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HABILITATION THESIS Pavel R˚Uzicka Representations Of HABILITATION THESIS Pavel R˚uˇziˇcka Representations of Distributive Algebraic Lattices Department of Algebra Prague, August 2017 i To My Parents Contents Introduction v Basic concepts xvii Chapter 1. Lifting of distributive lattices by locally matricial algebras 1 1. Introduction 2 2. Notation and terminology 3 3. The category aux revised 4 4. The correspondence B: dsem ! bool revised 6 5. Representation of distributive lattices revised 8 6. Lifting of the functor C with respect to Idc 13 7. Existence and non-existence of liftings 16 Chapter 2. Distributive congruence lattices of congruence-permutable algebras 19 1. Introduction 20 2. Preliminaries 22 3. V-distances of type n 23 4. An even weaker uniform refinement property 24 = 5. Failure of WURP in Conc F, for F free bounded lattice 27 6. Representing distributive algebraic lattices with at most @1 compact elements as submodule lattices of modules 31 7. Representing distributive algebraic lattices with at most @1 compact elements as normal subgroup lattices of groups 32 8. Representing distributive algebraic lattices with at most @0 compact elements as `-ideal lattices of `-groups 33 9. Functorial representation by V-distances of type 2 35 Chapter 3. Free trees and the optimal bound in Wehrung's solution of the Congruence Lattice Problem 41 1. Introduction 42 2. Diluting functors 43 3. Free Distributive Extension is Diluting 45 4. Free Trees 47 5. The optimal bound in Wehrung's Theorem 48 Chapter 4. Countable chains of distributive lattices and dimension groups 53 iii iv CONTENTS 1. Introduction 54 2. Notation and terminology 55 3. The construction 56 Chapter 5. Construction and realization of some wild refinement monoids 63 1. Introduction 64 2. Preliminaries 65 3. Partial H-maps and their applications 66 4. Non-cancellative refinement monoids 70 5. The monoid A2n, B2n, and C2n 75 6. Some linear algebra 82 7. The example of Bergman and Goodearl 85 8. Representing the monoids B2n 91 Chapter 6. Boolean ranges of Banaschewski functions 95 1. Introduction 96 2. Preliminaries 97 3. The lattice 100 4. A Banaschewski function on S 101 5. The counter-example 102 6. Representing S in a subspace lattice 105 7. Non existence of 3-frames 108 8. Coordinatizability 111 9. Maximal Abelian regular subalgebras 115 Acknowledgements 117 Bibliography 119 Introduction v vi INTRODUCTION All monoids in the thesis are supposed to be commutative. The stable equivalence on a monoid M, denoted by ∼s, is the least congruence on M such that the quotient M s := M= ∼s is cancellative. The congruence is defined by x ∼s y if there exists z 2 M such that x + z = y + z, for all x; y 2 M. The correspondence M 7! M s extends canonically to a functor that we denote by (−)s. Directed5 Abelian groups (−)∗ (−)+ CancellativeO monoids (−)s Monoids Id r Id h_; 0i-semilatticesH (−)c Id ) Ö w Algebraic lattices Figure 1. Partially ordered Abelian groups, monoids, and algebraic lattices There is an universal map (−)∗ : M ! M ∗ sending monoids to Abelian groups. Moreover the algebraic order on a monoid M induces a partial order on the target Abelian group M ∗; such that the image of the monoid corresponds to the positive cone of M ∗. The construction of the partially ordered Abelian group M ∗ for a given monoid M is an analogy of the con- struction of the field of fractions of a given commutative ring. We consider the set of formal differences between pairs of elements from M and an equiv- alence relation, say ∼∗, on them. The equivalence is given by x − y ∼∗ z − u provided that there is w 2 M such that x + u + w = z + y + w. The map (−)∗ is dermined by x 7! [ x − 0 ]∼∗ , x 2 M. Again, the correspondence is canonically functorial. Notice that the partially ordered Abelian group M ∗ is directed, that is, it is, as a group, generated by the positive cone. It is straightforward to see that this is equivalent to the partial order on M ∗ being upwards directed. Let G+ := fp 2 G j 0 ≤ pg denote the positive cone of a partially or- dered Abelian group G. Observing that an order preserving homomorphism G ! H maps the positive cone G+ of G into the positive cone H+ of H, we see that there is a functor (−)+ from the category of partially ordered + Abelian groups to monoids. Moreover, the composition (−) ◦ (−)∗ is nat- urally equivalent to the functor ∼s. INTRODUCTION vii We denote by ≍ the least congruences on M such that M=≍ is a h_; 0i- semilattice and we set r(M) := M=≍. As in the previous cases, the corre- spondence M ! r(M) extends a functor. The ideal lattice Id(S) of a h_; 0i-semilattice S is an algebraic lattice and, conversely, compact