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, 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.     

Epimorphisms, dominions and permutative semigroups

We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity $x_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}}We show that any right semicommutative semigroup satisfying the minimum condition on principal right ideals is saturated. Further we show that if a supersaturated semigroup S has a globally idempotent ideal U which satisfies a nontrivial permutation identity x₁x₂?x_n=x_i1x_i2?x_inx_{1}x_{2}\ldots x_{n}=x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}} such that i ₁=1 and i _n ≠n, then U is also supersaturated.     

Epimorphisms, dominions and seminormal identities

We find some sufficient conditions on semigroup identities in which both sides contain repeated variables to be preserved under epis in conjunction with any seminormal permutation identity.     

Absolute flatness and amalgamation in pomonoids

Absolute flatness and amalgamation for partially ordered monoids (briefly pomonoids) were first considered in the mid 1980s by S.M. Fakhruddin in two research articles. Though the study of absolute flatness for pomonoids was revived by X. Shi, S. Bulman-Fleming and others after a dormancy period of almost two decades—resulting in the appearance of several research articles on the subject since 2005—amalgamation in pomonoids was never reconsidered until the recent past when S. Bulman-Fleming and the author produced two research articles on the subject. The primary objectives of these papers were to show that imposition of order subjects the amalgamation of monoids to severe restrictions and to prove that partially ordered groups (briefly pogroups) are amalgamation bases in the class of all pomonoids. Proceeding further, we establish in this article the amalgamation property for the class of pogroups. (The property was first proved for the class of groups by O. Schreier, Abh. Math. Semin. Univ. Hamb. 5:161–183, 1927.) In addition, we show that absolutely flat commutative pomonoids are (strong) amalgamation bases in the category of commutative pomonoids. (A similar result was proved by Fakhruddin for weak amalgamation.) The special amalgamation property and the existence of pushouts in the category of pomonoids, which have been instrumental in proving our main results, are also established.     

Epimorphisms,dominions and $$\mathcal {H}$$H-commutative semigroups

Semigroup Forum - In the present paper, a series of results and examples that explore the structural features of $$\mathcal {H}$$-commutative semigroups are provided. We also generalize a result of...     

Categoricity,amalgamation, and tameness

Theorem: For each 2 ≤ k < ω there is an -sentence ϕ_k such that (1) ϕ_k is categorical in μ if μ≤ℵ_k−2; (2) ϕ_k is not ℵ_k−2-Galois stable (3) ϕ_k is not categorical in any μ with μ>ℵ_k−2; (4) ϕ_k has the disjoint amalgamation property (5) For k > 2 (a) ϕ_k is (ℵ₀, ℵ_k−3)-tame; indeed, syntactic first-order types determine Galois types over models of cardinality at most ℵ_k−3; (b) ϕ_k is ℵ_m-Galois stable for m ≤ k − 3 (c) ϕ_k is not (ℵ_k−3, ℵ_k−2). The first author is partially supported by NSF grant DMS-0500841.     

Natural Classes,Left Exact Preradicals,and Flat Epimorphisms

We define a function between the lattice of left exact preradicals in R-mod and the complete boolean lattice R-nat of natural classes in R-mod in such a way that the image of this function results to be a complete sublattice of R-nat. The image of this function is R-nat if and only if R is left semiartinian. We prove that every flat ring epimorphism from R to S induces a function between the lattice of left exact preradicals in R-mod and the lattice of left exact preradicals in S-mod. In this context, we obtain a lattice decomposition of S-nat for each left exact preradical.     

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.     

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.     

Homological Epimorphisms,Compactly Generated t-Structures and Gorenstein-Projective Modules

The aim of this paper is two-fold.Given a recollement(T′,T,T′′,i*,i_*,i~!,j!,j*,j*),where T′,T,T′′are triangulated categories with small coproducts and T is compactly generated.First,the authors show that the BBD-induction of compactly generated t-structures is compactly generated when i*preserves compact objects.As a consequence,given a ladder(T′,T,T′′,T,T′) of height 2,then the certain BBD-induction of compactly generated t-structures is compactly generated.The authors apply them to the recollements induced by homological ring epimorphisms.This is the first part of their work.Given a recollement(D(B-Mod),D(A-Mod),D(C-Mod),i*,i_*,i~!,j!,j*,j_*) induced by a homological ring epimorphism,the last aim of this work is to show that if A is Gorenstein,AB has finite projective dimension and j! restricts to D~b(C-mod),then this recollement induces an unbounded ladder(B-Gproj,A-Gproj,C-Gproj) of stable categories of finitely generated Gorenstein-projective modules.Some examples are described.     

Homomorphisms and amalgamation

We prove that for each finite core graph G, the class of all graphs admitting a homomorphism into G is a pseudo-amalgamation class, in the sense of Fraı̈ssé. This fact gives rise to a countably infinite universal pseudo-homogeneous graph which shares some of the properties of the infinite random graph. Our methods apply simultaneously to G-colourings in several classes of relational structures, such as the classes of directed graphs or hypergraphs.