1. Some Aspects of Quasi-Stone Algebras, II Solutions and Simplifications of Some Problems from Sankappanavar and Sankappanavar Jonathan David Farley, D.Phil. lattice.theory@gmail.com Institut für Algebra Johannes Kepler Universität Linz A-4040 Linz Österreich Joint work with Sara-Kaja Fischer Universität Bern Bern, Switzerland Buy! This is a subliminal message.

2. Definition A quasi-Stone algebra (QSA) is a bounded distributive lattice L with a unary operator ‘ such that: 0’=1 and 1’=0 (x  y)’=x’  y’ (x  y’)’=x’  y’’ x  x’’ x’  x’’=1 *** Jonathan D. Farley V Sara-Kaja Fischer

7. A Natural Example of a Quasi- Stone Algebra *** Jonathan D. Farley V Sara-Kaja Fischer

8. A Natural Example of a Quasi- Stone Algebra *** Jonathan D. Farley V Sara-Kaja Fischer 

9. A Natural Example of a Quasi- Stone Algebra *** Jonathan D. Farley V Sara-Kaja Fischer  The 1-element quasi-Stone algebra

10. A Natural Example of a Quasi- Stone Algebra *** Jonathan D. Farley V Sara-Kaja Fischer  The 1-element quasi-Stone algebra (Also known as the Farley algebra.)

11. Example of a Fuzzy Quasi-Stone Algebra *** Jonathan D. Farley V Sara-Kaja Fischer

12. Example of a Fuzzy Quasi-Stone Algebra *** Jonathan D. Farley V Sara-Kaja Fischer 

13. Example of a Fuzzy Quasi-Stone Algebra *** Jonathan D. Farley V Sara-Kaja Fischer  The 0.999999….-element quasi-Stone algebra

14. 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 x=0. This makes L a quasi-Stone algebra, which we call special. *** Jonathan D. Farley V Sara-Kaja Fischer

15. 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 x=0. This makes L a quasi-Stone algebra, which we call special. *** Jonathan D. Farley V Sara-Kaja Fischer

16. Special Quasi-Stone Algebras 0’=1 and 1’=0 (x  y)’=x’  y’ (x  y’)’=x’  y’’ x  x’’ x’  x’’=1 *** Jonathan D. Farley V Sara-Kaja Fischer

17. Problems of Sankappanavar and Sankappanavar from 1993 1.Subdirectly irreducibles a.A problem about simple algebras b.A problem about injectives 2.Amalgamation a.Refutation of a claim of Gaitán b.Coproducts *** Jonathan D. Farley V Sara-Kaja Fischer

18. 1.SUBDIRECTLY IRREDUCIBLES *** Jonathan D. Farley V Sara-Kaja Fischer

19. Finite Subdirectly Irreducible Quasi-Stone Algebras Let m and n be natural numbers. Let A n be the Boolean lattice with n atoms. Let  m be the Boolean lattice with m atoms with a new top element adjoined. Let Q mn be  m  A n viewed as a special quasi- Stone algebra. *** Jonathan D. Farley V Sara-Kaja Fischer

20. Finite Subdirectly Irreducible Quasi-Stone Algebras Let m and n be natural numbers. Let A n be the Boolean lattice with n atoms. Let  m be the Boolean lattice with m atoms with a new top element adjoined. Let Q mn be  m  A n viewed as a special quasi-Stone algebra. *** Jonathan D. Farley V Sara-Kaja Fischer Q 20 Q 11

21. Finite Subdirectly Irreducible Quasi-Stone Algebras Theorem (Sankappanavar and Sankappanavar, 1993). The set of finite subdirectly irreducible quasi-Stone algebras is {Q mn : m,n  0}. The set of finite simple quasi-Stone algebras is {Q 0n : n  0}. *** Jonathan D. Farley V Sara-Kaja Fischer Q 20 Q 11 Q 02

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

23. 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 q U *** Jonathan D. Farley V Sara-Kaja Fischer

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

25. 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 234567  *** Jonathan D. Farley V Sara-Kaja Fischer

26. 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 234567  The clopen up-sets are the finite sets not containing  and P itself. *** Jonathan D. Farley V Sara-Kaja Fischer

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

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

29. Gaitán’s Representation for Quasi- Stone Algebras *** Jonathan D. Farley V Sara-Kaja Fischer P

30. Gaitán’s Representation for Quasi- Stone Algebras *** Jonathan D. Farley V Sara-Kaja Fischer a b c  {b}{b} {a,b}{a,b} {a,b,c}{a,b,c} {b,c}{b,c} {c}{c} P L

31. Gaitán’s Representation for Quasi- Stone Algebras 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 b c  {b}{b} {a,b}{a,b} {a,b,c}{a,b,c} {b,c}{b,c} {c}{c} P L

32. Gaitán’s Representation for Quasi- Stone Algebras 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 b c  {b}{b} {a,b}{a,b} {a,b,c}{a,b,c} {b,c}{b,c} {c}{c} P L x

33. Fischer’s 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)=  Y:={p  P : p  y for some y  Y}. *** Jonathan D. Farley V Sara-Kaja Fischer

34. 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. *** Jonathan D. Farley V Sara-Kaja Fischer  1 2 3 4567 

35. P5 Definition. 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:B  I such that h=g  f. A class of algebras has enough injectives if every algebra can be embedded into an injective algebra. *** Jonathan D. Farley V Sara-Kaja Fischer A B I f g h

36. 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 Q 01 have enough injectives? Solution (***). No: this follows almost from the definitions!! This variety does not have the congruence extension property. *** Jonathan D. Farley V Sara-Kaja Fischer A B I f g h Q 01 Q 10

37. P5 Problem (Sankappavar and Sankappanavar, 1993). Does the variety generated by Q 01 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. *** Jonathan D. Farley V Sara-Kaja Fischer A B I f g h A B I f g A/θA/θ

38. 2.AMALGAMATION *** Jonathan D. Farley V Sara-Kaja Fischer

39. 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:L  M 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.d  f=e  g. *** Jonathan D. Farley V Sara-Kaja Fischer L N M K g e d f

40. The Amalgamation Property Definition. A variety has the amalgamation property if every algebra is an amalgamation base. *** Jonathan D. Farley V Sara-Kaja Fischer L N M K g e d f

41. 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 Q 01 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. *** Jonathan D. Farley V Sara-Kaja Fischer

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

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

44. Coproducts A coproduct of two objects A and B in a category consists of an object C and morphisms f:A  C and g:B  C such that, whenever D is an object and h:A  D, k:B  D morphisms, there is a unique morphism e:C  D such that e  f=h and e  g=k. *** Jonathan D. Farley V Sara-Kaja Fischer AB C D fg e h k

45. 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. *** Jonathan D. Farley V Sara-Kaja Fischer M N A

46. 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 k  K and l  L. Then the co-product of K and L in the category of quasi- Stone algebras is N/θ. *** Jonathan D. Farley V Sara-Kaja Fischer

47. 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. *** Jonathan D. Farley V Sara-Kaja Fischer K L

48. 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 D. Farley V Sara-Kaja Fischer