On subgroups of demushkin groups

It can be shown immediately by group-theoretic methods that a subgroup of finite index of a Demushkin group is itself a Demushkin group.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 349–354, September, 1968.The author extends his thanks to A. V. Yakovlev for his guidance and interest in the author's work and also D. K. Faddeev for valuable instructions and B. B. Lur'e for help in the editing of this article.     

Epimorphisms of knot groups onto free products

Let k be a knot in S³. There is an epimorphism from π₁(S³−k) onto a free product of two nontrivial cyclic groups sending a meridian to an element of length two iff k has property Q (Topology of Manifolds, Markham, Chicago, IL, 1970, pp. 195-199) that is if there is a closed surface F in S³ containing k, such that k is imprimitive in H₁(X) and in H₁(Y) where X and Y are the closures of the components of S³−F. We give answers to questions of Simon (1970) about properties Q, Q∗ and Q∗∗. Epimorphisms from knot groups onto torus knot groups are also studied and some results on property P and surgery are included.     

Epimorphisms of free monoids

A matrix characterization is obtained for the epimorphisms in the category of finitely generated free monoids. It follows from our result that it is effectively decidable whether a given morphism is an epimorphism. The corresponding question for monomorphisms has been answered in the algorithm of Sardinas and Patterson.     

Epimorphisms of uniform frames

It is shown that some familiar properties of epimorphisms in the category of frames carry over to the categories of uniform and complete uniform frames. This is achieved by suitably enriching certain frame homomorphisms to uniform frame homomorphisms.     

Epimorphisms and ℋ-coextensions

Conclusions Theorems 3.4 and 4.3 have consequences of a similar form: we have compact monoids built from objects in a smaller category of compact monoids, and an epimorphism in the larger category is surjective provided the induced morphism (in 3.4 the restriction to the idempotents) is surjective. In 3.2 and 3.3 we make use of the fact that for the smaller category CL this will always be the case. Thus 3.4 and 4.3 could be completed, in a sense, if we know more about epimorphisms of compact semilattices and about A-epimorphisms. As for 3.4, another type of completion would be an extension to the nonabelian case. Since it is not known (see [2], [17]) whether constructions analogous to those in 3.1 yield Hausdorff semi-groups for the non-Lawson case, this problem may be difficult.In closing, the author would like to thank Professor J. H. Carruth for his helpful advice and encouragement in the research leading to this paper.     

Epimorphisms Between Linear Orders

We study the relation on linear orders induced by order preserving surjections. In particular we show that its restriction to countable orders is a bqo.     

Epimorphisms,dominions and semigroups

We investigate the preservation of epimorphism-related properties of semigroups under the taking of morphic images, of ideals and adjunction of an identity.Presented by Boris M. Schein.     

Epimorphisms of Separated Superconvex Spaces

We give characterizations of the epimorphisms in the categories of (pre)separated convex and separated superconvex spaces.     

Epimorphisms of separated totally convex spaces

We give characterizations of the epimorphisms in the categories of open and separated totally convex and (pre)separated absolutely convex spaces.     

Epimorphisms in Certain Varieties of Algebras

We prove a lemma which, under restrictive conditions, shows that epimorphisms in V are surjective if this is true for epimorphisms from irreducible members of V. This lemma is applied to varieties of orthomodular lattices which are generated by orthomodular lattices of bounded height, and to varieties of orthomodular lattices which are generated by orthomodular lattices which are the horizontal sum of their blocks. The lemma can also be applied to obtain known results for discriminator varieties.     

Tilting Modules Arising from Ring Epimorphisms

We show that a tilting module T over a ring R admits an exact sequence 0?→?R?→?T ₀?→?T ₁?→?0 such that \(T_0,T_1\in\text{Add}(T)\) and Hom _R (T ₁,T ₀)?=?0 if and only if T has the form S?⊕?S/R for some injective ring epimorphism λ : R?→?S with the property that \(\text{Tor}_1^R(S,S)=0\) and pdS _R ?≤?1. We then study the case where λ is a universal localization in the sense of Schofield (1985). Using results by Crawley-Boevey (Proc Lond Math Soc (3) 62(3):490–508, 1991), we give applications to tame hereditary algebras and hereditary noetherian prime rings. In particular, we show that every tilting module over a Dedekind domain or over a classical maximal order arises from universal localization.     

Epimorphisms, dominions and seminormal identities II

We associate two natural numbers with a seminormal identity depending on the first and the last occurrences of two two-letter subwords of the left hand side which are not subwords of its right hand side. Using these natural numbers, we find some classes of heterotypical identities where both sides contain repeated variables that are preserved under epis in conjunction with a seminormal identity.     

Separation and Epimorphisms in Quasi-Uniform Spaces

We study some categorical aspects of quasi-uniform spaces (mainly separation and epimorphisms) via closure operators in the sense of Dikranjan, Giuli, and Tholen. In order to exploit better the corresponding properties known for topological spaces we describe the behaviour of closure operators under the lifting along the forgetful functor T from quasi-uniform spaces to topological spaces. By means of appropriate closure operators we compute the epimorphisms of many categories of quasi-uniform spaces defined by means of separation axioms and study the preservation (reflection) of epimorphisms under the functor T.