Monomorphisms and epimorphisms in pro-categories

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. We give characterizations of monomorphisms (respectively, epimorphisms) in pro-category pro-C, provided C has direct sums (respectively, pushouts).Let E(C) (respectively, M(C)) be the subcategory of C whose morphisms are epimorphisms (respectively, monomorphisms) of C. We give conditions in some categories C for an object X of pro-C to be isomorphic to an object of pro-E(C) (respectively, pro-M(C)).A related class of objects of pro-C consists of X such that there is an epimorphism X→P∈Ob(C) (respectively, a monomorphism P∈Ob(C)→X). Characterizing those objects involves conditions analogous (respectively, dual) to the Mittag-Leffler property. One should expect that the object belonging to both classes ought to be stable. It is so in the case of pro-groups. The natural environment to discuss those questions are balanced categories with epimorphic images. The last part of the paper deals with that question in pro-homotopy.     

Pure morphisms in pro-categories

Pure epimorphisms in categories pro-C, which essentially are just inverse limits of split epimorphisms in C, were recently studied by J. Dydak and F.R.R. del Portal in connection with Borsuk’s problem of descending chains of retracts of ANRs. We prove that pure epimorphisms are regular epimorphisms whenever C has weak finite limits, or pullbacks, or copowers. This improves the results of the above paper, and the results of the present authors on pure subobjects in accessible categories. We also turn to pure monomorphisms in pro-C, essentially just inverse limits of split monomorphisms in C, and prove that they are regular monomorphisms whenever C has finite products or pushouts.     

Strong pro-fibrations and ANR objects in pro-categories

The notions of pro-fibration and approximate pro-fibration for morphisms in the pro-category pro-Top of topological spaces were introduced by S. Mardeši? and T.B. Rushing. In this paper we introduce the notion of strong pro-fibration, which is a pro-fibration with some additional property, and the notion of ANR object in pro-Top, which is approximately an ANR-system, and we consider the full subcategory ANR of pro-Top whose objects are ANR objects. We prove that the category ANR satisfies most of the axioms for fibration category in the sense of H.J. Baues if fibrations are strong pro-fibrations and weak equivalences are morphisms inducing isomorphisms in the pro-homotopy category pro-H(Top) of topological spaces. We give various applications. First of all, we prove that every shape morphism is represented by a strong pro-fibration. Secondly, the fibre of a strong pro-fibration is well defined in the category ANR, and we obtain an isomorphism between the pro-homotopy groups of the base and total systems of a strong pro-fibration, and hence obtain the pro-homotopy sequence of a strong pro-fibration. Finally, we also show that there is a homotopy decomposition in the category ANR.     

Isomorphisms preserving invariants

Let V and W be finite dimensional real vector spaces and let G ì GL(V){G \subset {\rm GL}(V)} and H ì GL(W){H \subset {\rm GL}(W)} be finite subgroups. Assume for simplicity that the actions contain no reflections. Let Y and Z denote the real algebraic varieties corresponding to \mathbbR[V]^G{\mathbb{R}[V]^G} and \mathbbR[W]^H{\mathbb{R}[W]^H}, respectively. If V and W are quasi-isomorphic, i.e., if there is a linear isomorphism L : V → W such that L sends G-orbits to H-orbits and L ⁻¹ sends H-orbits to G-orbits, then L induces an isomorphism of Y and Z. Conversely, suppose that f : Y → Z is a germ of a diffeomorphism sending the origin of Y to the origin of Z. Then we show that V and W are quasi-isomorphic, This result is closely related to a theorem of Strub [8], for which we give a new proof. We also give a new proof of a result of Kriegl et al. [3] on lifting of biholomorphisms of quotient spaces.     

Foundational Isomorphisms in Continuous Differentiation Theory

We provide an elementary proof of existence for the Foundational Isomorphism in each of the categories of convergence spaces, compactly generated topological spaces and sequential convergence spaces. This isomorphism embodies the germ of differentiation and its inverse the germ of integration.     

Isomorphisms between Artin-Schreier towers

We give a method for efficiently computing isomorphisms between towers of Artin-Schreier extensions over a finite field. We find that isomorphisms between towers of degree over a fixed field can be computed, composed, and inverted in time essentially linear in . The method relies on an approximation process.

