Some Aspects of QuasiStone Algebras, II Solutions and Simplifications of Some Problems from Sankappa

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Some Aspects of Quasi-Stone Algebras,
IISolutions and Simplifications of Some Problems
from Sankappanavar and Sankappanavar
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• Jonathan David Farley, D.Phil.
• lattice.theory_at_gmail.com
• Institut für Algebra
• Johannes Kepler Universität Linz
• A-4040 Linz
• Österreich
• Joint work with
• Sara-Kaja Fischer
• Universität Bern
• Bern, Switzerland

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Definition

• A quasi-Stone algebra (QSA) is a bounded
  distributive lattice L with a unary operator
  such that
• 01 and 10
• (x?y)x?y
• (x?y)x?y
• x?x
• x ? x1

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A Natural Example of a Quasi-Stone Algebra
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A Natural Example of a Quasi-Stone Algebra
?
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A Natural Example of a Quasi-Stone Algebra
?
The 1-element quasi-Stone algebra
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A Natural Example of a Quasi-Stone Algebra
?
The 1-element quasi-Stone algebra
(Also known as the Farley algebra.)
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Example of a Fuzzy Quasi-Stone Algebra
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Example of a Fuzzy Quasi-Stone Algebra
?
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Example of a Fuzzy Quasi-Stone Algebra
?
The 0.999999.-element quasi-Stone algebra
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Special Quasi-Stone Algebras

• Let L be a bounded distributive lattice. For any
  x in L, let x be 0 if x?0 and 1 if x0.
• This makes L a quasi-Stone algebra, which we call
  special.

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Special Quasi-Stone Algebras

• Let L be a bounded distributive lattice. For any
  x in L, let x be 0 if x?0 and 1 if x0.
• This makes L a quasi-Stone algebra, which we call
  special.

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Special Quasi-Stone Algebras

• 01 and 10
• (x?y)x?y
• (x?y)x?y
• x?x
• x ? x1

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Problems of Sankappanavar and Sankappanavar from
1993

• Subdirectly irreducibles
• a. A problem about simple algebras
• b. A problem about injectives
• Amalgamation
• a. Refutation of a claim of Gaitán
• b. Coproducts

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1. Subdirectly irreducibles
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Finite Subdirectly Irreducible Quasi-Stone
Algebras

• Let m and n be natural numbers. Let An be the
  Boolean lattice with n atoms.
• Let Âm be the Boolean lattice with m atoms with a
  new top element adjoined.
• Let Qmn be Âm ? An viewed as a special
  quasi-Stone algebra.

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Finite Subdirectly Irreducible Quasi-Stone
Algebras

• Let m and n be natural numbers. Let An be the
  Boolean lattice with n atoms.
• Let Âm be the Boolean lattice with m atoms with a
  new top element adjoined.
• Let Qmn be Âm ? An viewed as a special
  quasi-Stone algebra.

Q11
Q20
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Finite Subdirectly Irreducible Quasi-Stone
Algebras

• Theorem (Sankappanavar and Sankappanavar, 1993).
  The set of finite subdirectly irreducible
  quasi-Stone algebras is Qmn m,n?0.
• The set of finite simple quasi-Stone algebras is
  Q0n n?0.

Q11
Q02
Q20
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P7

• Problem (Sankappanavar and Sankappanavar,
  1993). Are there non-Boolean simple quasi-Stone
  algebras?
• Solution (Celani and Cabrer, 2009). Yes, using a
  complicated example of Adams and Beazer.
• An uncomplicated example can be constructed using
  Priestley duality.

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Priestley Duality for Bounded Distributive
Lattices

• A partially ordered topological space P is
  totally order-disconnected if, whenever p and q
  are in P and p is not less than or equal to q,
  then there exists a clopen up-set U containing p
  but not q.

q
U
p
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Priestley Duality for Bounded Distributive
Lattices

• A Priestley space is a compact, totally
  order-disconnected partially ordered topological
  space.
• Every compact, Hausdorff totally-disconnected
  space (i.e., the space of prime ideals of a
  Boolean algebra) is a Priestley space.

