jonathan david farley, d.phil. lattice.theory@gmail institut für algebra

Some Aspects of Quasi-Stone Algebras, II

Solutions and Simplifications of Some Problems from Sankappanavar and Sankappanavar

Jonathan David Farley, [email protected]

Institut für AlgebraJohannes Kepler Universität Linz

A-4040 LinzÖsterreich

Joint work withSara-Kaja FischerUniversität 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:

0’=1 and 1’=0(xy)’=x’y’(xy’)’=x’y’’xx’’x’ x’’=1

*** Jonathan D. Farley V Sara-Kaja Fischer

A Natural Example of a Quasi-Stone Algebra




The 1-element quasi-Stone algebra



The 1-element quasi-Stone algebra

(Also known as the Farley algebra.)

Example of a Fuzzy Quasi-Stone Algebra


Example of a Fuzzy Quasi-Stone Algebra


The 0.999999….-element quasi-Stone algebra

Special Quasi-Stone Algebras

• Let L be a bounded distributive lattice. For any x in L, let x’ be 0 if x0 and 1 if x=0.

• This makes L a quasi-Stone algebra, which we call special.


Special Quasi-Stone Algebras0’=1 and 1’=0(xy)’=x’y’(xy’)’=x’y’’xx’’x’ x’’=1


Problems of Sankappanavar and Sankappanavar from 1993

1. Subdirectly irreduciblesa. A problem about simple algebrasb. A problem about injectives

2. Amalgamationa. Refutation of a claim of Gaitánb. Coproducts


1. SUBDIRECTLY IRREDUCIBLES


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.



• 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.


Q20

Q11


Theorem (Sankappanavar and Sankappanavar, 1993). The set of finite subdirectly irreducible quasi-Stone algebras is {Qmn : m,n0}.

The set of finite simple quasi-Stone algebras is {Q0n : n0}.


Q20

Q11 Q02

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.


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.

p

qU



• 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.


A Priestley SpaceLet 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


A Priestley SpaceLet 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.



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


A Topological Representation of Quasi-Stone AlgebrasTheorem (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):={pP : 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.


Gaitán’s Representation for Quasi-Stone Algebras


P



a

bc

{b}

{a,b}

{a,b,c}

{b,c}

{c}

P L


P\E()=P\={a,b,c}P\E({b})=P\{a,b}={c}P\E({a,b})=P\{a,b}={c}P\E({c})=P\{c}={a,b}P\E({b,c})=P\{a,b,c}= P\E({a,b,c})=P\{a,b,c}=


a

bc

{b}

{a,b}

{a,b,c}

{b,c}

{c}

P L


P\E()=P\={a,b,c}P\E({b})=P\{a,b}={c}P\E({a,b})=P\{a,b}={c}P\E({c})=P\{c}={a,b}P\E({b,c})=P\{a,b,c}= P\E({a,b,c})=P\{a,b,c}= *** Jonathan D. Farley V Sara-Kaja Fischer

a

bc

{b}

{a,b}

{a,b,c}

{b,c}

{c}

P L

x

Fischer’s Representation for Congruences of Quasi-Stone

AlgebraLet 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)=Y:={pP : py for some y Y}.


Fischer’s Representation for Congruences of Quasi-Stone Algebra

The quasi-Stone algebra L corresponding to this Priestley space P with E=PxP is simple:

Any non-empty closed subset Y corresponding to a congruence must contain all maximal elements by Fischer’s criterion: E(Y)=Y.

Hence Y must contain too. But L is not Boolean by Nachbin’s theorem.Thus Fischer’s criterion yields a solution to the problem P7 of Sankappanavar

and Sankappanavar’s 1993 paper; Celani and Cabrer’s 2009 example was the first, but it is much more complicated.


1 2 3 4 5 6 7

P5Definition. An algebra I is injective if for all algebras A, B and morphisms

f:A B and h:A I,where f is an embedding, there exists a morphism g:BI such that

h=gf.A class of algebras has enough injectives if every algebra can be

embedded into an injective algebra.


A

B

I

f g

h

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.


A

B

I

f g

h

Q01Q10

P5Problem (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 f:A B be the embedding. Let θ be a congruence of A. Embed A/θ into an injective I. Then the kernel of g extends θ. QED.


A

B

I

f g

hA

B

I

f g

A/θ

2. AMALGAMATION


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 f:LM and g:L N, there exist an algebra K in the class and embeddings d:M K and e:N K such that

d f=e g.


L

N

M

K

g

e

d

f

The Amalgamation Property

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


L

N

M

K

g

e

d

f

P3Problem (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.


P3Fischer 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”.*** Jonathan D. Farley V Sara-Kaja Fischer

Gaitán’s Claim• Gaitán also claimed that the class of finite quasi-Stone

algebras has the amalgamation property.• We discovered, however, that Gaitán’s 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.


Coproducts

• A coproduct of two objects A and B in a category consists of an object C and morphisms f:AC and g:BC such that, whenever D is an object and h:A D, k:BD morphisms, there is a unique morphism e:C D such that e f=h and e g=k.


A BC

D

f g

eh k

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.


MN

A

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 θ be the congruence of N generated by all pairs(k’K,k’N) and (l’L,l’N)

for all kK and lL.Then the co-product of K and L in the category of quasi-

Stone algebras is N/θ.


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.


K

L

Summary and Next Steps• Fischer’s representation of the duals of congruences gives us a

simpler solution to the 1993 problem of Sankappanavar and Sankappanavar than Celani and Cabrer’s 2009 example.

• We solved Sankappanavar and Sankappanavar’s 1993 problem about injectives (which is trivial: one can apply a result of Kollár 1980).

• We showed that Gaitan’s “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?


jonathan david farley, d.phil. lattice.theory@gmail institut für algebra

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