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Projective Quantales and Quantale Modules

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Projective Quantales and Quantale Modules Masaryk University Faculty of Science Projective Quantales and Quantale Modules Master's Thesis Radek Slesingerˇ Advisor: doc. RNDr. Jan Paseka, CSc. Brno, May 2008 Prohl´aˇsen´ı Prohlaˇsuji,ˇzejsem tuto diplomovou pr´acivytvoˇrilsamostatnˇea veˇsker´e pouˇzit´ezdroje a literaturu v pr´aciˇr´adnˇecituji a uv´ad´ımv seznamu. V Brnˇedne 16. kvˇetna2008 Acknowledgement I would like to thank my advisor Dr. Jan Paseka for directing me during my work on this thesis and providing me with many valuable suggestions and comments that enabled me to complete the thesis. Contents Introduction 2 1 Preliminaries 3 1.1 Sup-lattices . 3 1.2 Quantales . 4 1.3 Quantale Modules . 5 1.4 Projectivity . 6 1.5 Supercontinuous and Superalgebraic Lattices . 7 2 Projective Quantales 11 2.1 Free Quantales . 11 2.2 Supercontinuous Quantales . 12 2.3 Several Elementary Properties of Projective Quantales . 13 2.4 More General View at Projective Quantales . 15 2.4.1 K-flat Projective Quantales . 15 2.4.2 K-coherent quantales . 18 2.4.3 Concrete Instances . 19 3 Projective Quantale Modules 21 3.1 Free Modules . 21 3.2 Projective Unital Modules . 22 3.2.1 Decomposition of Unital Modules . 22 3.2.2 Projective Unital Supercontinuous Modules . 25 3.3 Relation to Results Obtained for S-posets . 27 A Computations of Sup-lattice Endomorphisms 29 A.1 Results . 29 A.2 Source Code . 31 Bibliography 34 1 Introduction The notion of quantale, which designates a complete lattice equipped with associative binary multiplication distributing over arbitrary joins, was used for the first time by C. J. Mulvey in 1986. However, multiplicative ordered structures were studied already in 1930s in the form of lattices of ring ide- als. During the previous two decades, quantales and quantale modules have found their application in areas of logic, functional analysis, and computer science. Because of a certain analogy between quantales and rings, or quantale and ring modules, respectively, properties like injectivity and projectivity suggest to be studied. Since 1950s a large theory of projective and injective modules over rings has been developed while the problematics of projective rings seems untouched. The case of injective quantales has already been solved in the article [13], showing that the only injective quantale is the trivial one. The authors of the article also attempted to describe projective quantales, however, the proof of the main theorem of the article contained a mistake and even the statement was not valid as can be easily shown. Nevertheless, part of the results can be still used. Also projectivity of sets equipped with an action of a semigroup, both unordered and partially ordered case, has already been studied, and this thesis will attempt to adapt a part of these existing results for some special types of quantale modules. The thesis consists of three chapters. The first chapter introduces basic algebraic and lattice-theoretic notions and general concepts further needed in the following parts. Chapter Two deals with projectivity of quantales. It presents the contemporary knowledge of quantale projectivity and intro- duces several conditions that projective quantales need to satisfy. The third chapter turns to projectivity of modules over quantales. Following a part of an article that deals with projectivity of certain partially ordered sets, it provides a characteristics of projectivity of a specific class of unital quantale modules. 2 Chapter 1 Preliminaries This chapter introduces the notions of sup-lattices, quantales and quantale modules (much more on them can be found e.g. in [10] and [4]), the concept of projectivity, and several lattice-theoretic concepts (making use of [3] and [2]) that will be used in both following chapters. 