ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. Let $L$ be a lattice and $M$ a bounded distributive lattice. Let $\mathop{\rm Con}L$ denote the congruence lattice of $L$, $P(M)$ the Priestley dual space of $M$, and $L^{P(M)}$ the lattice of continuous order-preserving maps from $P(M)$ to $L$ with the discrete topology. It is shown that $\mathop{\rm Con}(L^{P(M)})\cong(\mathop{\rm Con}L)_{\Lambda}^{P(\mathop{\rm Con}M)}$, the lattice of continuous order-preserving maps from $P(\mathop{\rm Con}M)$ to $\mathop{\rm Con}L$ with the Lawson topology. Various other ways of expressing $\mathop{\rm Con}(L^P)$ as a lattice of continuous functions or semilattice homomorphisms are presented. Indeed, a link is established between semilattice homomorphisms from a semilattice $S$ into a bounded distributive lattice $T$ (or its ideal lattice) and continuous order-preserving maps from $P(T)$ into the ideal lattice of $S$ with the Scott, Lawson, or discrete topology. It is also shown that, in general, $\mathop{\rm Con}(L^{P(M)})\not\cong(\mathop{\rm Con}L)^{P(\mathop{\rm Con}M)}$, solving a problem of E. T. Schmidt (independently solved by Grätzer and Schmidt).

AMS Subject Classification (1991): 06B10, 08A30, 08B26, 06B30, 06F30, 22A26, 06E15, 06B35, 06A12, 08B25, 18A30, 18A35

Keyword(s): Priestley power, Boolean power, (generalized) function lattice, congruence lattice, algebraic lattice, semilattice, (bounded) distributive lattice, Priestley space, Scott topology, Lawson topology, exponentiation

Received September 4, 1995 and in revised form February 7, 1996. (Registered under 5707/2009.)

Abstract. Strietz [4, 5] has shown that $\mathop{\rm Equ}(A)$, the lattice of all equivalences of a finite set $A$, has a four-element generating set. We extend this result for many infinite sets $A$; even for all sets if there are no inaccessible cardinals. Namely, we prove that if $A$ is a set consisting of at least four elements and there is no inaccessible cardinal $\le |A|$, then the {\it complete} lattice $\mathop{\rm Equ}(A)$ can be generated by four elements. This result is sharp in the sense that $\mathop{\rm Equ}(A)$ cannot be generated by three elements.

AMS Subject Classification (1991): 06B99, 06C10

Keyword(s): Lattice, equivalence, equivalence lattice, generating set, inaccessible cardinal

Received September 1, 1995 and in revised form January 12, 1996. (Registered under 5708/2009.)

Abstract. Two projective geometric theorems are characterized by lattice identities.

AMS Subject Classification (1991): 06C05, 51A30

Received January 2, 1995 and in revised form January 5, 1996. (Registered under 5709/2009.)

Abstract. A classical improvement of the fundamental Minkowski--Blichfeldt theorem in geometry of numbers is the Siegel--Bombieri formula: if we know that for a bounded measurable set $A\subset{\msbm R}^n$ the algebraic difference $A-A$ contains no non-zero points of a full dimensional point lattice $\Lambda\subset {\msbm R}^n$, then $V(A)(d\Lambda -V(A))$ is equal (roughly speaking) to a non-negative Fourier series generated by $\Lambda $ and $A$, where $d\Lambda $ is the determinant of $\Lambda $ and $V(A)$ is the measure of $A$. In the paper both improvements and extensions (to arbitrary dimensional point lattices $L$) of this result are proved. These are achieved by using two principially new tools. First, a series of five consecutive inequalities between the cardinality of the set $(A-A)\cap L$ and the number 1, as well as exact conditions of equality in four of them, are proved. (These among other things show that the condition ``$(A-A)\cap L$ contains only the zero vector" can be satisfied only by quite special $A$.) Second, a special technique is worked out to extend the Fourier analytic methods of Siegel and Bombieri to point lattices $L$ of any dimensions.

