Skew Boolean Algebras

Skew Boolean algebras

Ganna Kudryavtseva

University of Ljubljana Faculty of Civil and Geodetic Engineering IMFM, Ljubljana

Non-commutative structures 2018 Portoroˇz,May 2018 1. Stone duality for SBAs and SBIAs 2. Stone duality for Boolean inverse semigroups and pseudogroups 3. Generalized inverse ∗-semigroups 4. Dualizing object approach to Stone duality for SBAs 5. Free SBAs 6. Free SBIAs SKEW BOOLEAN ALGEBRAS Historical note

I 1949 - Pascual Jordan: work on non-commutative lattices.

I early 1970’s - a series of works by Boris Schein

I late 1970’s and early 1980’s - Robert Bignall and William Cornish studied non-commutative Boolean algebras

I 1989 - Jonathan Leech: modern study of skew lattices.

I 1995 - Jonathan Leech and Robert Bignall: the ﬁrst paper devoted to skew Boolean intersection algebras.

I 2012-2013 - Stone duality was extended to skew Boolean algebras and distributive skew lattices.

I 1989 - present many aspects of skew lattices and skew Boolean algebras have been studied.

I A. Bauer, R. Bignall, K. Cvetko-Vah, M. Gehrke, S. van Gool, M. Kinyon, GK, M. V. Lawson, J. Leech, J. Pita-Costa, A.Salibra, M.Spinks is an (incomplete) list of researchers who have contributed to the topic. Skew Boolean algebras (SBAs) A skew Boolean algebra is an algebra (S; ∧, ∨, \, 0) of type (2, 2, 2, 0) such that for any a, b, c, d ∈ S: 1. (associativity) a ∨ (b ∨ c) = (a ∨ b) ∨ c, a ∧ (b ∧ c) = (a ∧ b) ∧ c; 2. (absorption) a ∨ (a ∧ b) = a,(b ∧ a) ∨ a = a, a ∧ (a ∨ b) = a, (b ∨ a) ∧ a = a; 3. (distributivity) a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c), (b ∨ c) ∧ a = (b ∧ a) ∨ (c ∧ a); 4. (properties of 0) a ∨ 0 = 0 ∨ a = a, a ∧ 0 = 0 ∧ a = 0; 5. (properties of relative complement) (a \ b) ∧ b = b ∧ (a \ b) = 0, (a \ b) ∨ (a ∧ b ∧ a) = a = (a ∧ b ∧ a) ∨ (a \ b); 6. (normality) a ∧ b ∧ c ∧ d = a ∧ c ∧ b ∧ d. Associativity and absorption imply that (A, ∧, ∨) is a skew lattice and the absorption axiom implies 7. (idempotency) a ∨ a = a, a ∧ a = a. Question. What if normality is dropped and one requires only S is symmetric, has 0 and S/D is a Boolean lattice. These are generalizations of SBAs. Study their structure. Do they form a variety? How are they related to SBAs?

Skew Boolean algebras (SBAs)

Equivalently: a skew lattice S is Boolean (can be endowed with the structure of an SBA) if and only if

S is symmetric, has 0 and each principal subalgebra x↓ = {y ∈ S : y ≤ x} forms a Boolean lattice. IF AND ONLY IF S is symmetric, has 0, the operation ∧ is normal and S/D is a Boolean lattice. Question. What if normality is dropped and one requires only S is symmetric, has 0 and S/D is a Boolean lattice. These are generalizations of SBAs. Study their structure. Do they form a variety? How are they related to SBAs? Left-handed SBAs An SBA S is left-handed if the normality axiom is replaced by: (60)(left normality) x ∧ y ∧ z = x ∧ z ∧ y, and a dual axiom holds for right-handed SBAs. In this talk, we restrict attention to left-handed SBAs. The general case can be easily obtained applying

S ' S/R ×S/D S/L.

for the canonical diagram

S S/R ,

S/L S/D

This is the Kimura Factorization Theorem, extended from regular bands to skew lattices. LSBAs as Boolean left normal bands

I A band (S, ∧) is left normal if x ∧ y ∧ z = x ∧ z ∧ y for any x, y, z ∈ S.