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A Priestley Space

• Let P be the set 1,2,3,?? where the open
  sets are
• any set U not containing ?
• any co-finite set V containing ?.

1
2
3
4
5
6
7
?
?
?
?
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A Priestley Space

• Let P be the set 1,2,3,?? where the open
  sets are
• any set U not containing ?
• any co-finite set V containing ?.

1
2
3
4
5
6
7
?
?
?
?
The clopen up-sets are the finite sets not
containing ? and P itself.
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Priestley Duality for Bounded Distributive
Lattices

• Theorem (Priestley). The category of bounded
  distributive lattices 0,1-homomorphisms is
  dually equivalent to the category of Priestley
  spaces continuous, order-preserving maps.

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A Topological Representation of Quasi-Stone
Algebras

• Theorem (Gaitán). Let P be a Priestley space and
  let E be an equivalence relation on P with the
  property that
• Equivalence classes are closed.
• For every clopen up-set U of P,
• E(U)p?P pEu for some u ? U
• is a clopen up-set of P and P\E(U) is a clopen
  up-set of P.
• Then the lattice of clopen up-sets of P with the
  operator U P\E(U) is a quasi-Stone algebra,
  and every quasi-Stone algebra is isomorphic to
  such an algebra.

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Gaitáns Representation for Quasi-Stone Algebras
P
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Gaitáns Representation for Quasi-Stone Algebras
P
L
a,b,c
b
c
a,b
b,c
a
b
c
?
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Gaitáns Representation for Quasi-Stone Algebras
P
L
a,b,c
b
c
a,b
b,c
a
b
c
?

• P\E(?)P\?a,b,c
• P\E(b)P\a,bc
• P\E(a,b)P\a,bc
• P\E(c)P\ca,b
• P\E(b,c)P\a,b,c ?
• P\E(a,b,c)P\a,b,c ?

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Gaitáns Representation for Quasi-Stone Algebras
P
L
a,b,c
b
c
a,b
b,c
a
b
c
?

• P\E(?)P\?a,b,c
• P\E(b)P\a,bc
• P\E(a,b)P\a,bc
• P\E(c)P\ca,b
• P\E(b,c)P\a,b,c ?
• P\E(a,b,c)P\a,b,c ?

x
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Fischers Representation for Congruences of
Quasi-Stone Algebra

• Let L be a quasi-Stone algebra with Priestley
  dual space P and equivalence relation E.
• Fischer proved that every congruence of L
  corresponds to a closed subset Y of P such that
• E(Y)?Yp?P p?y for some y ? Y.

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Fischers Representation for Congruences of
Quasi-Stone Algebra

• The quasi-Stone algebra L corresponding to this
  Priestley space P with EPxP is simple
• Any non-empty closed subset Y corresponding to a
  congruence must contain all maximal elements by
  Fischers criterion E(Y)?Y.
• Hence Y must contain ? too. But L is not Boolean
  by Nachbins theorem.
• Thus Fischers criterion yields a solution to the
  problem P7 of Sankappanavar and Sankappanavars
  1993 paper Celani and Cabrers 2009 example was
  the first, but it is much more complicated.

1
2
3
4
5
6
7
?
?
?
?
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P5

• Definition. An algebra I is injective if for all
  algebras A, B and morphisms
• fA ? B and hA ? I,
• where f is an embedding, there exists a morphism
  gB?I such that hg?f.
• A class of algebras has enough injectives if
  every algebra can be embedded into an injective
  algebra.

h
A
I
f
g
B
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P5

• A class of algebras has enough injectives if
  every algebra can be embedded into an injective
  algebra.
• Problem (Sankappavar and Sankappanavar, 1993).
  Does the variety generated by Q01 have enough
  injectives?
• Solution (). No this follows almost from the
  definitions!!
• This variety does not have the congruence
  extension property.

h
A
I
Q01
Q10
f
g
B
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P5

• Problem (Sankappavar and Sankappanavar, 1993).
  Does the variety generated by Q01 have enough
  injectives?
• Solution (). Any variety with enough
  injectives has the congruence extension property
  Let A be a subalgebra of B and let fA ? B be the
  embedding. Let ? be a congruence of A. Embed A/?
  into an injective I. Then the kernel of g
  extends ?. QED.

h
A
A
I
A/?
I
g
f
g
f
B
B
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2. Amalgamation
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Amalgamation Bases

• Definition. An algebra L is an amalgamation base
  with respect to a class if, for all M and N in
  the class and embeddings fL?M and gL ? N, there
  exist an algebra K in the class and embeddings
  dM ? K and eN ? K such that
• d ? fe ? g.

