On non-commutative Stone duality

The aim of the proposed talk is to survey some recent developments on non-commutative generalizations of Stone duality. We present the speaker's recent work where Stone duality is generalized to right-handed skew Boolean algebras. One approach is via  assigning to these algebras étale spaces over Boolean spaces. The other approach stems from the dualizing object view of the classical Stone duality. The former approach leads to dual equivalences, and the latter one to a series of dual adjunctions between appropriate categories. Different non-commutative objects, that generalize Boolean algebras, are Boolean inverse semigroups, that are inverse semigroups with Boolean algebras of idempotents, satisfying the property that joins of all pairs of compatible elements exist. Mark Lawson's generalization of Stone duality links them on topological side with Boolean groupoids. The latter is an étale topological groupoid, whose space of identities is a Boolean space. Jointly with Mark Lawson, we found a common generalization of the mentioned work of each of us.  Our approach is based on Yamada's structure result for generalized inverse semigroups. The objects that generalize both right-handed skew Boolean algebras and Boolean inverse semigroups in the minimal possible way turn out to be right Boolean generalized inverse semigroups. These are right generalized inverse semigroups whose idempotents form a skew Boolean algebra and such that joins of compatible pairs of elements exist.