For an inverse semigroup S, the set of all isomorphisms betweeninverse subsemigroups of S is an inverse monoid under composition which is denoted by (S) and called the partial automorphism monoid of S. Kirkwood [7] and Libih [8] determined which groups have Clifford partial automorphism monoids. Here we investigate the structure of inverse semigroups whose partial automorphism monoids belong to certain other important classes of inverse semigroups. First of all, we describe (modulo so called “exceptional” groups) all inverse semigroups S such that (S) is completely semisimple. Secondly, for an inverse semigroup S, we find a convenient description of the greatest idempotent-separating congruence on (S), using a well-known general expression for this congruence due to Howie, and describe all those inverse semigroups whose partial automorphism monoids are fundamental.

In this paper we construct examples which answer three questions in the general area of noncommutative Noetherian local rings and rings of finite global dimension. The examples are formed in the same basic way, beginning with a commutative polynomial ring A over a field k and a k-derivation δ of A, taking the skew polynomial ring R = A[x;δ] and localizing at a prime ideal of the form IR, where I is a prime ideal of A invariant under δ. The localization is possible by a result of Sigurdsson [13].

All the graphs considered in this paper are connected finite undirected graphs without loops and multiple edges.

By the vertex-neighbourhood (v-neighbourhood) of any vertex x in the graph G we mean the subgraph induced by the set of all vertices adjacent to x. Analogously by the edge-neighbourhood (e-neighbourhood) of any edge f with end vertices x, y we mean the subgraph (f) (or (xy)) induced by the set of all vertices which are adjacent to at least one vertex of the pair x, y and which are different from x, y.

An algebra A = (L; ∨, ∧, *, +, 0, 1) of type (2, 2, 1, 1, 0, 0) is a doublep-algebra if (L; ∨, ∧, 0, 1) is a (0, l)-lattice in which * and + are unary operations of pseudocomplementation and dual pseudocomplementation determined by the respective requirements that x ≤ a* be equivalent to x ∧ a = 0, and that x ≥ a+ if and only if x ∨ a = 1.

A Banach space (X, ∥.∥) is said to have the Banach-Saks property (B.S.P.) if, for every bounded sequence (xn) in X, we can choose a subsequence () of (xn) such that the sequence

converges in the X-norm. This property, that a Banach space may enjoy or not, has been extensively studied.

Let (X, Σμ) denote a complete a-finite measure space and T: X → X a measurable (T-1A ε Σ each A ε Σ) point transformation from X into itself with the property that the measure μ°T-1 is absolutely continuous with respect to μ. Given any measurable, complex-valued function w(x) on X, and a function f in L2(μ), define WTf(x) via the equation

About fifteen years ago I. M. Isaacs and S. D. Smith [9] gave several character-theoretic characterizations of finite p-solvable groups G with p-length one, where p is a prime number. They proved that for a finite group G with a Sylow p-subgroup P, the following four conditions (a)–(d) are equivalent.

An ordered semigroup S will be called principally ordered if, for every x ɛ S, there exists

x* = max {y ɛ S; xyx ≤ x}.

Here we shall be concerned with the case where S is regular. We begin by listing some basic properties that arise from the above definition. As usual, we shall denote by V(x) the set of inverses of x ɛ S.

In this paper we continue the examination of the question of similarity of operators A and B begun in reference [3]. In that article, a similarity result was obtained based on a measure of closeness, or proximity, of the uniformly continuous semigroups etA and etB, t>0. The operators considered were elements of ℬ(ℋ), the algebra of bounded operators on a Hilbert space ℬWe now wish to relax this requirement and replace ℬ(ℋ) by a complex Banach algebra ℬ with unit I. In Section 2 we give a necessary condition for the similarity of A, B ∈ ℋ. We then give a condition sufficient to guarantee A and B are approximately similar (as defined in reference [5]). In Section 3 we restrict our attention to the case where ℋ = ℋ(ℋ). There we give a condition which guarantees A, B ∈ ℋ(ℋ) are intertwined by a Fredholm operator. This leads naturally into a discussion of proximity-similarity in the Calkin algebra si. This is the subject of Section 4. Following reference [7] we define a metric p on N(ℋ), the normal elements of ℋ We show (N(ℋ), p) is a complete metric space and that the unitary orbit of ℋ (N(ℋ) p)is the p-connected component of a in N (ℋ).

We make the convention that if a is an element of a semigroup Q then by writing a–1 it is implicit that a lies in a subgroup of Q and has inverse a–1 in this subgroup; equivalently, a ℋ a2 and a–1 is the inverse of a in Ha.

A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as a–1b where a, b ∈ S and, in addition, every element of S satisfying a weak cancellation condition which we call square-cancellable lies in a subgroup of Q. The notions of right order and semigroup of right quotients are defined dually; if S is both a left order and a right order in Q then S is an order in Q and Q is a semigroup of quotients of S.

Let w be a strictly positive function on ℂ and let , respectively denote the Banach spaces of those entire functions φ(z) with ∣φ(z)∣= O(w(z)) and ∣φ(z)∣ = o(w(z)). In this generality, these spaces may contain only constants, but for many functions w(z) these will be interesting Banach spaces with norm

.

We study two specific problems.

