Numerical Semigroup Rings and Almost Prüfer v-Multiplication Domains

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ? ?₀, D[Γ] be the semigroup ring of Γ over D (and hence D ? D[Γ] ? D[X]), and D + X ⁿ K[X] = {a + X ⁿ g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X ⁿ K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X ⁿ K[X].     

Semiprimitivity of Orthodox Semigroup Algebras

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R₀[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.     

Almost Prüfer v-Multiplication Domains and Related Domains of the Form D + D S [Γ*]

Let D be an integral domain, S be a (saturated) multiplicative subset of D such that D ? D _S , Γ be a numerical semigroup with Γ ? ?₀, Γ* = Γ?{0}, X be an indeterminate over D, D + XD _S [X] = {a + Xg ∈ D _S [X]∣a ∈ D and g ∈ D _S [X]}, and D + D _S [Γ*] = {a + f ∈ D _S [Γ]∣a ∈ D and f ∈ D _S [Γ*]}; so D + D _S [Γ*] ? D + XD _S [X]. In this article, we study when D + D _S [Γ*] is an APvMD, an AGCD-domain, an AS-domain, an AP-domain, or an AB-domain.     

Every semitopological semigroup compactification of the group H + [0,1] is trivial

Let G=H ₊ [0, 1] be the topological group of all orientation-preserving selfhomeomorphisms of the closed interval [0,1] endowed with the usual compact open topology. We show that every weakly almost periodic function on G is constant. Consequently, G does not admit nontrivial (weakly) continuous representations by linear isometries in reflexive Banach spaces. September 6, 1999     

On the Commuting Graph of Dihedral Group

Let Γ be a non-abelian group and Ω ? Γ. We define the commuting graph G = 𝒞(Γ, Ω) with vertex set Ω and two distinct elements of Ω are joined by an edge when they commute in Γ. In this article, among some properties of commuting graphs, we investigate distant properties as well as detour distant properties of commuting graph on D_2n. We also study the metric dimension of commuting graph on D_2n and compute its resolving polynomial.     

Total outer-connected domination numbers of trees

Let G=(V,E) be a graph without an isolated vertex. A set DV(G) is a total dominating set if D is dominating, and the induced subgraph G[D] does not contain an isolated vertex. The total domination number of G is the minimum cardinality of a total dominating set of G. A set DV(G) is a total outer-connected dominating set if D is total dominating, and the induced subgraph G[V(G)−D] is a connected graph. The total outer-connected domination number of G is the minimum cardinality of a total outer-connected dominating set of G. We characterize trees with equal total domination and total outer-connected domination numbers. We give a lower bound for the total outer-connected domination number of trees and we characterize the extremal trees.     

Split-By-Nilpotent Extensions Algebras and Stratifying Systems

Let Γ and Λ be artin algebras such that Γ is a split-by-nilpotent extension of Λ by a two sided ideal I of Γ. Consider the change of rings functors G: =_ΓΓ_Λ ?_Λ ? and F: =_ΛΛ_Γ ?_Γ ?. In this article, by assuming that I _Λ is projective, we find the necessary and sufficient conditions under which a stratifying system (Θ, ≤) in modΛ can be lifted to a stratifying system (GΘ, ≤) in mod(Γ). Furthermore, by using the functors F and G, we study the relationship between their filtered categories of modules; and some connections with their corresponding standardly stratified algebras are stated (see Theorem 5.12, Theorem 5.15 and Theorem 5.18). Finally, a sufficient condition is given for stratifying systems in mod(Γ) in such a way that they can be restricted, through the functor F, to stratifying systems in mod(Λ).     

Stably free cancellation for abelian group rings

We show that if Γ is a finitely generated abelian group, then every stably free module over Z[Γ] is free.     

Integral Closure of Graded Integral Domains

Let Γ be a torsion-free cancellative commutative monoid and let R = ?_α∈ΓR_α be a commutative Γ-graded ring. We show that if R is a graded Noetherian domain, then its integral closure is a graded Krull domain. This is a graded analog of the Mori–Nagata theorem. We also show that for a graded Strong Mori domain, its complete integral closure is a graded Krull domain but its integral closure is not necessarily a graded Krull domain.     

Hamiltonian groups are color-graph-hamiltonian

A group Γ is said to be color -graph -hamiltonian if Γ has a minimal generating set Δ such that the Cayley color graph D_Δ(Γ) is hamiltonian. It is shown that every hamiltonian group is color -graph -hamiltonian.     

