Morita Equivalence of Subrings with Applications to Inverse Semigroup Algebras
Allen Zhang
TL;DR
This work develops general, verifiable criteria to establish Morita equivalence between substructures and ambient algebras in the setting of rings with local units, extending prior desingularization techniques. It unifies Morita theory across several combinatorial and algebraic frameworks by translating subring, partial action, and inverse semigroup data into Morita-equivalence conditions for associated $R$-algebras, including Leavitt path algebras and labelled Leavitt path algebras. Key contributions include a constructive criterion for subring Morita equivalence, Morita-transfer results for partial subactions and inverse subsemigroups, enlargement-based transfers, and a desingularization theorem for labelled Leavitt path algebras. The results yield practical consequences such as explicit Morita classifications for graph algebras (e.g., DAGs and functional graphs) and a systematic desingularization approach that preserves Morita class while regularizing the underlying combinatorial objects.
Abstract
We develop a technique to show the Morita equivalence of certain subrings of a ring with local units. We then apply this technique to develop conditions that are sufficient to show the Morita equivalence of subalgebras induced by partial subactions on generalized Boolean algebras and, subsequently, strongly $E^{\ast}$-unitary inverse subsemigroups. As an application, we prove that the Leavitt path algebra of a graph is Morita equivalent to the Leavitt path algebra of certain subgraphs and use this to calculate the Morita equivalence class of some Leavitt path algebras. Finally, as the main application, we prove a desingularization result for labelled Leavitt path algebras.