In [7] the level, sublevel, and product level of finite dimensional central division algebras D over a field F were calculated when F is a local or global field. In Theorem 1.4 of this paper we calculate the same quantities if all finite extensions K of F satisfy ū(K) ≤2, where ū is the Hasse number of a field as defined in [2]. This occurs, for example, if F is an algebraic extension of the function field R(x) where R is a real closed field or hereditarily Euclidean field (see [4]).

The concept of (V*) set was introduced, as a dual companion of that of (V)-set, by Pelczynski in his important paper [14]. In the same paper, the so called properties (V) and (V*) are defined by the coincidence of the (V) or (V*) sets with the weakly relatively compact sets. Many important Banach space properties are (or can be) defined in the same way; that is, by the coincidence of two classes of bounded sets. In this paper, we are concerned with the study of the class of (V*) sets in a Banach space, and its relationship with other related classes. To this general study is devoted Section I. A (as far as we know) new Banach space property (we called it property weak (V*)) is defined, by imposing the coincidence of (V*) sets and weakly conditionally compact sets. In this way, property (V*) is decomposed into the conjunction of the weak (V*) property and the weak sequential completeness. In Section II, we specialize to the study of (V*) sets in Banach lattices. The main result in the section is that every order continuous Banach lattice has property weak (V*), which extends previous results of E. and P. Saab ([16]). Finally, Section III is devoted to the study of (V*) sets in spaces of Bochner integrable functions. We characterize a broad class of (V*) sets in L1(μ, E), obtaining similar results to those of Andrews [1], Bourgain [6] and Diestel [7] for other classes of subsets. Applications to the study of properties (V*) and weak (V*) are obtained. Extension of these results to vector valued Orlicz function spaces are also given.

Prime rings came into prominence when Posner characterized prime rings satisfying a polynomial identity [9]. The scarcity of invertible central elements made it difficult to generalize results from central simple and primitive algebras to prime rings. For example, we do not automatically have tensor products at our disposal. In [5], the first author introduced the Martindale ring of quotients Q(R) of a prime ring R in his theorem characterizing prime rings satisfying a generalized polynomial identity (GPI). Q(R) is a prime ring containing R whose center C is a field called the extended centroid of R. The central closure of R is the subring RC of Q(R) generated by R and C. RC is a closed prime ring since its extended centroid equals its center C. Hence we have a useful procedure for proving results about an arbitrary prime ring R. We first answer the question for closed prime rings and then apply to R the information obtained from RC. It should be noted that simple rings and free algebras of rank at least 2 are closed prime rings. For these reasons, closed prime rings are natural objects to study.

Throughout the paper, rings are associative rings with identity. A ring is called right duo if every right ideal is two-sided, and it is called right p.p. if every principal right ideal is projective. A left duo (p.p.) ring is denned similarly, and a duo (p.p.) ring will mean a ring which is both right and left duo (p.p.). There is a right p.p. ring that is not left p.p. (see Chase [2[). Small [9] proved that right p.p. implies left p.p. if there are no infinite sets of orthogonal idempotents, and Endo [5, Proposition 2] has shown the same implication in the case where each idempotent in the ring is central. Since Courter [3, Theorem 1.3] noted that every idempotent in a right duo ring is central, we can simply speak of right duo p.p. rings. A typical example of a right duo ring which is not left duo is the following. Let F be a field and F(x) the field of rational functions over F. Let R = F(x)× F(x) as an additive group and define the multiplication as follows:

Then R is a local artinian ring with c(RR) = 2 and c(RR)= 3. Thus R is right duo but not left due.

Consider the vector space of m-tuples of n by n matrices

.

The linear group GLn(C) acts on Xm, n by simultaneous conjugation. The corresponding ring of polynomial invariants

will be denoted by C(n, m) and is called the ring of matrix invariants of m-tuples of n by n matrices. C. Procesi has shown in [8] that C(n, m) is generated by traces of products of the corresponding generic matrices and, as such, coincides with the center of the trace ring of m generic n by n matrices R (n, m) which is also the ring of equivariant maps from Xm, n to Mn(ℂ).

Let R be a semiprime ring (possibly without 1). The symmetric ring of quotients of R is defined as the set of equivalence classes of essentially defined double centralizers (ƒ, g) on R; see [1], [8]. So, by definition, ƒ is a left R-module homomorphism from an essential ideal I of R into R, g is a right R-module homomorphism from an essential ideal J of R into R, and they satisfy the balanced condition ƒ(x)y = xg(y) for x ∈ Iand y ∈ J. This ring was used by Kharchenko in his investigations on the Galois theory of semiprime rings [4] and it is also a useful tool for the study of crossed products of prime rings [7]. We denote the symmetric ring of quotients of a semiprime ring R by Q(R).

Let Δ = {z ∈ ℂ ⃒ ⃓z⃓ <1) and H(Δ) the set of analytic functions on Δ. We recall the definition of subordination between two functions, say ƒ and g, analytic on Δ: this means that f(0)= g(0) and there is a function ρ ∈ H (Δ) such that ρ(0) = 0, ⃒ ρ(z)⃒<1 if z ∈ Δ, and f(z) ≡ g(ρ(z)). Subordination between f and g will be denoted by

f<g. The Hadamard product (or convolution) of two functions and in H(Δ)is the function f * g ∈ H(Δ)definedas f * g (z)= .