A Ramsey theorem for trees

We prove a Ramsey theorem for trees. The infinite version of this theorem can be stated: if T is a rooted tree of infinite height with each node of T having at least one but finitely many immediate successors, if n is a positive integer, and if the collection of all strongly embedded, height-n subtrees of T is partitioned into finitely many classes, then there must exist a strongly embedded subtree S of T with S having infinite height and with all the strongly embedded, height-n subtrees of S in the same class.     

Automatic subsemigroups of free products

We consider the automaticity of subsemigroups of free products of semigroups, proving that subsemigroups of free products, with all generators having length greater than one in the free product, are automatic. As a corollary, we show that if S is a free product of semigroups that are either finite or free, then any finitely generated subsemigroup of S is automatic. In particular, any finitely generated subsemigroup of a free product of finite or monogenic semigroups is automatic.     

Su una condizione di finitezza per semigruppi finitamente generati

We prove that a finitely generated semigroupS is finite if and only if there exists an integern such that in each sequence ofn elements ofS there exist two different non empty factors with the same value inS. We prove this result using only elementary facts concerning the canonical form of an element of a finitely generated semigroup.     

Palindromic Width of Finitely Generated Solvable Groups

We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated 3-step solvable group has finite palindromic width. More generally, we show the finiteness of the palindromic width for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary solvable groups of step ≥3, we prove that if G is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for normal subgroups, then the palindromic width of G is finite. We also prove that the palindromic width of ??? with respect to the set of standard generators is 3.     

On Finite Presentability of Direct Products of Semigroups

A finite semigroup S is said to preserve finite generation (resp., presentability) in direct products, provided that, for every infinite semigroup T, the direct product S × T is finitely generated (resp., finitely presented) if and only if T is finitely generated (resp., finitely presented). The main result of this paper is a constructive necessary and sufficient condition for S to preserve both finite generation and presentability in direct products. The condition is that certain graphs, (s), one for each s S, are all connected. The main result is illustrated in three examples, one of which exhibits a 4-element semigroup that preserves finite generation but not finite presentability in direct products.1991 Mathematics Subject Classification: 20M05, 05C25The first author is financially supported by the Sub-Programa Ciência e Tecnologia do 2° Quadro Comunitário de Apoio (grant number BD/ 15623/98). The author also acknowledges the support of the Centro de Álgebra da Universidade de Lisboa and of the Projecto Praxis 2/2.1/MAT/73/94. The second author acknowledges partial financial support from the Nuffield Foundation.     

Green index in semigroups: generators, presentations, and automatic structures

The Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S?T under the natural actions of T on S via right and left multiplication. This partitions the complement S?T into T-relative -classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schützenberger group. If the Rees index |S?T| is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows how generating sets for S may be used to obtain generating sets for T and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from presentations of T and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of group-embeddable semigroups).     

On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.     

Operator norm localization property of relative hyperbolic group and graph of groups

In this article we study the spaces which have operator norm localization property. We prove that a finitely generated group Γ which is strongly hyperbolic with respect to a collection of finitely generated subgroups {H₁,…,Hn} has operator norm localization property if and only if each Hi, i=1,2,…,n, has operator norm localization property. Furthermore we prove the following result. Let π be the fundamental group of a connected finite graph of groups with finitely generated vertex groups GP. If GP has operator norm localization property for all vertices P then π has operator norm localization property.     

Finite character of finitely stable domains

A commutative domain is finitely stable if every nonzero finitely generated ideal is stable, i.e. invertible over its endomorphism ring. A domain satisfies the local stability property provided that every locally stable ideal is stable.We prove that a finitely stable domain satisfies the local stability property if and only if it has finite character, that is every nonzero ideal is contained in at most finitely many maximal ideals. This result allows us to answer the open problem of whether every Clifford regular domain is of finite character.     

Quasi-injective dimension

Following our previous work about quasi-projective dimension [11], in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module M over a local ring R is finite, then it is equal to the depth of R. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then R is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then R is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.     

Wakamatsu tilting modules with finite injective dimension

Let R be a left Noetherian ring, S a right Noetherian ring and _R ω a Wakamatsu tilting module with S = End( _R ω). We introduce the notion of the ω-torsionfree dimension of finitely generated R-modules and give some criteria for computing it. For any n ? 0, we prove that l.id _R (ω) = r.id _S (ω) ? n if and only if every finitely generated left R-module and every finitely generated right S-module have ω-torsionfree dimension at most n, if and only if every finitely generated left R-module (or right S-module) has generalized Gorenstein dimension at most n. Then some examples and applications are given.     

