Let L be a complemented, χ-complete modular lattice. A theorem of Amemiya and Halperin (see [l], Theorem 4.3) asserts that if the intervals [O, a] and [O, b], a, bεL, are upper χ-continuous then [O, a∪b] is also upper χ-continuous. Roughly speaking, in L upper χ-continuity is additive. The following question arises naturally: is χ-completeness an additive property of complemented modular lattices? It follows from Corollary 1 to Theorem 1 below that the answer to this question is in the negative.

A complemented modular lattice is called a Von Neumann geometry or continuous geometry if it is complete and continuous. In particular a complete Boolean algebra is a Von Neumann geometry. In any case in a Von Neumann geometry the set of elements which possess a unique complement form a complete Boolean algebra. This Boolean algebra is called the centre of the Von Neumann geometry. Theorem 2 shows that any complete Boolean algebra can be the centre of a Von Neumann geometry with a homogeneous basis of order n (see [3] Part II, definition 3.2 for the definition of a homogeneous basis), n being any fixed natural integer.

Un anneau partiellement ordonné A est un anneau sur lequel est définie une relation d'ordre partiel telle que:

1. 1. a≥b entraîne a+c≥+c pour tout c∈A.

2. 2. a≥o et b≥o entraînent ab≥o.

Si la relation d'ordre est une relation d1 ordre total, l'anneau A est dit un anneau totalement ordonné.

Il est bien connu (Théorème d'artin-Schreier [l]) que, pour qu'un corps commutatif K puisse être totalement ordonné, il faut et il suffit que la relation entraîne a1=…=an=o. Ce théorème a été généralisé par T. Szele [4] qui a montré que, pour qu'un corps quelconque K puisse être totalement ordonné, il faut et il suffit que le demi-groupe additif et multiplicatif S, engendré par les éléments de K qui sont des carrés non nuls, ne contienne pas I'élément zéro de K. Ce résultat a été étendu au cas d1 un anneau d'intégrité par R.E. Johnson [2] de la manière suivante.

P. Scherk a posé le problème suivant (Canadian Mathematical Bulletin, vol.4, no. 3, 1961, p. 306, problème 49):

Let x(t) and y(t) be two differentiable functions defined in some interval. Does the function have the Darboux property?

La réponse est négative. En effet, Witold Wilkosz (Some properties of derivative functions, Fundam. Math. vol.2, 1921, p. 145−154) a montré (p. 146−147 loc. cit.) que la fonction

The object of this paper is to give simple criteria for the existence of direct limits in categories and for the permuting of a functor with direct limits.

The notion of direct limit of a diagram that we shall use here is essentially that of Kan (4), which is more general than the usual notion of direct limit of a directed diagram.

Our treatment is based on the fact (Lemma 2) that the usual process for constructing the direct limit of a diagram of modules, which consists in taking a direct sum of the modules in the diagram and then considering a certain homomorphic image of this direct sum (3, p. 220), is essentially, once certain notions have been properly generalized, the only process for constructing the direct limit of any diagram in any category.

Utumi has shown [3, Claim 5.1] that for a certain class of rings the associated maximal rings of right quotients are isomorphic to the endomorphism rings of modules over division rings. We shall prove a generalization of this theorem and then show how it is obtained as a corollary. The following proofs do not depend on Utumi's paper; instead, they make extensive use of results proved in [1]. The terminology and notations employed here are the same as in [1].

I wish to thank Dr. B. Banaschewski for his suggestions and helpful criticism.

LEMMA: If J is a left ideal with zero left annihilator in a ring R then a maximal ring of right quotients of R is also a maximal ring of right quotients of J.

In this note we obtain some simple criteria which show that in certain algebraic number fields factorization of elements is not unique. All the arguments depend only on the most elementary ideas (except in §6) so that probably many of the results are not new. However the proofs are short and direct and therefore should be of some interest.

We now list some basic facts, whose proofs can be found in any of the books listed in the bibliography, at the same time fixing our notation.

Marshall Hall [l] shows how projective planes of very general structure may be constructed and at the same time exhibits an extensive class which are non-Desarguesian. Here we shall indicate how his method of free extension can be generalized to yield a class of planes which seem to be distinct from those which he obtains.

A partial plane is a system consisting of two distinct sets of elements, a set of "points" P, Q,… and a set of "lines" l, m,…, and a relation between these two sets, called "incidence", such that for any two distinct points, there is at most one line incident with both (or, equivalently, for any two distinct lines, there is at most one point incident with both). A partial plane is complete if every two distinct points are joined by a line and every two distinct lines intersect in a point.

A recent publication entitled "Magic Squares and Cubes", by W. S. Andrews, published by the Dover Publications, New York, is an excellent authoritative book on magic squares, and magic cubes. In fact, chapter XIV entitled "Magic Octrahedroids", has shown examples of the extension into four-dimensional space. Andrews' method seems to be that of extending symmetrical considerations, which he calls reversions, in forming higher dimensional forms of magic squares. Neither he, nor any other author, to the best of my knowledge, has extended magic squares to higher than four dimensions.

La Hire must be given full credit for his method of breaking down magic squares into component squares, and conversely constructing magic squares from component squares. The approach of this author is to extend La Hire's method for n-dimensional space by means of a relation:

In a recent paper [1], L. Moser and E.L. Whitney have proved the following.

Theorem: The number of compositions of n into parts=1, 2, 4 or 5 (mod 6) and involving an even number of parts=4 or 5 (mod 6) exceeds by n the number of compositions of n into parts=1, 2, 4 or 5 (mod 6) and involving an odd number of parts=4 or 5 (mod 6).

Their method of proof utilizes the notion of weighted compositions and the method of generating series. They remark that they have not been able to find a direct combinatorial proof. The purpose of this note is to give a direct proof of a more general result.