Dans [2] nous avons introduit, en nous inspirant des notions équivalentes pour les anneaux [3], les notions de monoïde maximal Q(S) et de monoïde complet B(S) des quotients d'un monoïde donné S, la premiére construction n'étant valide que lorsque S est commutatif, et nous avons montré que dans ce cas, on a toujours les inclusions (en général strictes) de monoïdes suivantes: S ⊂ Q(S) ⊂ B(S).

Let R be an associative ring (not necessarily with identity).

R is a left П-ring if it has the following property: Let M be a finitely generated left R-module, N a submodule of M and ϕ:N→M an epimorphism. Then ϕ is an isomorphism.

The relationship between bases and isomorphisms (i.e. linear homeomorphisms) between complete metrizable linear spaces has been studied with great interest by Arsove and Edwards (see [1] and [2]). We prove (Theorem 1) that in the case of B-complete barrelled spaces, similar generalized bases imply existence of an isomorphism. This result was also proved by Dyer and Johnson [4], so we do not give a proof. We show (Theorem 6) that if one assumes that the bases are Schauder and similar, then Theorem 1 holds for countably barrelled spaces. We use Theorem 1 to advantage (Theorems 2-5) to show that one can improve some results due to Davis [3].

H. T. Croft's Research Problems, 1967, contains the following problem due to J. H. Conway. "Let A = {a1, a2,…, aN} be a finite set of integers, and define

and

Prove that A—A always has more members than A+A, unless A is symmetric about 0."Marica in [1] showed that the conjecture is false for the set A = {1, 2, 3, 5, 8, 9, 13, 15, 16}. In this case A+A has 30 elements and A—A has 29 elements.

The study of group types was completed by Meldrum [1]. The concept of ring type described here is based on analogous definitions.

The series R = R0 ⊃ R1 ⊃ … ⊃ Rα = Rα+1is the lower central series for the ring R if Rγ+1=RRγ + RγR for ordinal number γ and Rγ∩δ<γ Rδ if γ is a limit ordinal. The upper central series for R is the series 0=J0 ⊂ J1 ⊂ … ⊂jβ = Jβ+1 where Jγ+1={x ∊ R:xR + Rx ⊆ Jγ} for every ordinal number γ and Jγ = ∪ ∩δ<γ Jδ if γ is a limit ordinal. The length of the upper central series is the smallest ordinal number γ for which Jβ = Jβ+1.The length of the lower central series is defined similarily. We shall say the ring has type (β,α) if the length of the upper central series is β and the length of the lower central series is α.

Aczel has conjectured, [1, p. 448], the possibility of developing a net theory for structures more general than quasigroups. Steps in this direction have been taken by Havel who considers nets associated with multigroupoids [2], The work presented here introduces a generalization of 3-nets and their algebraization which is wide enough to encompass most algebraic structures based on a single binary operation.

A presentation of a group G is an exact sequence of groups

where F is a free group. Let l→S ⊆ F→G→1 be another presentation of G involving the same free group F.

A mapping f from a metric space (X,d) into itself is said to be contractive if x≠y implies d(f(x),f(y))<d(x,y). Theorems of Edelstein [2] state that a contractive selfmapfofa metric space X has a fixed point if, for some x0, the sequence {fn(x0)} of iterates at x0 has a convergent subsequence; moreover, the sequence {fn(x0)} converges to the unique fixed point of f. Nadler [3] observes that, from the point of view of applications, it is usually as difficult to verify the condition (for some x0 …) as it is to find the fixed point directly.

Throughout this note, we adopt the graph-theoretical terminology and notation of Harary [3]. A graph G is hypohamiltonianif G is not hamiltonian but the deletion of any point u from G results in a hamiltonian graph G-u. Gaudin, Herz, and Rossi [2] proved that the smallest hypohamiltonian graph is the Petersen graph. Using a computer for a systematic search, Herz, Duby, and Vigué [4] found that there is no hypohamiltonian graph with 11 or 12 points. However, they found one with 13 and one with 15 points. Sousselier [4] and Lindgren [5] constructed independently the same sequence of hypohamiltonian graphs with 6k+10 points. Moreover, Sousselier found a cubic hypohamiltonian graph with 18 points. This graph and the Petersen graph were the only examples of cubic hypohamiltonian graphs until Bondy [1] constructed an infinite sequence of cubic hypohamiltonian graphs with 12k+10 points. Bondy also proved that the Coxeter graph [6], which is cubic with 28 points, is hypohamiltonian.

