Skew Boolean algebras are non-commutative generalizations of Boolean algebras and are idempotent counterparts of Boolean inverse semigroups. By means of a non-commutative Stone duality, left-handed skew Boolean algebras correspond to \'etale spaces over Boolean spaces in a similar way as Boolean inverse semigroups correspond to \'etale groupoids with Boolean spaces of identities. Skew Boolean intersection algebras are skew Boolean algebras which are also meet-semilattices with respect to the natural partial order and are analogues of Boolean inverse meet-semigroups. Under the non-commutative Stone duality for left-handed skew Boolean algebras, left-handed skew Boolean intersection algebras correspond to Hausdorff \'etale spaces.

Beside a general overview, I will discuss two recent results: on the structure of free skew Boolean algebras (joint work with Jonathan Leech) as well as on the structure of free skew Boolean intersection algebras and the subtle bijection between their ultrafilters and pointed partitions of non-empty subsets of the generating set. This is parallel to the bijection between ultrafilters of a free generalized Boolean algebra and non-empty subsets of the generating set and shrinks to this bijection if one additionally imposes the commutativity axiom. Using this bijection, one can express some combinatorial characteristics of finite free skew Boolean intersection algebras in terms of Bell numbers and Stirling numbers of the second kind. Under the canonical inclusion into the $k$-generated free algebra, where $k\geq n$, an atom of the $n$-generated free algebra decomposes into an orthogonal join of atoms of the $k$-generated free algebra in an agreement with the containment order on the respective pointed partitions. For countably many generators, this leads to the `partition analogue' of the Cantor tree whose boundary is the `partition variant' of the Cantor set.