In conjunction with an earlier work by Leong (1974a), this paper completes the solution of the isomorphism problem for finite nilpotent groups of class two with cyclic centre. A canonical decomposition for 2-groups of such type is obtained and proved.

Optimality conditions without constraint qualifications are given for the convex programming problem: Maximize f(x) such that g(x) ∈ B, where f maps X into R and is concave, g maps X into Rm and is B-concave, X is a locally convex topological vector space and B is a closed convex cone containing no line. In the case when B is the nonnegative orthant, the results reduce to some of those obtained recently by Ben-Israel, Ben-Tal and Zlobec.

The conjugacy of Cartan subalgebras of a Lie algebra L over an algebraically closed field under the connected automorphism group G of L is inherited by those G-stable ideals B for which B/Ci is restrictable for some hypercenter Ci of B. Concequently, if L is a restrictable Lie algebra such that L/Ci restrictable for some hypercenter Ci of L, and if the Lie algebra of Aut L contains ad L, then the Cartan subalgebras of L are conjugate under G. (The techniques here apply in particular to Lie algebras of characteristic 0 and classical Lie algebras, showing how the conjugacy of Cartan subgroups of algebraic groups leads quickly in these cases to the conjugacy of Cartan subalgebras.)

Ordered vector spaces more general than vector lattices are represented as function spaces with ‘almost’ the pointwise order. The ‘pseudo-lattice’ property which is necessary (and sufficient in the finite-dimensional case) for the representation is also studied in relation to the lattice property.

We establish a general algebraic independence theorem for the solutions of a certain kind of functional equations. As a particular application, we prove that for any real irrational ζ, the numbers are algebraically independent, for multiplicatively independent algebraic αi with 0<|<| <1.

Let G be a finite and u(G) the group of all invertible transformations (polynomial permutations) of the form x→a1 x1→ xk a2⃛ar xkr ar+1 (aiε G, x runs through G). Continuing investigations of H. Lausch of groups satisfying u(G) = {X→axk b} we show here that this condition implies that G is the direct product of its {2, 3}-Hall subgroup and its {2, 3}′-Hall subgroup H where H is nilpoint of class ≤2. Essentially all non-nilpoint groups G of order 2m 3n are described having the property u(G)= {x→axk b}

Richmond and Szekeres (1977) have conjecturned that certain of the coefficients in the power series expansions of certain infinite products vanish. In this paper, we prove a general family of results of this nature which includes the above conjectures.

The existence of Nash equilibrium points of game between n players has been established under the conditions dual to known ones.

The coproduct of a family of Kleene algebras is determined firstly by describing the maximal homomorphic image of a De Morgan algebra in the subvariety of Kleene algebras and, secondly, by characterizing the categorical product in the dual category of Kleene spaces.

We present a new class of topological spaces called SL-spaces, on which every Borel measure has a Lindelöf support. The class contains all metacompact spaces. However, a θ-refinable space is not necessarily an SL-space.

Let A be a universal algebra. A function ϕ Ak-A is called a t-local polynomial function, if ϕ can ve interpolated on any t places of Ak by a polynomial function— for the definition of a polynomial function on A, see Lausch and Nöbauer (1973), Let Pk(A) be the set of the polynomial functions, LkPk(A) the set of all t-local polynmial functions on A and LPk(A) the intersection of all LtPk(A), then . If A is an abelian group, then this chain has at most five distinct members— see Hule and Nöbauer (1977)— and if A is a lattice, then it has at most three distinct members— see Dorninger and Nöbauer (1978). In this paper we show that in the case of commutative rings with identity there does not exist such a bound on the length of the chain and that, in this case, there exist chains of even infinite length.

Let A be a finite dimensional algebra over a field F. Let R and S be biregular algebras over F such that 1R ∈ R and 1S ∈ S. We show that if R/P≃A≃ S/M for each primitive ideal P in A and each primitive ideal M in S then End FR≃ End S implies R≃S.

We are interested in determining whether two spaces are coabsolute by comparing their Boolean algebras of regular closed sets. It is known that when the spaces are compact Hausdorff they are coabsolute precisely when the Boolean algebras of regular closed sets are isomorphic; but in general this condition is not strong enough to insure that the spaces be coabsolute. In this paper we show that for paracompact Hausdorff spaces, the spaces are coabsolute when the Boolean algebra isomorphism and its inverse ‘preserve’ local finiteness, and for locally compact paracompact Hausdorff spaces, the spaces are coabsolute when the collections of compact regular closed subsets are ‘isomorphic’.