On the compact-regular coreflection of a stably compact locale
... We begin by recalling the definitions and facts concerning nuclei needed in the development that follows (see e.g. [12, Section II-2]) and establishing some terminology and notation. We then consider Scott continuous nuclei. In the category of topological spaces and continuous maps, the natural noti ...
... We begin by recalling the definitions and facts concerning nuclei needed in the development that follows (see e.g. [12, Section II-2]) and establishing some terminology and notation. We then consider Scott continuous nuclei. In the category of topological spaces and continuous maps, the natural noti ...
Monotone complete C*-algebras and generic dynamics
... A C -algebra is separably representable when it has an isometric -representation on a separable Hilbert space. As a consequence of more general results, I showed that if a monotone complete factor is separably representable (as a C -algebra) then it is a von Neumann algebra. So, in these circumstan ...
... A C -algebra is separably representable when it has an isometric -representation on a separable Hilbert space. As a consequence of more general results, I showed that if a monotone complete factor is separably representable (as a C -algebra) then it is a von Neumann algebra. So, in these circumstan ...
A Cut-Invariant Law of Large Numbers for Random Heaps
... CNRS Laboratory IRIF (UMR 8243), University Paris Diderot - Paris 7, Paris, France ...
... CNRS Laboratory IRIF (UMR 8243), University Paris Diderot - Paris 7, Paris, France ...
Locally Finite Constraint Satisfaction Problems
... linear identities iff each of its finite subtemplates admit such polymorphisms. This proves a series of results analogous to finite CSP theory: for example, if a locally finite template T has a majority polymorphism or a Maltsev polymorphism then specific polynomial time algorithms correctly solve f ...
... linear identities iff each of its finite subtemplates admit such polymorphisms. This proves a series of results analogous to finite CSP theory: for example, if a locally finite template T has a majority polymorphism or a Maltsev polymorphism then specific polynomial time algorithms correctly solve f ...
STS: A STRUCTURAL THEORY OF SETS
... be considered as an actual existence in its own respect. These two sides of a structure might not match each other perfectly: maybe an actual object is \richer" than its potential counterpart. But, following Cantor (or at least my interpretation of his views), I believe that these two aspects are i ...
... be considered as an actual existence in its own respect. These two sides of a structure might not match each other perfectly: maybe an actual object is \richer" than its potential counterpart. But, following Cantor (or at least my interpretation of his views), I believe that these two aspects are i ...
Compactifications and Function Spaces
... is such that f −1 (r) is infinite and not compact, then there is some x ∈ αX \ X so that x ∈ cl(f −1 (r)). This shows that whether a function f extends to a compactification αX depends on whether there are enough “points” in αX \ X to capture the “behavior” of f at “∞”. This is very close to the pre ...
... is such that f −1 (r) is infinite and not compact, then there is some x ∈ αX \ X so that x ∈ cl(f −1 (r)). This shows that whether a function f extends to a compactification αX depends on whether there are enough “points” in αX \ X to capture the “behavior” of f at “∞”. This is very close to the pre ...
NON-SPLIT REDUCTIVE GROUPS OVER Z Brian
... discriminant. Thus, by Minkowski’s theorem that every number field K 6= Q has a ramified prime, Spec(Z) has no nontrivial connected finite étale covers. Hence, π1 (Spec(Z)) = 1, so all Z-tori are split. Every Chevalley group G is a Z-model of its split connected reductive generic fiber over Q, and ...
... discriminant. Thus, by Minkowski’s theorem that every number field K 6= Q has a ramified prime, Spec(Z) has no nontrivial connected finite étale covers. Hence, π1 (Spec(Z)) = 1, so all Z-tori are split. Every Chevalley group G is a Z-model of its split connected reductive generic fiber over Q, and ...
A survey of totality for enriched and ordinary categories
... category A is said to be total if it is locally small - so that we have a Yoneda embedding Y : A -> [A°p, Set] where Set is the category of small sets - and if this embedding Y admits a left adjoint Z. Totality for these ordinary categories has been further investigated by Tholen [22], Wood [24 ...
... category A is said to be total if it is locally small - so that we have a Yoneda embedding Y : A -> [A°p, Set] where Set is the category of small sets - and if this embedding Y admits a left adjoint Z. Totality for these ordinary categories has been further investigated by Tholen [22], Wood [24 ...
Ideals
... we will use to prove the two most important results of this section. Let us begin with the fact that prime is a notion stronger than maximal. You may want to recall what is the general result for an arbitrary commutative ring and compare with respect to the case of a ring of integers (in general, ma ...
... we will use to prove the two most important results of this section. Let us begin with the fact that prime is a notion stronger than maximal. You may want to recall what is the general result for an arbitrary commutative ring and compare with respect to the case of a ring of integers (in general, ma ...
Class Field Theory - Purdue Math
... Let K be a number field, A = OK , and p a prime ideal of K. The localization Ap is a discrete valuation ring whose normalized valuation we denote by ordp or νp . To describe this valuation more explicitly, let π be a generator of the unique maximal ideal of Ap . Then every x ∈ K ∗ can be uniquely wr ...
... Let K be a number field, A = OK , and p a prime ideal of K. The localization Ap is a discrete valuation ring whose normalized valuation we denote by ordp or νp . To describe this valuation more explicitly, let π be a generator of the unique maximal ideal of Ap . Then every x ∈ K ∗ can be uniquely wr ...
Vector bundles and torsion free sheaves on degenerations of elliptic
... all vector bundles over an affine line are trivial. Therefore to define a vector bundle over P1 one only has to prescribe its rank r and a gluing matrix M ∈ GL(r, k[z, z −1 ]). Changing bases in free modules over k[z] and k[z −1 ] corresponds to the transformations M 7→ T −1 M S, where S and T are i ...
... all vector bundles over an affine line are trivial. Therefore to define a vector bundle over P1 one only has to prescribe its rank r and a gluing matrix M ∈ GL(r, k[z, z −1 ]). Changing bases in free modules over k[z] and k[z −1 ] corresponds to the transformations M 7→ T −1 M S, where S and T are i ...
Haar Measures for Groupoids
... the space of complex-valued continuous functions with compact support, in which we work for the duration of the paper. We also use several measure theory definitions to build to the definition and idea of a Haar measure on a topoogical group. The remainder of the chapter is devoted to the rigorous c ...
... the space of complex-valued continuous functions with compact support, in which we work for the duration of the paper. We also use several measure theory definitions to build to the definition and idea of a Haar measure on a topoogical group. The remainder of the chapter is devoted to the rigorous c ...
Nearrings whose set of N-subgroups is linearly ordered
... If there exists k ∈ Ker ψ \ Im ψ r , then there is c ∈ N \ Im ψ and j < r such that k = ψ j (c). {0} = N ∗ 0 = N ∗ ψ j+1 (c) = ψ j+1 (N ∗ c) = ψ j+1 (N ) shows that ψ j+1 = 0 and Ker ψ j+1 = N = Ker ψ r . We note that r is the smallest integer such that ψ r = 0. Necessarily, Im ψ r−1 ⊆ Ker ψ. If the ...
... If there exists k ∈ Ker ψ \ Im ψ r , then there is c ∈ N \ Im ψ and j < r such that k = ψ j (c). {0} = N ∗ 0 = N ∗ ψ j+1 (c) = ψ j+1 (N ∗ c) = ψ j+1 (N ) shows that ψ j+1 = 0 and Ker ψ j+1 = N = Ker ψ r . We note that r is the smallest integer such that ψ r = 0. Necessarily, Im ψ r−1 ⊆ Ker ψ. If the ...