     

Isomorphisms between Jacobson graphs

Let \(R\) be a commutative ring with a non-zero identity and \(\mathfrak {J}_R\) be its Jacobson graph. We show that if \(R\) and \(R'\) are finite commutative rings, then \(\mathfrak {J}_R\cong \mathfrak {J}_{R'}\) if and only if \(|J(R)|=|J(R')|\) and \(R/J(R)\cong R'/J(R')\) . Also, for a Jacobson graph \(\mathfrak {J}_R\) , we obtain the structure of group \(\mathrm {Aut}(\mathfrak {J}_R)\) of all automorphisms of \(\mathfrak {J}_R\) and prove that under some conditions two semi-simple rings \(R\) and \(R'\) are isomorphic if and only if \(\mathrm {Aut}(\mathfrak {J}_R)\cong \mathrm {Aut}(\mathfrak {J}_{R'})\) .     

Isomorphisms of Linear Semigroups

A linear semigroup is a subsemigroup of the semigroup of all endomorphisms of a vector space over a (not necessarily commutative) field. In this note it is shown that every isomorphism of linear semigroups that contain all rank-one-operators is induced by a semilinear bijection of the corresponding vector spaces, unless these vector spaces have dimension 1.     

ISOMORPHISMS OF CIRCULANT DIAGAPHS

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Isomorphisms of one-relator semigroup algebras

All isomorphisms between comutative semigroup algebras K[S] and K[T] are found when K is a field and S, T have two generators subject to a single homogeneous defining relation     

Isomorphisms of global representation rings

In this paper, we study properties of isomorphisms of global rings that preserve the standard bases.     

Isomorphisms of Brin-Higman-Thompson groups

Let m, m′, r, r′, t, t′ be positive integers with r, r′ ? 2. Let \(\mathbb{L}_r \) denote the ring that is universal with an invertible 1×r matrix. Let \(M_m (\mathbb{L}_r^{ \otimes t} )\) denote the ring of m × m matrices over the tensor product of t copies of \(\mathbb{L}_r \) . In a natural way, \(M_m (\mathbb{L}_r^{ \otimes t} )\) is a partially ordered ring with involution. Let \(PU_m (\mathbb{L}_r^{ \otimes t} )\) denote the group of positive unitary elements. We show that \(PU_m (\mathbb{L}_r^{ \otimes t} )\) is isomorphic to the Brin-Higman-Thompson group tV _r,m ; the case t=1 was found by Pardo, that is, \(PU_m (\mathbb{L}_r )\) is isomorphic to the Higman-Thompson group V _r,m . We survey arguments of Abrams, Ánh, Bleak, Brin, Higman, Lanoue, Pardo and Thompson that prove that t′V _r′,m′ ≌ tV _r,m if and only if r′ =r, t′ =t and gcd(m′, r′?1) = gcd(m, r?1) (if and only if \(M_{m'} (\mathbb{L}_{r'}^{ \otimes t'} )\) and \(M_m (\mathbb{L}_r^{ \otimes t} )\) are isomorphic as partially ordered rings with involution).     

Isomorphisms ofp-adic group rings

SupposeP is the ring ofp-adic integers,G is a finite group of orderp ⁿ , andPG is the group ring ofG overP. IfV _p (G) denotes the elements ofPG with coefficient sum one which are of order a power ofp, it is shown that the elements of any subgroupH ofV _p (G) are linearly independent overP, and if in additionH is of orderp ⁿ , thenPGPH. As a consequence, the lattice of normal subgroups ofG and the abelianization of the normal subgroups ofG are determined byPG.     

Isomorphisms Between Quasi-Banach Algebras

In this paper, the author proves the Hyers-Ulam-Rassias stability of homo-morphisms in quasi-Banach algebras. This is used to investigate isomorphisms between quasi-Banach algebras.     

Isomorphisms of Kac-Moody groups

We give the solution of the isomorphism problem for Kac-Moody groups over algebraically closed fields of any characteristic. In particular, we prove a conjecture of Kac and Peterson and compute the automorphism group of a Kac-Moody group over an algebraically closed field of characteristic zero. Mathematics Subject Classification (2000) 17B40, 20E36, 20E42, 20G15, 22E65, 51E24     

Isomorphisms of Group Algebras

Let G₁ and G₂ be locally compact groups. If T is an algebraisomorphism of L¹(G₁) onto L¹(G₂) with ||T|| (1+3), then G₁and G₂ are isomorphic. This improves on earlier results, and,in a certain sense, is best possible. However, the main conjecturethat the groups are isomorphic if ||T|| < 2 remains unsolvedexcept for abelian groups and for connected groups. Similarresults are given for the measure algebra M(G) and for the algebraC(G) of continuous functions when the group G is compact.