Constructing quantales and their modules from monoidal
... One might ask how closely the three main examples of quantales and their modules are related to each other. For example, is it necessa.ry to directly verify that the objects in question are in fact quantales or modules, or do they arise from a single construction? This question was partly answered ( ...
... One might ask how closely the three main examples of quantales and their modules are related to each other. For example, is it necessa.ry to directly verify that the objects in question are in fact quantales or modules, or do they arise from a single construction? This question was partly answered ( ...
The Theory of Polynomial Functors
... 2 Of course, as we discovered in due time, polynomial functors provide much too weak a notion. Over more general base rings than Z, they are subsumed by numerical functors. ...
... 2 Of course, as we discovered in due time, polynomial functors provide much too weak a notion. Over more general base rings than Z, they are subsumed by numerical functors. ...
Constructible Sheaves, Stalks, and Cohomology
... Since étale maps are open, such a local ring should factor through the local ring of x. So we can concentrate on the case of a local scheme S = Spec(R), with x = s the closed point. By the characterization of étale morphisms to spectra of fields, in case R = k is a field, we should expect that the l ...
... Since étale maps are open, such a local ring should factor through the local ring of x. So we can concentrate on the case of a local scheme S = Spec(R), with x = s the closed point. By the characterization of étale morphisms to spectra of fields, in case R = k is a field, we should expect that the l ...
AUTOMORPHISM GROUPS AND PICARD GROUPS OF ADDITIVE
... modules N . Then Pic(C) is naturally isomorphic to AutA (C). Such construction agrees with the classical definition of Picard groups if C is the whole category of modules or the category of projective modules, and we will have the classical Picard group Pic A of the ring A in these cases, as develop ...
... modules N . Then Pic(C) is naturally isomorphic to AutA (C). Such construction agrees with the classical definition of Picard groups if C is the whole category of modules or the category of projective modules, and we will have the classical Picard group Pic A of the ring A in these cases, as develop ...
HIGHER HOMOTOPY OF GROUPS DEFINABLE IN O
... Λ((e, x, s), t) = λ00 ((e, (x, s)), t − s) for t > s which is easily proved continuous, moreover for t ≤ s we have p ◦ Λ((e, x, s), t) = p ◦ λ0 ((e, (x, s)), s − t) = g 0 ((x, s), s − t) = f (x, t) and similarly for s < t; and finally Λ((e, x, s), s) = λ0 ((e, (x, s)), 0) = e. ...
... Λ((e, x, s), t) = λ00 ((e, (x, s)), t − s) for t > s which is easily proved continuous, moreover for t ≤ s we have p ◦ Λ((e, x, s), t) = p ◦ λ0 ((e, (x, s)), s − t) = g 0 ((x, s), s − t) = f (x, t) and similarly for s < t; and finally Λ((e, x, s), s) = λ0 ((e, (x, s)), 0) = e. ...
PDF
... For an ordinal α, 0α -computably categorical structures are usually called ∆0α+1 categorical structures. An equivalent definition of ∆0β -categoricity, which also works for limit ordinals β, is that a computably presentable structure is ∆0β -categorical if any two of its computable presentations are ...
... For an ordinal α, 0α -computably categorical structures are usually called ∆0α+1 categorical structures. An equivalent definition of ∆0β -categoricity, which also works for limit ordinals β, is that a computably presentable structure is ∆0β -categorical if any two of its computable presentations are ...
Recursive Domains, Indexed Category Theory and Polymorphism
... in terms of that of the codomain, whilst the former retains the property of final functors that v ...
... in terms of that of the codomain, whilst the former retains the property of final functors that v ...
18 Divisible groups
... where ∆, j are given by ∆(x) = (x, −x) and j(a, b) = a + b. The condition on f, g is equivalent to saying that ∆(A ∩ B) lies in the kernel of f ⊕ g : A ⊕ B → G. Consequently, we get an induced map on the quotient: f +g : A+B → G. Theorem 18.6. A group G is divisible if and only if it satisfies the f ...
... where ∆, j are given by ∆(x) = (x, −x) and j(a, b) = a + b. The condition on f, g is equivalent to saying that ∆(A ∩ B) lies in the kernel of f ⊕ g : A ⊕ B → G. Consequently, we get an induced map on the quotient: f +g : A+B → G. Theorem 18.6. A group G is divisible if and only if it satisfies the f ...
Amenability for dual Banach algebras
... this impression. We first show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus finite-dimensional. We then develop two notions of amenability—Connes amenability and strong Connes amenability—which take the w∗ -topology on dual Banach algebras into acc ...
... this impression. We first show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus finite-dimensional. We then develop two notions of amenability—Connes amenability and strong Connes amenability—which take the w∗ -topology on dual Banach algebras into acc ...
Three Lectures on Automatic Structures
... this, there are well-studied classes of structures (such as abelian groups, Boolean algebras, linearly ordered sets, algebras, and graphs) for which it is impossible to give simple isomorphism invariants. Thus, in general, there is no solution to the isomorphism problem. The elementary equivalence p ...