I The natural partial order on S: a ≤ b iﬀ a ∧ b = b ∧ a = a.

I x D y iﬀ x ∧ y = x and y ∧ x = y (in fact D = L). The quotient over D is the maximum commutative quotient of S.

I S decomposes as a strong semilattice of left zero semigroups where the components are precisely the D-classes.

I A left normal band (S, ∧) is Boolean if S/D is a Boolean algebra and joins of compatible pairs of elements exist.

I Fact: the category of LSBAs is equivalent to the category of Boolean left normal bands. Structure of ﬁnite LSBAs

I Finite primitive LSBAs: (aka ﬁnite left zero semigroups) (k + 1)L = {0,..., k} (∧ is left zero multiplication). This generalizes the Boolean algebra 2 = {0, 1}.

I Structure of ﬁnite LSBAs. Let S be a ﬁnite LSBA. Then

k2 k3 km+1 S ' 2 × 3L × · · · × (m + 1)L

for some m where all ki ≥ 0. This generalizes the fact that any ﬁnite Boolean algebra is isomorphic to some 2k . Skew Boolean intersection algebras (SBIAs)

Let (S; ∧, ∨, \, 0) be an SBA.

I S has intersections if the meet of a and b with respect to ≤ exists for any a, b ∈ S.

I a u b — the intersection of a and b.

I If S is ﬁnite a u b, always exists.

I If (S; ∧, ∨, \, 0) has intersections, (S; ∧, ∨, u, \, 0) is a skew Boolean intersection algebra (SBIA).

Proposition (Bignall and Leech) Let (S; ∧, ∨, \, u, 0) be an algebra of type (2, 2, 2, 2, 0). Then it is an SBIA if and only if (S; ∧, ∨, \, 0) is an SBA, (S, u) is a semilattice and the following identities hold:

x u (x ∧ y ∧ x) = x ∧ y ∧ x; x ∧ (x u y) = x u y = (x u y) ∧ x. Question Let S be a LSBIA. Then S∗ is a Boolean space so that there is its dual GBA, B. Is it possible to extend the assignment S 7→ B to a functor? What are adjoints, if exist? What can be said about the relationship between S/D and B?

Non-commutative Stone dualities

Theorem 1. (GK) The category of LSBAs is dually equivalent to the category of ´etale spaces over (locally compact) Boolean spaces.

Theorem 2 (GK; Bauer and Cvetko-Vah) The category of LSBIAs is dually to the category of Hausdorﬀ ´etale spaces over Boolean spaces. Non-commutative Stone dualities

Theorem 1. (GK) The category of LSBAs is dually equivalent to the category of ´etale spaces over (locally compact) Boolean spaces.

Theorem 2 (GK; Bauer and Cvetko-Vah) The category of LSBIAs is dually to the category of Hausdorﬀ ´etale spaces over Boolean spaces.

Question Let S be a LSBIA. Then S∗ is a Boolean space so that there is its dual GBA, B. Is it possible to extend the assignment S 7→ B to a functor? What are adjoints, if exist? What can be said about the relationship between S/D and B? Stone duality for SBAs: idea of proof

I S- LSBA, S/D - GBA, construct its dual Boolean space (S/D)∗: spectrum = space of ultraﬁlters.

I Upgrade the above to S: consider ultraﬁlters of S together with their projections to (S/D)∗. ∗ ∗ ∗ I Obtain S and p : S → (S/D) .

I Important: p is an ´etalemap, that is, a local homeomorphism, that is, locally, S∗ looks the same as (S/D)∗. ∗ ∗ I Therefore, (S , p, (S/D) ) is an ´etalespace. ↓ I Not surprising: for any a ∈ S the set a = {b ∈ S : b ≤ a} is a Boolean algebra, due to normality.