Let p be an odd prime, n an integer not divisible by p and α a positive integer. For any integer h with (h,pα)=l, is defined as any solution of the congruence (mod,pα). The Kloosterman sum Ap α(n) (see for example [4]) is defined by

(1.1)

where the dash (') indicates that the letter of summation runs only through a reduced residue system with respect to the modulus.

Let A be an ideal of the commutative ring R with identity. There is a canonical homomorphism ϕA from the polynomial ring R[X] onto (R/A)[X], obtained by reducing all coefficients modulo A. If fR[X], then we say that f is reducible (irreducible) modulo A if ϕA(f) is reducible (irreducible) in (R/A)[X].

Maeda ([1, p. 29, Problème 1]) pose la question suivante: Est-ce qu'un treillis ⏊-symétrique et relativement semi-orthocomplémenté est nécessairement M-symétrique? Nous donnons une réponse affirmative à cette question.

Soit L un treillis, et a,b,y ∊ L. Si (a,b)M et y≤b, alors (y V a,b)M.

In answer to a question raised by Leo Moser, A. Meir proved some years ago that every plane arc of unit length lies in some closed semidisk of radius ½. His elegant, unpublished argument is reproduced here with his kind permission.

In 1892, Sylvester [7] proved that in the set of integers n, n+l,…, n+k—1, n> k > 1, there is a number containing a prime divisor greater than k. This theorem was rediscovered, in 1929, by Schur [6]. More recent results include an elementary proof by Erdös [1] and a proof of the following theorem by Faulkner [2]: Let pk be the least prime ≥2k; if n≥pk then has a prime divisor ≥pk with the exceptions and In that paper the author uses some deep results of Rosser and Schoenfeld [5] on the distribution of primes.

Letf:S1→S1 be an orientation preserving C1-diffeomorphism. Denote by the flow on S1 x S1 which is the suspension of f(see Smale [5]).

We consider the problem of approximating by a smoother foliation.

A line-coloring of a graph G is an assignment of colors to the lines of G so that adjacent lines are colored differently; an n-line coloring uses n colors. The line-chromatic number χ'(G) is the smallest n for which G admits an n-line coloring.

In 1959, Moser [4] posed the following problem: how should a pair of n-sided dice be loaded (identically) so that, on throwing the dice, the frequency of the most frequently occurring sum is as small as possible? This can be recast in the following form: determine for each n(≥1), the polynomial Pn(x) which minimizes the maximum coefficient in the polynomial subject to the conditions that the coefficients of Pn(x) are nonnegative and sum to unity.

The Hermite conjugate Poisson integral of a given f ∊ L1(μ), dμ(y)= exp(—y2) dy, was defined by Muckenhoupt [5, p. 247] as

where

If the Hermite conjugate function operator T is defined by (Tf) a.e., then one of the main results of [5] is that T is of weak-type (1, 1) and strongtype (p,p) for all p>l.

It follows from [1, p. 377, Lemma 1] that a noncommutative subdirectly irreducible ring, with no nonzero nilpotent elements, cannot possess any proper zero-divisors. From [2, p. 193, Corollary 1] a subdirectly irreducible distributive lattice, with more than one element, is isomorphic to the chain with two elements. Hence we can say that a subdirectly irreducible distributive lattice with 0 possesses no proper zero-divisors.

The following theorem is the principal result of this paper.

Let (M, d) be a metric space and T a self-mapping of M satisfying the condition for x,y ∊ M

1

where a, b, c, e,f are nonnegative and we set α=a+b+c+e+f.