Arithmetic groups with isomorphic finite quotients

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the fibers are of unbounded size.     

Special Lie superalgebras with maximality condition for subalgebras

We consider finitely generated Lie superalgebras over a field of characteristic zero satisfying Capelli identities. We prove that any such an algebra with the maximality condition for abelian subalgebras is finite dimensional. In particular, any special Lie superalgebra with the maximality condition for its subalgebras has a finite dimension. We also prove that the universal enveloping algebra U(L) of special Lie superalgebra L is Noetherian if and only if $\dim L<\infty$ .     

On Automatic Rees Matrix Semigroups

We consider a Rees matrix semigroup S = M[U; I, J; P] over a semigroup U, with I and J finite index sets, and relate the automaticity of S with the automaticity of U. We prove that if U is an automatic semigroup and S is finitely generated then S is an automatic semigroup. If S is an automatic semigroup and there is an entry p in the matrix P such that pU ¹ = U then U is automatic. We also prove that if S is a prefix-automatic semigroup, then U is a prefix-automatic semigroup.     

On generators and relations for unions of semigroups

Finite generation and presentability of general unions of semigroups, as well as of bands of semigroups, bands of monoids, semilattices of semigroups and strong semilattices of semigroups, are investigated. For instance, it is proved that a band Y of monoids S _α (α∈ Y ) is finitely generated/presented if and only if Y is finite and all S _α are finitely generated/presented. By way of contrast, an example is exhibited of a finitely generated semigroup which is not finitely presented, but which is a disjoint union of two finitely presented subsemigroups. January 21, 2000     

Notes on the products of the lower topology and Lawson topology on posets

In this paper, we investigate the relation between the lower topology respectively the Lawson topology on a product of posets and their corresponding topological product. We show that (1) if S and T are nonsingleton posets, then Ω(S×T)=Ω(S)×Ω(T) iff both S and T are finitely generated upper sets; (2) if S and T are nontrivial posets with σ(S) or σ(T) being continuous, then Λ(S×T)=Λ(S)×Λ(T) iff S and T satisfy property K, where for a poset L, Ω(L) means the lower topological space, Λ(L) means the Lawson topological space, and L is said to satisfy property K if for any x∈L, there exist a Scott open U and a finite F⊆L with x∈U⊆↑F.     

A note on the volume flux group of four-manifolds

We show that closed orientable smooth four-manifolds with non-trivial volume flux group and fundamental group of subexponential growth type are finitely covered by a manifold homeomorphic to S³×S¹, S²×T² or a nil-manifold. We also show that if a compact complex surface has non-trivial volume flux group then it has zero minimal volume.     

On the cohomology of highly connected covers of finite Hopf spaces

Relying on the computation of the André-Quillen homology groups for unstable Hopf algebras, we prove that if the mod p cohomology of both the fiber and the base in an H-fibration is finitely generated as algebra over the Steenrod algebra, then so is the mod p cohomology of the total space. In particular, the mod p cohomology of the n-connected cover of a finite H-space is always finitely generated as algebra over the Steenrod algebra.     

Theories of linear order

LetT be a complete theory of linear order; the language ofT may contain a finite or a countable set of unary predicates. We prove the following results. (i) The number of nonisomorphic countable models ofT is either finite or 2^ω. (ii) If the language ofT is finite then the number of nonisomorphic countable models ofT is either 1 or 2^ω. (iii) IfS ₁(T) is countable then so isS _n(T) for everyn. (iv) In caseS ₁(T) is countable we find a relation between the Cantor Bendixon rank ofS ₁(T) and the Cantor Bendixon rank ofS _n(T). (v) We define a class of modelsL, and show thatS ₁(T) is finite iff the models ofT belong toL. We conclude that ifS ₁(T) is finite thenT is finitely axiomatizable. (vi) We prove some theorems concerning the existence and the structure of saturated models. Most of the results in this paper appeared in the author’s Master of Science thesis which was prepared at the Hebrew University under the supervision of Professor H. Gaifman.