Let S be a compact, topological semigroup with identity. Suppose D, L and R are the D, L and R classes of some x ∈ S. Let (L, α., L/H), (R, β, R/H), (D, γ, D/H) and (D, δ, D/R) by the fibre spaces gotten where α, β γ an δ are the natural maps. It is shown that (D, γ, D/H) has topologically the same structure as the fibre space associated with (L, α, L/H) by R. Also if (L, α, L/H) is locally trivial (locally a cartesian product) then so is (D, δ, D/R) and if both (L, α, L/H) and (R, β, R/H) are locally trivial then so is (D, γ, D/H).

We introduce the concept of a variety of topological groups and of a free topological group F(X, ) of on a topological space X as generalizations of the analogous concepts in the theory of varieties of groups. Necessary and sufficient conditions for F(X, ) to exist are given and uniqueness is proved. We say the topological group FM,(X) is moderately free on X if its topology is maximal and it is algebraically free with X as a free basis. We show that FM(X) is a free topological group of the variety it generates and that if FM(X) is in then it is topologically isomorphic to a quotient group of F(X, ). It is also shown how well known results on free (free abelian) topological groups can be deduced. In the algebraic theory there are various equivalents of a free group of a variety. We examine the relationships between the topological analogues of these. In the appendix a result similar to the Stone-Čech compactification is proved.

In a finite soluble group G, the Fitting (or nilpotency) length h(G) can be considered as a measure for how strongly G deviates from being nilpotent. As another measure for this, the number v(G) of conjugacy classes of the maximal nilpotent subgroups of G may be taken. It is shown that there exists an integer-valued function f on the set of positive integers such that h(G) ≦ f(v(G)) for all finite (soluble) groups of odd order. Moreover, if all prime divisors of the order of G are greater than v(G)(v(G) - l)/2, then h(G) ≦3. The bound f(v(G)) is just of qualitative nature and by far not best possible. For v(G) = 2, h(G) = 3, some statements are made about the structure of G.

Dependence structures (in the finite case, matroids) arise when one tries to abstract the properties of linear dependence of vectors in a vector space. With the help of a theorem due to P. Hall and M. Hall, Jr concerning systems of distinct representatives of families of finite sets, it is proved that if B1 and B2 are bases of a dependence structure, then there is an injection σ: B1 → B2 such that (B2 / {σ(e)}) ∩ {e} is a basis for all e in B1. A corollary is the theorem of R. Rado that all bases have the same cardinal number. In particular, it applies to bases of a vector space. Also proved is the fact that if B1 and B2 are bases of a dependence structure then given e in B1 there is an f in B2 such that both (B1 / {e}) ∩ {f} and (B2 / {f}) ∩ {e} are bases. This is a symmetrical kind of replacement theorem.

Let R be a relation on a set X, and if A ⊂ X set RA = {x ∣ (x, α) ∈ R for some α ∈ A} and AR = {x ∣ (α, x) ∈ R for some α ∈ A}. Also A is called an antiset in case no two distinct elements of A are related. If A is a collection of antisets, then we generate a topology T(A) by taking sets of the form RA or AR (or X or ø) as subbasic open sets. Then conditions are given under which this topology satisfies separation axioms, or is compact or connected. For example. Theorem: Let A contain the singletons. If for each x ∈ X and y ∈ X \ x, there is a z ∈ X such that (x, z) ∈ R ((z, x) ∈. R) and (y, z) ∉ R ((z, y) ∉ R), then T(A) is a T1-topology. The conditions used to obtain compactness or connectedness are analogous to the conditions used to get the same properties for the order topology on a totally ordered set. Finally, by modifying the definition of T(A) slightly, we obtain conditions so that if X is a tree and R the cutpoint order, then T(A) is the original topology.

For every infinite cardinal a we let Ca, be the set of all real-valued continuous functions on a product of a closed unit intervals with the supmetric. It is shown that Ca has separability degree a. Further, the classical theorem of Banach and Mazur is generalized by showing that every metric space of separability degree a is isometric to a subspace of ca

Let G be the free product of groups Gα, [Gα] the cartesian subgroup of G and k [Gα] the intersection of [G] with the k–th term of the lower central series for G. Then the k[Gα] form a descending chain of subgroups of ] and it is shown that if the intersection of all the subgroups in this chain is trivial then G and hence each Gα, is residually nilpotent. This answers a question of S. Moran.

