摘要:

The usual proofs of the well-known set-theoretical theorem “Given one-one maps f: A → B and g:B → A, there exists a one-one onto map h:A → B” actually produce a map h:A → B contained in the relation f U g−1. Considering Tarski's Fixpoint Theorem as the implicit basic ingredient of such proofs. We examine several classical proofs/starting with Dedekind (1887), and illuminate their common feature by means of the categorical notion of a natural fixpoint. We consider a categorical form (CBT) of the theorem (with h ⊆ f Ug−1) in a variety of contexts, obtaining some examples of categories where CBT holds and others where it fails. Among other results we prove for a toposE, (1) CBT holds ifEis Boolean, and conversely ifEhas a natural number object; (2) The Axiom of Choice inEimplies a dual version of CBTI and conversely ifEhas splitting supports and a natural number object.

ISSN:1607-3606    

DOI:10.1080/16073606.1986.9632106

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

New proofs of universal coefficient theorems for generalized homology theories (cf. ∮ 2, ∮ 3) including L. G. Brown's result, relating Brown-Douglas-Fillmore'sExt(X) with complex K-theory are presented. They are all based on a theorem asserting the existence of a chain functor for a generalized homology theory (cf. ∮ 1), which was originally designed for the construction of strong homology theories on strong shape categories.

ISSN:1607-3606    

DOI:10.1080/16073606.1986.9632107

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

The nil radical, N(M) of a Γ-ring M was defined by Coppage and Luh [3], and shown by Groenewald [4] to be a special radical. We define s-prime ideals of M and show that N(M) is equal to the intersection of the s-prime ideals of M. If R is a ring, the nil radical of R considered as a Γ-ring with Γ = R is equal to the upper nil radical of R. We also give a sufficient condition for the equality N(R)* = N(M), where R is the right operator ring of M, and N(R) is its upper nil radical.

ISSN:1607-3606    

DOI:10.1080/16073606.1986.9632108

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

This paper generalizes the concept of a power alge bra to that of a power structure, and gives three application of power structures to logic.

ISSN:1607-3606    

DOI:10.1080/16073606.1986.9632109

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

Our objective was to embed schemes of finite type over a field k in a suitably small Cartesian closed category. Two types of globalized versions of an ind-affine scheme were proposed: locally ind-affine ringed spaces and ind-schemes obtained by taking the inductive limit of closed subschemes of a locally ind-affine ringed space in ringed spaces. First in ε case some reasonably general conditions implying that translations, basic open subsets and closed subsets of an ind-affine scheme are again ind-affine schemes were obtained. Certain immersive properties of locally ind-affine ringed spaces are shown. As an adjunct we then determine a class of locally ind-affine ringed spaces which since they patch appropriately are ind-schemes. A restriction of locally ind-affine ringed spa1 leads to the category of locally ind-affine schemes (containing the category of schemes of finite type over k) which is see1 to be Cartesian closed with respect to the contravariant variable. Possible extensions to the covariant variable are studied.

ISSN:1607-3606    

DOI:10.1080/16073606.1986.9632110

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

An abelian p-group C is said to be essentially finitely indecomposable (efi) if given any decomposition of G as the direct sum of a family of subgroups, there exists a positive integer n such that all but at moat a finite number of subgroups of this family are bounded by n. We look at examples and related questions. We prove that a reduced abelian p-group G is efi if and only if G modulo its elements of infinite height is efi. In the proof of this we obtain the following result which is of independent interest: Let A be a reduced p-group with a summand K such that K is a direct sum of cyclic groups. Let B be a basic subgroup of A. Then B contains a subgroup C such that C is a summand of A and the final rank of C is equal to the final rank of K.

ISSN:1607-3606    

DOI:10.1080/16073606.1986.9632111

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

Astandard extension(resp.standard completion) is a function Z assigning to each poset P a (closure) system ZP of subsets such that x ⋚ y iff x belongs to every Z ε ZP with y ε Z. A poset P is Z -completeif each Z ε 2P has a join in P. A map f: P → P′ is Z—continuousif f−1[Z′] ε ZP for all Z′ ε ZP′, and a Z—morphismif, in addition, for all Z ε ZP there is a least Z′ ε ZP′ with f[Z] ⊆ Z′. The standard extension Z iscompositiveif every map f: P → P′ with {x ε P: f(x) ⋚ y′} ε ZP for all y′ ε P′ is Z -continuous. We show that any compositive standard extension Z is the object part of a reflector from IPZ, the category of posets and Z -morphisms, to IRZ, the category of Z -complete posets and residuated maps. In case of a standard completion Z, every Z -continuous map is a Z -morphism, and IR2 is simply the category of complete lattices and join—preserving maps. Defining in a suitable way so-calledZ -embeddingsand morphisms between them, we obtain for arbitrary standard extensions Z an adjunction between IPZand the category of Z -embeddings. Many related adjunctions, equivalences and dualities are studied and compared with each other. Suitable specializations of the function 2 provide a broad spectrum of old and new applications.

ISSN:1607-3606    

DOI:10.1080/16073606.1986.9632112

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

Given a pair of short exact sequences

ISSN:1607-3606    

DOI:10.1080/16073606.1986.9632113

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

In this paper the topological methods introduced by Kaplansky and the theory of linear compactifications are used to prove a result classifying the maximal immediate extensions of a valued field. Results on the existence of a complete discrete rank n valued field of characteristic 0 with prescribed residue class field of characteristic p > 0 are discussed. By applying results of Endler and Ribenboim the existence of a valued field of characteristic 0 and having prescribed residue field of characteristic p > 0 when the value group has finite rank but need not be discrete is demonstrated.

ISSN:1607-3606    

DOI:10.1080/16073606.1986.9632114

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor

摘要:

Various familiar concepts of algebraic categories (such as primitive and quasiprimitive categories of algebras as well as varietal, monadic, regular monadic, and regular categories) are contrasted with each other and with the new notion of essentially algebraic categories.

ISSN:1607-3606    

DOI:10.1080/16073606.1986.9632115

出版商:Taylor & Francis Group    

年代:1986

数据来源: Taylor