantics of quantum logic. We study effect algebras which obtain suprema of countable monotone sequences – so-called \(\omega \)-effect algebras. This assumption is necessary to capture basic (non-discrete) probabilistic concepts. We establish a free \(\omega \)-completion of effect algebras (i.e., a left adjoint to the functor that forgets the existence of \(\omega \)-suprema) and the existence of a tensor product in the category of \(\omega \)-effect algebras. These results are obtained by means of so-called test spaces. Test spaces form a category that contains effect algebras as a reflective subcategory, but provides more space for constructions.