... this, there are well-studied classes of structures (such as abelian groups, Boolean algebras, linearly ordered sets, algebras, and graphs) for which it is impossible to give simple isomorphism invariants. Thus, in general, there is no solution to the isomorphism problem. The elementary equivalence p ...
Lecture Notes for Math 614, Fall, 2015
... Before we begin the systematic development of our subject, we shall look at some very simple examples of problems, many unsolved, that are quite natural and easy to state. Suppose that we are given polynomials f and g in C[x], the polynomial ring in one variable over the complex numbers C. Is there ...
... Before we begin the systematic development of our subject, we shall look at some very simple examples of problems, many unsolved, that are quite natural and easy to state. Suppose that we are given polynomials f and g in C[x], the polynomial ring in one variable over the complex numbers C. Is there ...
lecture notes
... Such expressions are not very fortunate, and according the the Italian mathematician F. Severi today’s Modern Algebra will soon become the Classic Algebra of tomorrow. Actually, when one compares, e.g, the XXI Century Algebra with the XVI Century Algebra, and one eliminates differences common to all ...
... Such expressions are not very fortunate, and according the the Italian mathematician F. Severi today’s Modern Algebra will soon become the Classic Algebra of tomorrow. Actually, when one compares, e.g, the XXI Century Algebra with the XVI Century Algebra, and one eliminates differences common to all ...
higher algebra
... 10.5 Finitely generated modules over a PID . . . . . . . . . . . . . 327 10.6 The uniqueness of the Invariant Factors. . . . . . . . . . . . . 331 10.7 Applications of the Theorem on PID Modules. . . . . . . . . . 333 10.7.1 Classification of Finite Abelian Groups. . . . . . . . . . 333 10.7.2 The R ...
... 10.5 Finitely generated modules over a PID . . . . . . . . . . . . . 327 10.6 The uniqueness of the Invariant Factors. . . . . . . . . . . . . 331 10.7 Applications of the Theorem on PID Modules. . . . . . . . . . 333 10.7.1 Classification of Finite Abelian Groups. . . . . . . . . . 333 10.7.2 The R ...
![arXiv:1510.01797v3 [math.CT] 21 Apr 2016 - Mathematik, Uni](http://s1.studyres.com/store/data/003897650_1-5601c13b5f003f1ac843965b9f00c683-300x300.png)
arXiv:1510.01797v3 [math.CT] 21 Apr 2016 - Mathematik, Uni
... vector space A◦ of A• is the subspace of A∗ consisting of those linear forms whose kernel contains a cofinite ideal. This extension is no longer an inverse equivalence to the (restricted) dual algebra functor, but only a left adjoint. In addition, A• underlies a Hopf algebra if A does. Attempts have ...
... vector space A◦ of A• is the subspace of A∗ consisting of those linear forms whose kernel contains a cofinite ideal. This extension is no longer an inverse equivalence to the (restricted) dual algebra functor, but only a left adjoint. In addition, A• underlies a Hopf algebra if A does. Attempts have ...
Fascicule 1
... Equational Logic in the category Alg Σ: its aim is to describe the fully invariant congruence generated by (u, v), whereas the coequalizer rule takes the congruence that (u, v) generates for granted. We therefore present our main logic, called the Quasi-Equational Logic, without the coequalizer rule ...
... Equational Logic in the category Alg Σ: its aim is to describe the fully invariant congruence generated by (u, v), whereas the coequalizer rule takes the congruence that (u, v) generates for granted. We therefore present our main logic, called the Quasi-Equational Logic, without the coequalizer rule ...
On the Domination and Total Domination Numbers of Cayley Sum
... In [6], Lev proved that if S is a subset of a finite Abelian group G, then Cay+ (G, S) is connected if and only if S is not contained in a coset of a proper subgroup of G, except, perhaps, for the non-zero coset of a subgroup of index 2. In the following theorem, we find a bound for the total domina ...
... In [6], Lev proved that if S is a subset of a finite Abelian group G, then Cay+ (G, S) is connected if and only if S is not contained in a coset of a proper subgroup of G, except, perhaps, for the non-zero coset of a subgroup of index 2. In the following theorem, we find a bound for the total domina ...
Projective ideals in rings of continuous functions
... coz/i Π coz/2 = 0 , each ht must be the characteristic function of the corresponding supp/*. Consequently, feL — h2 is a continuous function that is 1 on pos/ and —1 on negf. Thus, the two sets are completely separated and (/, |/|) = (/). If one now defines feC(R) by f(x) = xy then, by [6, 2H] and P ...
... coz/i Π coz/2 = 0 , each ht must be the characteristic function of the corresponding supp/*. Consequently, feL — h2 is a continuous function that is 1 on pos/ and —1 on negf. Thus, the two sets are completely separated and (/, |/|) = (/). If one now defines feC(R) by f(x) = xy then, by [6, 2H] and P ...