1.1 Sup-lattices Definition. A sup-lattice (also complete join-semilattice) is a partially or- dered set L where each subset X ⊆ L has its supremum in L. In this thesis the symbol 1 will denote the top element and 0 will stand for the bottom one. Every sup-lattice is obviously also a complete lattice; the difference between the categories lies in morphisms. Definition. A map f : L ! K between two sup-lattices is called a sup- lattice homomorphism if it preserves arbitrary joins, that for any X ⊆ L it satisfies _ _ f X = ff(x) j x 2 Xg: The category of sup-lattices with sup-lattice homomorphisms will be denoted by SLat. When L is a sup-lattice, the set of all sup-lattice endomorphisms L ! L with pointwise ordering f ≤ g () f(x) ≤ g(x) for all x 2 L and pointwise computed joins forms a sup-lattice again. Let X and Y be posets and f : X ! Y , g : Y ! X be monotone maps. We say that f is left adjoint to g (and g is right adjoint to f) if f(x) ≤ y () x ≤ g(y). It is not difficult to check that f preserves all existing suprema, g preserves all existing infima, g is injective iff f is onto, and the adjoint maps are given uniquely. A right adjoint f∗ : K ! L can be assigned to every W sup-lattice homomorphism f : L ! K as f∗(y) = fx 2 L j f(x) ≤ yg. 3 1.2 Quantales Definition. A quantale Q is a sup-lattice equipped with associative binary multiplication · bound to suprema by the following rules. For any a 2 Q, X ⊆ Q, it satisfies: _ _ a · X = fa · x j x 2 Xg; _ _ X · a = fx · a j x 2 Xg: A quantale Q is called unital if (Q; ·) is a monoid. The multiplicative unit will be denoted by e. Provided that A and B are subsets of a quantale, A · B shall stand for the set fa · b j a 2 A; b 2 Bg. Well known examples of quantales include the following: • ideals of a ring (inclusion ordering and standard multiplication of ide- als), • binary relations on a set (inclusion ordering and composition of rela- tions), • for a sup-lattice L, Q(L) = ff : L ! L j f is a homomorphismg (with pointwise ordering and map composition). Definition. A quantale homomorphism is a sup-lattice morphism f : Q ! R between two quantales that in addition preserves multiplication. For any X ⊆ Q and x; y 2 Q: _ _ f X = ff(x) j x 2 Xg; f(x · y) = f(x) · f(y): If Q and R are unital quantales and f(eQ) = eR, f is called a unital homo- morphism. Within the text Quant will denote the category of quantales with quan- tale homomorphisms and UnQuant will stand for the category of unital quantales with unital quantale homomorphisms. Definition. An element x of a quantale Q is called • (strictly) right-sided if x·1 ≤ x (x·1 = x). (Strictly) left-sided elements are defined by analogy; • two-sided if it is both right- and left-sided; • idempotent if x2 = x. 4 When all elements of a quantale possess some of the properties mentioned above, we use the same name for the quantale. A frame is a quantale in which multiplication is the meet of two elements. It holds that frames are exactly two-sided idempotent quantales: as any elements x; y are two sided, x · y ≤ x and x · y ≤ y, thus x · y ≤ x ^ y. For the contrary, x ^ y = (x ^ y) · (x ^ y) ≤ (x ^ y) · y ≤ x · y. 1.3 Quantale Modules Definition. Let Q be a quantale. A sup-lattice M equipped with a left action ·: Q × M ! M is called a left Q-module if the following equalities hold for any q; r 2 Q, R ⊆ Q, m 2 M, and L ⊆ M: _ _ R · m = fr · m j r 2 Rg; _ _ q · L = fq · l j l 2 Lg; (q · r) · m = q · (r · m): If Q is unital and e · m = m for any m 2 M we talk about left unital Q-module. Right modules can be defined in an analogous way. For the rest of the thesis, a module will stand for a left module. Here are several instances of quantale modules: • Any (unital) quantale is a (unital) module over itself. Also the subset of all right-sided elements of a quantale forms a left module and vice versa. • Any sup-lattice L can be turned into a module as well by taking the quantale Q(L) and multiplication f · x = f(x). • As a generalization of the previous example, the set of all maps from a set X to a sup-lattice S forms a Q(S)-module with multiplication f · ' = f ◦ '. • If Q is a quantale, Q(Q) can be viewed as Q-module with multiplica- tion (q · f)(x) = q · f(x). Definition. When M and N are Q-modules, a module homomorphism is a sup-lattice homomorphism f : M ! N that preserves multiplication by elements of Q. That is, for any q 2 Q, m 2 M, X ⊆ M it holds that: _ _ f X = ff(x) j x 2 Xg; f(q · m) = q · f(m): For a given quantale Q, Q-Mod will denote the category of Q-modules with module homomorphisms. 5 1.4 Projectivity This section shall provide general concepts regarding projectivity that are not specific for either quantales or modules. Definition. An object P of a category C is called projective with respect to a class M of morphisms if for any morphism g : A ! B that belongs to M and any morphism f : P ! B there exists a morphism h: P ! A such that g ◦ h = f as illustrated in the diagram. P ~ @@ f h ~ @@ ~ @@ ~ @ A g / B In the language of category theory this says that the functor Hom(P; −): C! Set maps M-morphisms to epimorphisms, i.e.
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