Let Ω[x1, x2, …, xn] denote the algebra of polynomials in variables x1, x2, …, xn with coefficients from a fixed field Ω of characteristic zero, where n = 1, 2,…. There exists a natural projection which maps xn onto 0 and all the other variables onto themselves, for n = 1, 2, …. This enables one to construct the corresponding inverse limit which we here denote by Ω[x]. The algebra Ω[x] has a natural degree function defined on it.

To each idempotent v of a semigroup T, there is associated, in a natural way, a subsemigroup Tv of T. The subsemigroup Tv is simply the collection of all elements of T for which v acts as a two-sided identity. We refer to such a subsemigroup as an I-subsemigroup of T. We first establish some elementary properties of these subsemigroups with no restrictions on the semigroup in which they are contained. Then we turn our attention to the semigroup of all continuous selfmaps of a topological space. The I-subsemigroups of these semigroups are investigated in some detail and so are the a-monomorphisms [3, p. 518] from one such semigroup into another. Among other things, a relationship is established between I-subsemigroups and α-monomorphisms. An analogous theory exists for semigroups of closed selfmaps on topological spaces. A number of results are listed for these semigroups with the proofs often deleted since, in many cases, the situation is much the same as for semigroups of continuous functions.

We use standard graph notation and definitions, as in [1]: in particular Kn is the complete graph on n vertices and Kn, n is the regular complete bigraph of order 2n.

In [3], Levinson proved a duality theorem for linear programming in complex space. Ben-Israel [1] generalized this result to polyhedral convex cones in complex space. In this paper, we give a simple proof of Ben-Israel's result based directly on the duality theorem for linear programming in real space. The explicit relations shown between complex and real linear programs should be useful in actually computing a solution for the complex case. We also give a simple proof of Farkas' theorem, generalized to polyhedral cones in complex space ([1], Theorem 3.5); the proof depends only on the classical form of Farkas' theorem for real space.

It is proved that if is a formal power series with algebraic p-adic coefficients which satisfies an algebraic differential equation, then a constant y4 > 0 and a constant integer h1 ≧ 0 exist such that .

In an earlier paper [11] we discussed the problem of when an (m, n)-complete lattice L is isomorphic to an (m, n)-ring of sets. The condition obtained was simply that there should exist sufficiently many prime ideals of a certain kind, and illustrations were given from topology and elsewhere. However, in these illustrations the prime ideals in question were all principal, and it is desirable to find and study examples where this simplification does not occur. Such an example is the lattice Z(X) of all zero-sets of a topological space X; we refer to Gillman and Jerison [5] for the simple proof that Z(X) is a (2, σ)-ring of subsets of X, where we denote aleph-zero by σ.

The classification of groups according to the varieties they generate requires the study of a class of indecomposable elements. Such a class is the class of basic groups which have been studied in [4], [5] and [6]. A group is called basic if it is indecomposable qua group; that is, it is critical and indecomposable qua variety; that is, its variety is join-irreducible. In this note we consider the higher commutator structure of basic p-groups. Our main theme is the relation between the formal weight of the higher commutator subgroups and the class of the group. We obtain information about the power-commutator structure of a basic p-group, the kinds of laws that can hold in such a group and the varietal structure of groups of the form: Center-extended-by-X.

The purpose of this note is to prove the following result. Theorem A. Let G be a finite group with the following properties (i) V▵ G, &V& = 2n and V is elementary abelian, (ii) G/V ≃ GL(n, 2), (iii) CG(V) ⊆ V. If n ≧ 6 then G splits over V.

Among all non-self-adjoint operators, the shift has a special place in questions relating to invariant subspaces. It is therefore natural to attempt to use this fact to study other operators related in some way to the shift. Examples of this procedure are available in the work of Freeman [3] and Lim [5] where perturbations of the shift are studied.

Hanna Neumann was one of the pioneers of the algebraic theory of nearrings. In the functional analysis, the near-rings appear as the sets of non-linear mappings of a space into itself. The relations between the algebraic properties of these near-rings and the analytic structures of the spaces and mappings involved have not been fully investigated. The purpose of this note is to consider one of such problems.

Let K be a skewfield, E a left vector space over K, r an integer ≧ 1 and Gr(E) the set of all r-dimensional subspaces of E, called the Grassmannian of index r. The function d(A, B) = r — dim (A∩B) is a distance on Gr(E). If K′ is a skewfield and E′ a left vector space over K1, then any semilinear ismorphism u: E → E1(relative to an isomorphism K → K') induces a distance preserving bijection Gr(u):Gr(E) → Gr(E′). When E has finite dimension n and 2r = n, another example of such a mapping is obtained by taking K′ = Kop, E′ = E* and defining wr: Gr(E) → Gr(E*) to be wr(A) = {f ∈ E*&f(A) = 0}.

In [6] and [2] Markov and Graev introduced their respective concepts of a free topological group. Graev's concept is more general in the sense that every Markov free topological group is a Graev free topological group. In fact, if FG(X) is the Graev free topological group on a topological space X, then it is the Markov free topological group FM(Y) on some space Y if and only if X is disconnected. This, however, does not say how FG(X) and FM(X) are related.

Let be a non-cyclic free presentation of a group G, the lower central series of R. Then Rn/Rn+1, n ≧ 1, can be regarded as a right G-module by defining the action of G via conjugation in F. We wish to investigate the annihilators of these modules which we call higher relation modules.

One of the best known classes of non-Desarguesian planes is the calss of Hall planes (see Hall [2]). In [6] Hanna Neumann showed that the finite Hall planes of old order possess subplanes of order two (i.e., Fano subplanes)1. Kirkpatrick [5] has considered a type of plane which is generalization of the Hall planes and which he calls generalized Hall planes. In this paper we will give a sufficient condition that a finite generalized Hall plane possesses Fano subplanes. Some examples of odd order planes to which the condition applies shall be exhibited.

A right module A over a ring R is called finitely injective if every diagram of right R-modules of the form where X is finitely generated and the row is exact, can be imbedded in a commutative diagram The finitely injective modules turn out to have the nice property of being ‘strongly pure’ in all the containing modules and, in particular, are absolutely pure in the sense of Maddox [8]. For this reason we call them strongly absolutely pure (for short, SAP) modules. Several other characterizations of the SAP modules are obtained. It is shown that every module admits a ‘SAP hull’. The concepts of finitely M-injective and finitely quasi-injective modules are then investigated. A subclass of finitely quasi-injective modules, called the strongly regular modules, is studied in some detail and it is shown that, if R/J is right Noetherian, then the strongly regular right R-modules coincide with the semi-simple R-modules.

In his work [5] on subgroups of one relator groups, Moldavanski observed that if G is a one relator group whose defining relator R is cyclically reduced and has exponent sum zero on some generator occurring in it, then G is an HNN extension of a one relator group H whose defining relator is shorter than R. This observation, together with Britton's Lemma, can be used to give rather easy proofs of the basic results on one relator groups. To exposit this point of view, we give here a proof of the Freiheitssatz, the solvability of the word problem for one relator groups, and the theorem classifying elements of finite order in one relator groups. In particular, the solution obtained for the word problem is often easy to apply. We also give a proof of the “Spelling Theorem” of Newman [6].