L
M
f
g
d
N
K
e
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The Amalgamation Property

• Definition. A variety has the amalgamation
  property if every algebra is an amalgamation
  base.

L
M
f
g
d
N
K
e
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P3

• Problem (Sankappanavar and Sankappanavar, 1993).
  Investigate the amalgamation property for the
  subvarieties of quasi-Stone algebras.
• Gaitán (2000) stated that the proper subvarieties
  of quasi-Stone algebras containing Q01 do not
  have the amalgamation property. He used a
  difficult universal algebraic result of C.
  Bergman and McKenzie, which in turn depends on
  previous results of Bergman and results of
  Taylor.

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P3

• Fischer found intricate combinatorial proofs that
  the amalgamation property fails for some
  subvarieties, proofs that we feel we can extend
  to all subvarieties except the varieties of Stone
  algebras (which have AP).
• Note that the kinds of arguments that work for
  pseudocomplemented distributive lattices do not
  work here because we do not have the congruence
  extension property.
• This is not turning the crank.

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Gaitáns Claim

• Gaitán also claimed that the class of finite
  quasi-Stone algebras has the amalgamation
  property.
• We discovered, however, that Gaitáns published
  proof is wrong. It does not simply have a gap
  it is wrong.
• Hence it remains an open problem to show if the
  class of (finite) quasi-Stone algebras has the
  amalgamation property.
• A first step is to show that any two non-trivial
  quasi-Stone algebras can be embedded into some
  quasi-Stone algebra.

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Coproducts

• A coproduct of two objects A and B in a category
  consists of an object C and morphisms fA?C and
  gB?C such that, whenever D is an object and hA
  ? D, kB?D morphisms, there is a unique morphism
  eC? D such that e ? fh and e ? gk.

A
B
f
g
C
h
k
e
D
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Free Quasi-Stone Extension of a Bounded
Distributive Lattice

• Theorem (Gaitán 2000). Let M be a bounded
  distributive lattice. There exists a quasi-Stone
  algebra N containing L as a 0,1-sublattice,
  which is generated by M as a quasi-Stone algebra
  and is such that every lattice homomorphism from
  M to a quasi-Stone algebra A extends to a
  quasi-Stone algebra homomorphism from N to A.

M
N
A
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Coproducts of Quasi-Stone Algebras

• Theorem (). Let K and L be non-trivial
  quasi-Stone algebras. Let M be the coproduct of K
  and L in the category of bounded distributive
  lattices. Let N be the free quasi-Stone extension
  of M.
• Let ? be the congruence of N generated by all
  pairs
• (kK,kN) and (lL,lN)
• for all k?K and l?L.
• Then the co-product of K and L in the category of
  quasi-Stone algebras is N/?.

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Coproducts of Quasi-Stone Algebras

• The coproduct of K and L in the category of
  bounded distributive lattices has 9 elements.
  The coproduct in the category of quasi-Stone
  algebras has 324 elements.

L
K
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Summary and Next Steps

• Fischers representation of the duals of
  congruences gives us a simpler solution to the
  1993 problem of Sankappanavar and Sankappanavar
  than Celani and Cabrers 2009 example.
• We solved Sankappanavar and Sankappanavars 1993
  problem about injectives (which is trivial one
  can apply a result of Kollár 1980).
• We showed that Gaitans proof that the class of
  finite quasi-Stone algebras has the amalgamation
  property is wrong.
• We proved coproducts exist in the category of
  quasi-Stone algebras.
• What are the Priestley duals of principal
  congruences?
• What is the Priestley dual of a coproduct of
  quasi-Stone algebras?