A new linear expression in σ(ν), ν = l, 2, …, n, which vanishes identically is established. A linear expression in σ(ν)'s has been found for σ3(n). A similar expression in σ3(ν)'s has been proved for σ7(n) also, Ramanujan's τ(n) = P24(n-1) is given in three different ways as linear expressions in σ2k+1(n) and σK(ν)'s with k = 1, 3, 5 respectively. Again, the coefficient p48(n-2) is expressed as a linear expression in σ11(v)'S and σ5(ν)'s. In establishing these results advantage is taken of the general theorem, also established, that the coefficients of the square of a power series whose coefficients satisfy a certain functional equation are expressible as linear functions of the latter coefficients.

We associate with a graph (finite, undirected, without loops and multiple lines) a graph T(G), called the total graph of G. This new graph has the property that a one-to-one correspondence can be established between its points and the elements (points and lines) of G such that two points of T(G) are adjacent if and only if the corresponding elements of G are adjacent or incident. The object of this article is to prove the following theorem: If K(G1) = n, n ≥ 1, and λ(G2) = m, m ≥ 1, then K(T(G1)) ≥ n + 2 + [(n - 2)/3], λ(T(G1)) ≥ 2n, K(T(G2)) ≥ m + 1, and λ(T(G2)) ≥ 2m, where k(G) and λ(G) denote the connectivity and line-connectivity of the graph G.

It has been shown by D. Stephen that the number N of open sets in a non-discrete topology on a finite set with n elements is not greater than 3 × 2n-2.We show that for admissable topologies on a finite group N ≦ 2n/r, where r is the least order of its non-trivial normal subgroups. This is clearly a sharper bound.

Here we announce results which will appear in full in due course elsewhere. The aim is a study of varieties of metabelian groups. The technique is to study varieties of certain universal algebras, which are very similar to groups, called bigroups. For certain varieties of bigroups all the non-nilpotent join-irreducible subvarieties are determined, and this is used to reduce the same problem for varieties of metabelian groups to the case of prime-power exponent. Questions of distributivity of the lattice of metabelian varieties are also discussed.

It is proved that an orthocomplemented poset P is an orthomodular lattice if and only if it admits a suitable defined set of order preserving maps. These maps are called projections. They are, in fact, Just the projections of the Baer *-semigroup associated with the orthomodular lattice.

The following result is proved. Let X be a Schunk class and k a positive integer. Then the k–th nilpotent product of any two groups in X¯ is in X¯.

A complex analytic family of mappings P → M(α, P) from an abstract Riemann surface (analytic manifold) into itself is studied. The mapping M(α, P) is assumed to satisfy in local coordinates the autonomous differential equation = L(w), and the condition M(O, P) = P. Under certain assumptions of regularity of the reciprocal differential L in a domain D ⊂ S, we prove that for every fixed α, ∣a∣ < α, the mapping M(α, P) is conformal and one to one in D. Moreover, it is shown that the family of mappings M(α, P) satisfies the iteration equation M[a, M(b, P)] = M(a + b, P) and hence is an analytic group (analytic iteration).

Consider a stably stratified liquid, whose density varies exponentially with the vertical co-ordinate, that is bounded above by a free surface and below by a bed whose height depends on only one of the horizontal co-ordinates. Suppose that a gravity wave, that may be either a surface or an internal one, is travelling in a direction normal to the lines of constant depth. It is shown that if the frequency is below a certain value an infinite number of waves, all of the same frequency but having differing wave lengths, are generated and expressions for their amplitude are given in terms of the changes in depth which are assumed to be small.

For brevity the semigroups in the title are called orthodox semigroups. The finest inverse semigroup congruence on an orthodox semigroup is shown to have a simple form and conversely, regular semigroups whose finest inverse congruence has this simple form are shown to be orthodox. Next ideal extensions of orthodox semigroups by orthodox semigroups are shown to be also orthodox, whence a finite semigroup is orthodox if and only if each principal factor is orthodox and completely O-simple or simple. Finally it is determined which completely O-simple semigroups are orthodox.

We establish several infinite classes of regular graphs with the property that any two distinct vertices have a fixed number of other vertices joined to both of them. The graphs are found by constructing their incidence matrices, which correspond to certain Hadamard matrices.

It is a curious fact that, while the functions which preserve closures are exactly the closed continuous functions, the interior-preserving functions form a much smaller class than the open continuous functions. In fact, the interior-preserving maps are essentially homeomorphisms, surjections between discrete spaces, and “sums” of these.

A weak logical structure is defined as a set of boolean propositional logics in which one can define common operations of negation and implication. The set union of the boolean components of a weak logical structure is a logic of propositions which is an orthocomplemented poset, where orthocomplementation is interpreted as negation and the partial order as implication. It is shown that if one can define on this logic an operation of logical conjunction which has certain plausible properties, then the logic has the structure of an orthomodular lattice. Conversely, if the logic is an orthomodular lattice then the conjunction operation may be defined on it.