Factorization in dedekind domains with finite class group

LetD be a Dedekind domain. It is well known thatD is then an atomic integral domain (that is to say, a domain in which each nonzero nonunit has a factorization as a product of irreducible elements). We study factorization properties of elements in Dedekind domains with finite class group. IfD has the property that any factorization of an elementα into irreducibles has the same length, thenD is called a half factorial domain (HFD, see [41]). IfD has the property that any factorization of an elementα into irreducibles has the same length modulor (for somer>1), thenD is called a congruence half factorial domain of orderr. In Section I we consider some general factorization properties of atomic integral domains as well as the interrelationship of the HFD and CHFD property in the Dedekind setting. In Section II we extend many of the results of [41], [42] and [36] concerning HFDs when the class group ofD is cyclic. Finally, in Section III we consider the CHFD property in detail and determine some basic properties of Dedekind CHFDs. IfG is any Abelian group andS any subset ofG−[0], then {G, S} is called a realizable pair if there exists a Dedekind domainD with class groupG such thatS is the set of nonprincipal classes ofG which contain prime ideals. We prove that for a finite abelian groupG there exists a realizable pair {G, S} such that any Dedekind domain associated to {G, S} is CHFD for somer>1 but not HFD if and only ifG is not isomorphic toZ ₂,Z ₂,Z ₂ ⊕Z ₂, orZ ₃ ⊕Z ₃. The first author received support under the John M. Bennett Fellowship at Trinity University and also gratefully acknowledges the support of The University of North Carolina at Chapel Hill.     

Quiver Algebras of Coverings and Twisted Tensor Products

Given a covering Γ of a quiver Δ, we show that the quiver algebra K[Γ] of Γ over a field K is a twisted tensor product of the quiver algebra of the fibre of the covering viewed as a trivial quiver and the quiver algebra K[Δ]. To make sense of this, we first extend the theory of twisted tensor products of algebras to include algebras without units.     

MULTIPLICATIVELY CLOSED SETS OF IDEALS AND RESIDUAL DIVISION

Let R be a ring (commutative with identity), let Γ be a multiplicatively closed set of finitely generated nonzero ideals of R, for an ideal I of R let I _Γ = ∪ {I : G; G ∈ Γ}, and for an R-algebra A such that GA ≠ (0) for all G ∈ Γ let A ^Γ = ∪ {A : _T GA; G ∈ Γ}, where T is the total quotient ring of A. Then I _Γ is an ideal in R, I → I _Γ is a semi-prime operation (on the set I of ideals I of R) that satisfies a cancellation law, and it is a prime operation on I if and only if R = R ^Γ. Also, A ^Γ is an R-algebra and A → A ^Γ is a closure operation on any set A = {A; A is an R-algebra, R ? A, and if C is a ring between R and A, then regular elements in C remain regular in A}. Finally, several results are proved concerning relations between the ideals I _Γ and (IA)_ΓA and between the R-algebras R ^Γ and A ^Γ.

     

FACTORIZATION IN MONOID DOMAINS

In this paper, we determine necessary and sufficient conditions for the group ring D[G] to be a BFD (resp., an FFD, an SFFD). Also we give necessary and sufficient conditions for the monoid domain D[S] to be a BFD (resp., an FFD, an SFFD). In addition, we characterize when themonoid domain D[S] is a UFD in terms of 2-factoriality.     

Cannon–Thurston maps for hyperbolic free group extensions

This paper gives a detailed analysis of the Cannon–Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free group of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension E_Γ of F, and the main theorem of Dowdall–Taylor [DT14] states that in this situation E_Γ is hyperbolic if and only if Γ is purely atoroidal.     

Hecke Algebras of Group Extensions

Abstract

We describe the Hecke algebra ?(Γ,Γ₀) of a Hecke pair (Γ,Γ₀) in terms of the Hecke pair (N,Γ₀) where N is a normal subgroup of Γ containing Γ₀. To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S ? Γ/N satisfies S ^?1 S = Γ/N, we show that ? (Γ,Γ₀) is the twisted crossed product of ? (N,Γ₀) by S. This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products.     

Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

We study algebraic and topological properties of the convolution semigroup of probability measures on a topological groups and show that a compact Clifford topological semigroup S embeds into the convolution semigroup P(G) over some topological group G if and only if S embeds into the semigroup exp(G)\exp(G) of compact subsets of G if and only if S is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup S embeds into the functor-semigroup F(G) over a suitable compact topological group G for each weakly normal monadic functor F in the category of compacta such that F(G) contains a G-invariant element (which is an analogue of the Haar measure on G).     

An explicit spine for the Picard modular group over the Gaussian integers

Let Γ\D be an arithmetic quotient of a symmetric space of non-compact type. A spine D₀ is a Γ-equivariant deformation retraction of D with dimension equal to the virtual cohomological dimension of Γ. We explicitly construct a spine for the case of Γ=SU(2,1;Z[i]). The spine is then used to compute the cohomology of Γ\D with various local coefficients.