elements of an algebraic lattice L form a h_; 0i- semilattice, denoted by Lc. Both the correspondences extend to functors that are inverse to each other (up to obvious natural equivalences). Here are more ideal-type functors to consider. Firstly, the functor that assigns to a monoid M the algebraic lattice Id(M) of all o-ideals of M. Secondly, the functor G 7! Id(G+) which assigns to a directed Abelian group the algebraic lattice of all convex subgroups of G. All the introduced functors are depicted in Figure 1. Note that the diagram of functors is commutative (up to natural equivalences). Directed3 interpolation groups (−)∗ (−)+ Cancellative refinementO monoids (−)s Refinement monoids Id r Id Distributive h_H ; 0i-semilattices (−)c Id + Ö u Distributive algebraic lattices Figure 2. Directed interpolation groups, refinement monoids, and distributive algebraic lattices We will be interested in structures that are mapped by the ideal functor Id to algebraic lattices that are distributive. Starting from the bottom of Figure 2, these are distributive h_; 0i-semilattice (cf. [27, Section II.5]). In- deed, a h_; 0i-semilattice is distributive if and only if Id(S) is an algebraic distributive lattice. Next we consider the class of refinement monoids, i.e, the conical monoids that satisfy the Riesz refinement property. The maxi- mal semilattice quotient r(M) of a refinement monoid M is a distributive h_; 0i-semilattice and the lattice Id(M) of all o-ideals of M is distributive (cf. [25, lemma 2.4]. Finally, a directed Abelian group G is an interpolation group if and only if the positive cone G+ is a refinement monoid [21, Prop. 2.1]. In particular, the lattice Id(G) of all ideals (i.e, convex subgroups) of a directed interpolation group is again distributive. There are more structures in the picture as we tried to depict in Figure 3. Given a ring R, we denote by V (R) the monoid of all isomorphism classes viii INTRODUCTION Dir. interpol.5 @ groups jjjj K0 jj jjjj jjjj L j Regular rings − (−)+ ( )∗ Ô V Compl. mod. lattices Cancel.jjj5 ref. monoids − jj ( )sj jjjj jjjj Refinement monoids r Idc w Dist. 3h_; 0iH -semilattices Conc Id (−)c Id NId ' Con Ö x / Dist. alg. lattices Figure 3. Regular rings, refinement monoids, and distribu- tive h_; 0i-semilattices of finitely generated projective right R-modules with addition derived from direct sums. If the ring R is (Von-Neumann) regular, the monoid V (R) sat- isfies the Riesz refinement property (see [22, Corollary 2.7]). The partially ordered Abelian group V (R)∗, denoted by K0(R), is called the Grothendieck group of R. When we limit ourselves unital rings, it is appropriate to assign to a ring R a partially ordered Abelian group K0(R) with an order-unit corresponding to the isomorphism class [ R ] and study the category of par- tially ordered Abelian groups with order units (cf. [22, Chapter 15]). If the ring R is regular, then K0(R) is a directed interpolation group. We denote by L(R) the h_; 0i-semilattice of all right finitely gener- ated ideals of a ring R. For a regular ring, the h_; 0i-semilattice L(R) is closed under finite meets, therefore L(R) forms a lattice [22, Theorem 2.3]. Moreover, the lattice L(R) is modular and sectionally complemented (complemented if R is with an unit element). Congruences of sectionally complemented modular lattices correspond to their neutral ideals (see [27, Section III.3.10]). In particular, if R is a regular ring, then the lattice Con(L(R)) is isomorphic to the lattice NId(L(R)) of all neutral ideals of L(R). By [78, Lemma 4.2], an ideal of the lattice L(R) (for a regular ring R) is neutral if and only if it contains with each aR all principal ideals bR with bR ' aR. It follows that Con(L(R)) ' NId(L(R)) ' Id(R) (see [78, Lemma 4.3]), and so, the lattice Id(R) of two- sided ideals of a regular ring R is distributive. Moreover, combining [78, Corollary 4.4 and Proposition 4.6] we get the isomorphisms Conc(L(R)) ' INTRODUCTION ix r(V (R)) ' Idc(R) of distributive h_; 0i-semilattices, for every regular ring R. We have seen that a distributive algebraic lattice that is isomorphic to the lattice of two-sided ideals of a regular ring is at the same time isomor- phic to the congruence lattice of a modular sectionally complemented lattice. This brings a connection with the Congruence lattice problem, whether every distributive algebraic lattice is isomorphic to the congruence lattice of a lat- tice. The conjecture has an interesting history (see [86]) and remained open four over sixty years until the counter-example was found by F. Wehrung [83]. We will discuss the Congruence Lattice Problem in detail in Chapter 3.
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