Tensor Product in the Category of Effect Algebras

Abstract

We study a tensor product in the category of effect algebras and in the category of partially ordered Abelian groups with order unit. We show that the tensor product preserves all the constructions that are essentially colimits over a connected diagram. Further, we prove the construction of a universal group for an effect algebra preserves all tensor products. We establish the corresponding functor from the category of effect algebras to the category of unital Abelian po-groups as a strong monoidal functor. We note that the technique we use in establishing the result could be used in various similar situations. Finally, we show that the tensor product of effect algebras does not preserve the Riesz decomposition property, which was an open question for a while.