I Conversely, if (E, p, X ) is an ´etalespace over a Boolean space X , then the set of all sections over compact open sets is an LSBA, dual to (E, p, X ). Non-commutative Stone dualities: continued A skew lattice S is strongly distributive if it is

I normal,

I symmetric and

I S/D is a distributive lattice. Priestley Duality for distributive skew lattices (Bauer, Cvetko-Vah, Gehrke, van Gool, K) The category of left-handed strongly distributive skew lattices is dually equivalent to the category of ´etalespaces over Priestley spaces.

1. If S is strongly distributive, take the dual spectral space (S/D)+ of S/D with patch topology, and consider sections over increasing compact open sets. 2. The Booleanization of S is the set of all sections of (S/D)+ over all compact open sets. Pseudogroups and Boolean inverse semigroups

I Pseudogroup - inverse semigroup whose idempotents form a frame and compatible joins exist.

I Prototypical example - pseudogroup of homeomorphisms between opens of a topological space.

I Dual objects - localic ´etalegroupoids. This upgrades the duality between frames and locales. See works by Resende, Lawson and Lenz; K and Lawson.

I Boolean inverse semigroup - inverse semigroup whose idempotents form a Boolean algebra and compatible ﬁnite joins exist. These are dual to (topological) ´etalegroupoids with Boolean spaces of identities.

I Distributive inverse semigroup - idempotents form a distributive lattice and compatible joins exist. Dual objects - spectral groupoids. Some questions

I Question There is a forgetful functor from Boolean inverse semigroups to LSBAs (indeed, domain and range maps of an ´etalegroupoids are ´etalemaps). Does it have a left adjoint? Then how is it constructed?

I Question Similar question for distributive inverse semigroups is harder: duality for distributive inv. semigroups involves spectral groupoids, whereas that for strongly distributive skew lattices - ´etalespaces over Priestley spaces! Explore what can be done! Dualizing object approach to non-commutative Stone duality

The classical Stone duality is induced by a dualizing object, {0, 1}. Does this admit an extension to a non-commutative setting? ∗ I Let S be an LSBA and let Sn be the set of all non-zero SBA homomorphisms S → n + 1 with the subspace topology from S ∗ {0, 1,..., n} . Then Sn is a Boolean space. This gives rise to a functor Λn from the category of LSBAs to the category of Boolean spaces.

I Let X be a Boolean space and consider the set of all proper continuous maps X → {0, 1,..., n}. This inherits an SBA structure from n + 1 = {0, 1,..., n} and gives rise to a functor λn from the category of Boolean spaces to the category of SBAs. Dualizing object approach to non-commutative Stone duality

Theorem The functor Λn is a left adjoint to the functor λn, for each n ≥ 0. The category λn(BS) is a reﬂective subcategory of the category of skew Boolean algebras. The reﬂector (= left adjoint to the inclusion functor) is given by the functor λnΛn.

GK, A dualizing object approach to non-commutative Stone duality, J. Aust. Math. Soc., 2013. Skew Boolean algebras and Boolean inverse semigroups put together

I A left generalized inverse semigroup is a regular semigroup whose idempotents form a left normal band.

I Left generalized inverse semigrops are given by the following data: an inverse semigroup S and a presheaf of sets P over E(S). Then

(S, P) = {(s, e): e ∈ P(s−1s), s ∈ S}, where

(s, e)(t, f ) = (st, e|(st)−1st ), is a left generalized inverse semigroup and any left generalized inverse semigroup is of this form (essentially due to Yamada).

I One can deﬁne a Boolean left generalized inverse semigroup as such an (S, P) for which S is a Boolean inverse semigroup and P is a sheaf. A topological duality follows (K and Lawson). Generalized inverse ∗-semigroups

A semigroup S is called a right ∗-semigroup, if it is equipped with a unary operation s 7→ s∗, satisfying the following axioms: (S1) s∗∗ = s; (S2) s∗ ∈ V (s); (S3)( st)∗ = t∗(stt∗)∗; (S4) If e2 = e then e = e∗. ∗-semigroups were introduced by J. Funk, in 2007 in order to characterize the classifying topos β(S) in semigroup terms. Theorem (GK and M. V. Lawson, 2012) Let S be an inverse semigroup. The category of right generalized inverse ∗-semigroups T with T /γ ' S is equivalent to the classifying topos β(S). FREE SKEW BOOLEAN ALGEBRAS (Joint work with Jonathan Leech) Free generalized Boolean algebras: well known

