We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

Whitman [2] solved the word problem for lattices by giving an explicit construction of the free lattice, FL(X), on a given set of generators X.

The solution is the following:

For x, y ∊ X, and a, b, c, d ∊ FL(X),

(W1)

(W2)

(W3)

(W4)

where [p, q] = {x; p ≤ x ≤ q}.

The purpose of this note is to give a simple nonconstructive proof that the condition (W4) must hold in every projective (hence every free) lattice. Jonsson [1] has shown that in every equational class of lattices (Wl), (W2), and (W3) hold. Therefore the combination of these results gives a complete nonconstructive solution to the word problem for lattices.

As a step towards characterizing ID-groups (i.e., groups G such that, for every ring R without zero-divisors, the group ring RG has no zero-divisors), Rudin and Schneider defined Ω-groups, a possibly wider class than that of right-orderable groups, and proved that if every non-trivial finitely generated subgroup of a group G has a non-trivial H-group as an epimorphic image, then G is an ID-group. We prove that such groups are even Ω-groups and obtain the analogous result for right-orderable groups.

We determine the general solutions of the functional equation for ƒi: G → F (i = 1,2,3,4), where G is a 2-divisible group and F is a commutative field of characteristic different from 2. The motivation for studying this equation came from a result due to Dry gas [4] where he proved a Jordan and von Neumann type characterization theorem for quasi-inner products. Also, this equation is a generalization of the quadratic functional equation investigated by several authors in connection with inner product spaces and their generalizations. Special cases of this equation include the Cauchy equation, the Jensen equation, the Pexider equation and many more. Here, we determine the general solution of this equation without any regularity assumptions on ƒi.

The relationship between properties of the generalized inverse of A, A†, and of the adjoint of A, A*, are studied. The property that A†A and AA† commute, called (E4), is investigated. (E4) generalizes the property of A being EPr. A canonical form and a formula for A† are given if a matrix A is (E4). Results are in a Hilbert space setting whenever possible. Examples are given.

H1. Equip ℝ3 with the group law

(1.1)

where (z, t) stands for (x, y, t). This is a nilpotent Lie group, usually referred to as the first Heisenberg group, H1. In general Hk denotes ℝ2k+1 equipped with a similar group law, namely

In this paper we investigate the uniqueness of transcendental meromorphic function dealing with the shared values in some angular domains instead of the whole complex plane.

Paul Halmos expressed [3, p. 110] the general dissatisfaction with the usual proofs of this famous and important theorem. They all make it seem like an accidental product of a computation. A more conceptual proof was devised by N. P. Dekker. In spite of the elegance of his proof, the one offered below may have some claim to be regarded as "the reason the theorem is true".

An element x in a lattice L is join-reducible (meet-reducible) in L if there exist y, z∈L both distinct from x such that x=y⋁z (x=y⋀z); x is join-irreducible (meet-irreducible) in L if it is not join-reducible (meet-reducible) in L; x is doubly irreducible in L if it is both join- and meet-irreducible in L. Let J(L), M(L), and Irr(L) denote the set of all join-irreducible elements in L, meet-irreducible elements in L, and doubly irreducible elements in L, respectively, and ℓ(L) the length of L, that is, the order of a maximum-sized chain in L minus one.

P. Erdös and G. Szekeres [1] proved that from any points in the plane one can always choose n + 1 of them which are the vertices of a convex polygon, thus answering a question due to Miss Esther Klein (who later became Mrs. G. Szekeres).

Burgess [l] has pointed out that in the categories and (where . is the category of Ti. spaces), epi means onto. In this paper, Burgess1 technique will be used to show that epi has a different meaning in and that this meaning reduces to onto when the range is a T1 space.

Let $\text{Z}$ be the unital simple nuclear infinite dimensional ${{C}^{*}}$-algebra which has the same Elliott invariant as $\mathbb{C}$, introduced in [9]. A ${{C}^{*}}$-algebra is called $\text{Z}$-stable if $A\,\cong \,A\,\otimes \,\text{Z}$. In this note we give some necessary conditions for a unital simple ${{C}^{*}}$-algebra to be $\text{Z}$-stable.

We use the notations of [2], particularly for operators on classes of structures; in addition, P*r(K) (respectively, Pr(K)) denotes the class of all reduced products of families of structures in K (those with respect to a proper dual ideal, respectively).

Let us define a planar partition of a graph G as a pair {H1, H2} of subgraphs of G with the following properties

1. (i) Each of H1 and H2 includes all the vertices of G.

2. (ii) Each edge of G belongs to just one of H1 and H2.

3. (iii) H1 and H2 are planar graphs.

It is not required that H1 and H2 are connected. Moreover either of these graphs may have isolated vertices, incident with none of its edges.

We describe a graph having a planar partition as biplanar.

The purpose of this note is to offer a equalization of the concept of Goldie dimension and to prove a structure theorem (Theorem 3.1) for modules satisfying the conditions of this dualization. In this paper, all rings considered are associative with unit and all modules are unital.

In this note the stability theorem of Albert and Baker concerning the n-th difference equation is proved by using invariant means.

We study a class of irreducible modules for Affine Lie algebras which possess weight spaces of both finite and infinite dimensions. These modules appear as the quotients of "imaginary Verma modules" induced from the "imaginary Borel subalgebra".

Let R be a prime ring of characteristic not equal to two and let T be an automorphism of R. If U is a Lie ideal of R such that T is nontrivial on U and xxT — xTx is in the center of R for every x in U, then U is contained in the center of R.

In his classic memoir on the moment problem that bears his name, Stieltjes [2] exhibited

1

as an example of an indeterminate (Stieltjes) moment sequence.

Stieltjes also obtained the corresponding S-fraction and thus implicitly obtained the three-term recurrence formula satisfied by the corresponding orthogonal polynomials.

Let X be a compact metric space. By a packing in X we mean a subset S ⊆ X such that, for x, y ∈ S with x ≠ y, the distance d(x, y) ≥ 1. Since X is compact, any packing of X is finite. In fact, the set of numbers

{card(S): S is a packing in X}

is bounded. The cardinality of the largest packing in X will be called the packing number of X and will be denoted by ρ(X). If A(X) and P(X) denote the area and perimeter, respectively, of a compact convex subset X of the plane, then a special case of a result conjectured by H. Zassenhaus [6] and proved by N. Oler [l] is the following.