Ultragraph algebras via labelled graph groupoids, with applications to generalized uniqueness theorems

Ultragraphs give rise to labelled graphs. We realize algebras associated such graphs as groupoid algebras, generalizing a known algebra realization of ultragraph C*-algebras any ultragraph. Then, we characterize the shift space an tight spectrum inverse semigroup with via its graph. In purely algebraic setting, show that partial action used describe Leavitt path skew group ring is equivalent dual topological action, and use this Steinberg algebras. Finally, prove generalized uniqueness theorems for both their abelian core subalgebras.