3 Example. Let X = {x1, x2}. Then GBAX ' 2 where 2 = {0, 1}. Indeed, denote a1 = a{1} = x1 \ x2, a2 = a{2} = x2 \ x1 and a3 = a{1,2} = x1 ∧ x2. I These terms are pairwise distinct.

I a1 ∧ a2 = a1 ∧ a3 = a2 ∧ a3 = 0.

I x1 = (x1 \ x2) ∨ (x1 ∧ x2) and x2 = (x2 \ x1) ∨ (x1 ∧ x2).

I Thus any element of GBAX is a join of a subset of {a1, a2, a3}.

2n−1 Similarly, if |X | = n, GBAX ' 2 , atoms are in a bijections with non-empty subsets of X . Finite free LSBAs: an example

Atoms of LSBA2:

E 2 x2 \ x1 x2 ∧ x1 1 x1 \ x2 x1 ∧ x2

X [x1] \ [x2] [x2] \ [x1] [x1] ∧ [x2] Finite free LSBAs: another example

Atoms of LSBA3:

E 3 x3 \ (x1 ∨ x2) (x3 ∧ x1) \ x2 (x3 ∧ x2) \ x1 x3 ∧ x1 ∧ x2

2 x2 \ (x1 ∨ x3) (x2 ∧ x1) \ x3 (x2 ∧ x3) \ x1 x2 ∧ x1 ∧ x3

1 x1 \ (x2 ∨ x3) (x1 ∧ x2) \ x3 (x1 ∧ x3) \ x2 x1 ∧ x2 ∧ x3

X [x1] \ ([x2] ∨ [x3]) [x2] \ ([x1] ∨ [x3]) [x3] \ ([x1] ∨ [x2]) ([x1] ∧ [x2]) \ [x3] ([x1] ∧ [x3]) \ [x2] ([x2] ∧ [x3]) \ [x1] [x1] ∧ [x2] ∧ [x3] Finite free LSBAs: ﬁnite case

LSBAX — the free LSBA over the generating set X . Let X = {1, 2,..., n}.

I Let Y ⊆ X be a non-empty subset and y ∈ Y .

I (X , Y , y)– pointed non-empty subset of X .

I Atoms: ^ e(X , Y , y) = y ∧ {y : y ∈ Y \ ∨{y : y ∈ X \ Y }.

By left normality, this is well deﬁned and e(X , Y , y) = e(X , Z, z) if and only if Y = Z and y = z.

I Atoms are in a bijection with pointed non-empty subsets of X .

I Atomic D-classes = atoms of S/D are in a bijection with non-empty subsets of X . Free LSBAs: structure

Theorem (J. Leech and GK, 2015) Let n ≥ 1. 1. n (n) (n) (n) (1) 2 3 n LSBAn ' 2 × 3L × 4L × · · · × (n + 1)L , (n) (n) (n) (n) |LSBAn| = 2 1 3 2 4 3 ... (n + 1) n .

2. The number of atoms of LSBAn equals n n n n 1 + 2 + ··· + (n − 1) + n = n2n−1. 1 2 n − 1 n

n 3. The center of LSBAn is isomorphic to 2 . Let X be inﬁnite.

4. LSBAX is atomless.

5. The center of LSBAX equals {0}. FREE SKEW BOOLEAN INTERSECTION ALGEBRAS

Remark: This should be compared with D. Ellerman’s talk. Models for ’partition logic’ Free LSBIAs: ﬁnite case, idea

LSBIAX — the free LSBIA over the generating set X . I Recall: in free LSBAs atoms are encoded by pointed non-empty subsets of X .

