Abstract
We introduce a class of inverse semigroups built from directed graphs that we refer to as Leavitt inverse semigroups. These semigroups are closely related to graph inverse semigroups and Leavitt path algebras. We find a presentation for the Leavitt inverse semigroup of a graph in terms of generators and relations. We describe the structure of the Leavitt inverse semigroup and the Leavitt path algebra of a graph that admits a directed immersion into a circle. We show that two graphs that have isomorphic Leavitt inverse semigroups have isomorphic Leavitt path algebras and we classify graphs that have isomorphic Leavitt inverse semigroups. As a consequence, we show that Leavitt path algebras are 0-retracts of certain matrix algebras.
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