A non-commutative Priestley duality

Abstract

We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a noncommutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras.From the point of view of skew lattices, Leech showed early on that any strongly distributive skew lattice can be embedded in the skew lattice of partial functions on some set with the operations being given by restriction and socalled override. Our duality shows that there is a canonical choice for this embedding.Conversely, from the point of view of sheaves over Boolean spaces, our results show that skew lattices correspond to Priestley orders on these spaces and that skew lattice structures are naturally appropriate in any setting involving sheaves over Priestley spaces.

On skew Heyting algebras

“…A skew lattice is an algebra S = (S; ∧, ∨) of type (2,2) such that ∧ and ∨ are both idempotent and associative and they satisfy the following absorption laws:…”

“…By results in [1] and [12], any skew Boolean algebra is dual to a sheaf of rectangular bands over a locally-compact Boolean space. A further generalization given in [2] showed that any strongly distributive skew lattice (as defined below) is dual to a sheaf (of rectangular bands) over a locally compact Priestley space.…”

“…In passing to the non-commutative setting one needs to sacrifice either the top or the bottom of the algebra in order not to end up in the commutative setting. In the previous papers [1], [12] and [2] algebras with bottoms were considered, and hence the notion of distributivity was generalized to the notion of so-called strong distributivity. If one tried to define an implication operation in the setting of strongly distributive skew lattices with a bottom as a right adjoint to conjunction, that would force the skew lattice to also possess a top and hence be commutative, resulting in a usual Heyting algebra.…”

Free skew Boolean algebras

“…Proof. The orthosums on both sides of parts (1), (2) and (3) are well-defined because the family {D 1 , . .…”

“…Furthermore, the condition that D i is orthogonal to D j , for i = j, is equivalent to a i ∧ a j = 0, which is also an equality. It follows that the equalities in parts (1), (2) and (3), under the given assumptions, are expressible as quasi-identities (where + must be read as ∨). Likewise, the claim of (4), under the given assumptions, consists of two quasi-identities.…”