I In free LSBIAs: atoms are encoded by pointed partitions of non-empty subsets of X ! The construction: an example X = {x1, x2, x3}. We deﬁne:

I For α = x1|x2 and the marked block {x2}:

e(X , α, {x2}) = (x2 ∧ x1) \ (x3 ∨ (x1 u x2)).

I For α = x1|x2|x3 and the marked block {x2}:

e(X , α, {x2}) = (x2 ∧x1 ∧x3)\((x1 ux2)∨(x1 ux3)∨(x2 ux3)).

I For α = x1x2|x3 and the marked block {x1, x2}:

e(X , α, {x1, x2}) = ((x1 u x2) ∧ x3) \ (x1 u x2 u x3).

I For α = x1x2x3 (that is, one three-element block) and the marked block {x1, x2, x3}:

e(X , α, {x1, x2, x3}) = x1 u x2 u x3. The construction (GK, 2016) Let (X , α, A) be a pointed partition of a non-empty subset of X . Let α = {A1,..., Ak } and Y = dom(α). Deﬁne

e(X , α, A) = p \ (∨Q), where p = (uA)∧(∧{uAi : 1 ≤ i ≤ k}) = (uA)∧(uA1)∧· · ·∧(uAk ) and

Q = (X \ Y ) ∪ {u(Ai ∪ Aj ): 1 ≤ i < j ≤ k}. The containment order on partitions.

Let Z ⊆ Y . We deﬁne:( Z, α) (Y , β) if

dom(α) ⊆ dom(β) ⊆ dom(α) ∪ (Y \ Z)

and for any x, y ∈ dom(α): x α y if and only if x β y.

Example

Let Y = {x1, x2, x3, x4}, Z = {x1, x2, x3}. Then

(Z, x1|x2) (Y , x1|x2), (Y , x1|x2|x4), (Y , x1x4|x2), (Y , x1|x2x4). BUT:

(Z, x1|x2) 6 (Y , x1|x2x3|x4), (Y , x1x2), (Y , x1x2|x4), etc. Free LSBIAs: ﬁnite case, structure

Theorem (GK, 2016)

1. Atoms of LSBIAn are in bijection with pointed partitions of non-empty subsetes {1, 2,..., n}.

2. LSBIAn has precisely Bn+1 − 1 atomic D-classes. Thus B −1 LSBIAn/D' 2 n+1 .

3. Atoms of (LSBIAn)/D are in a bijections with partitions of non-empty subsets {1, 2,..., n}. 2n−1 4. The center of LSBIAn is isomorphic to 2 which is isomorphic to GBAn which is isomorphic to the maximum commutative quotient of LSBIAn. Free LSBIAs: ﬁnite case, structure, continued

The nth Bell number Bn - the number of partitions of an n n-element set. The Stirling number of the second kind k - the number of partitions of an n-element set into k non-empty subsets, n ≥ 1, 1 ≤ k ≤ n. Theorem (GK, 2016)

n+1 {n+1} {n+1} { 2 } 3 n+1 1. LSBIAn ' 2 × 3L × · · · × (n + 1)L , {n+1} {n+1} {n+1} |LSBIAn| = 2 2 3 3 ··· (n + 1) n+1 .

2. LSBIAn has precisely Bn+2 − 2Bn+1 atoms. Free LSBAs: countable generating set Inﬁnite partition tree: level i: partitions of subsets of [i] = {1, 2,..., i}. ([i], α) is connected with ([i + 1], β) iﬀ ([i], α) ([i + 1], β).

∅ ∅

1 ∅ 1

1|2 12 1 2 ∅ 1, 2

1|2|3 13|2 1|23 1|2 12|3 12312 1|3 13 1 2|3 23 2 3 ∅ 1, 2, 3

This tree is Cantorian (G. Michon, Les Cantors r´eguliers,1985, J. Pearson, J. Bellissard, 2009), so that its boundary is homeomorphic to the Cantor set. Basis of topology: sets [v] = all paths passing through v, where v ranges over the vertices. Thus

LSBIAN/D' GBAN.