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; ^; _; n; 0) of type (2; 2; 2; 0) such that for any a; b; c; d 2 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 n b) ^ b = b ^ (a n b) = 0, (a n b) _ (a ^ b ^ a) = a = (a ^ b ^ a) _ (a n 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# = fy 2 S : y ≤ xg forms a Boolean lattice. IF AND ONLY IF S is symmetric, has 0, the operation ^ is normal and S=D is a Boolean lattice. 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# = fy 2 S : y ≤ xg 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 2 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 = f0;:::; kg (^ is left zero multiplication). This generalizes the Boolean algebra 2 = f0; 1g. 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; ^; _; n; 0) be an SBA. I S has intersections if the meet of a and b with respect to ≤ exists for any a; b 2 S. I a u b | the intersection of a and b. I If S is finite a u b, always exists. I If (S; ^; _; n; 0) has intersections, (S; ^; _; u; n; 0) is a skew Boolean intersection algebra (SBIA). Proposition (Bignall and Leech) Let (S; ^; _; n; u; 0) be an algebra of type (2; 2; 2; 2; 0). Then it is an SBIA if and only if (S; ^; _; n; 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 2 S the set a = fb 2 S : b ≤ ag 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, f0; 1g.