Tensor Product in the Category of Effect Algebras
Dominik Lachman
TL;DR
Addressing the tensor product in the category of effect algebras, the paper develops a framework for how tensoring interacts with colimits and the universal-group construction. It proves that the functor $E\otimes -$ preserves connected colimits and shows that the Gr functor from $\mathbf{EA}$ to $\mathbf{POG}_u$ can be equipped with a strong monoidal structure, yielding $\mathrm{Gr}(E\otimes F) \cong \mathrm{Gr}(E)\otimes \mathrm{Gr}(F)$. It further demonstrates that tensoring in $\mathbf{EA}$ does not preserve the Riesz decomposition property (RDP) by transporting Wehrung's counterexample via the Gr–Γ adjunction. Together, these results establish a robust monoidal bridge between effect algebras and unital po-groups and point to practical directions for computing tensor products in concrete cases, such as specific interval 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.