AMS Subject Classification (1991): 11H16, 42B05

Received May 19, 1995 and in revised form March 5, 1996. (Registered under 5710/2009.)

Abstract. For any given positive integer $n$ let $\rho(n) = \max\sum _{i=1}^r 1/a_i $, where the maximum is taken over all integers $a_1, a_2, \cdots,a_r $ which satisfy $1\le a_1 < a_2 < \cdots < a_r \leq n$ and $[a_i, a_j] > n$ ($1\leq i< j\leq n$). In this paper we show that $\rho(n) < 1.0170166 $ for large $n$ and give two conjectures either of which implies Erdős' conjecture $\rho(n)\to1$ ($n\to\infty $).

AMS Subject Classification (1991): 11Y60, 11A99

Keyword(s): positive integer, least common multiple, Erdős conjecture

Received March 3, 1995. (Registered under 5711/2009.)

Abstract. We show that, up to isomorphism, there is exactly one topological nearring which is not zero symmetric and has an identity, whose additive group is the two-dimensional Euclidean group ${\msbm R}^2$. We determine the endomorphism semigroup and the automorphism group of this nearring. We determine its right, left and two-sided ideals. In particular, we show that it has exactly one nonzero proper two-sided ideal and that the corresponding quotient nearring is the field of real numbers. Finally, we investigate the structure of the multiplicative semigroup of this nearring.

AMS Subject Classification (1991): 16Y30

Received November 7, 1995. (Registered under 5712/2009.)

Abstract. Hilbert triple systems were introduced into mathematics because of the role they play in a certain part of infinite-dimensional geometry. Recently the theory of involutive $H^\ast $-triples emerged as a triple analogue of $H^\ast $-algebras which are a classical topic in the Banach algebra theory. Our intention is to compare those theories. Structure theorems for associative Hilbert triple systems and associative $H^\ast $-triples, as well as the Wedderburn type theorems for general triple systems of both types, suggest that there might be an intimate connection between these two theories. We prove that this is in fact the case. More explicitly, we give two ways of constructing a simple involutive $H^\ast $-triple from a simple Hilbert triple and then prove that every simple $H^\ast $-triple can be obtained from a simple Hilbert triple via one of the above mentioned constructions. These results can be applied in order to give structure theorems for alternative Hilbert triple system and $JH^\ast $-triples.

AMS Subject Classification (1991): 17A40, 17A60, 17C65, 17D05, 46H25, 46K15, 46K70

Keyword(s): Hilbert triple system, H^\ast, involutive-triple, Hilbert module, isomorphism pair, polarized triple, alternative triple, Jordan triple

Received June 2, 1994 and in revised from November 15, 1994. (Registered under 5713/2009.)

Abstract. Alternative algebras known as comtrans algebras have been used in the coordinatization of web structures, and in the formulation of quantum mechanics. Time reversal in quantum mechanics corresponds to transposition of comtrans algebras. The current paper presents an alternative axiomatization of comtrans algebras, namely as tercom algebras, that simplifies the description of transposition. A new class of simple comtrans or tercom algebras, corresponding to simple alternative Akivis algebras, is then obtained.

AMS Subject Classification (1991): 17D10, 17D99, 53A60

Received December 28, 1994 and in revised form November 17, 1995. (Registered under 5714/2009.)

Abstract. The well-known inequalities of G. H. Hardy and J. E. Littlewood [1] play very important role in the proof of many theorems concerning convergence or summability of orthogonal series. Recently, L. Leindler proved a number of new inequalities of Hardy--Littlewood type (see, for example, [3], [4], [5], and references cited there). The aim of the present paper is to prove a Hardy--Littlewood type inequality for double numerical series.

AMS Subject Classification (1991): 40A05

Received January 29, 1996. (Registered under 5715/2009.)

Abstract. It is given a characterization of weighted composition operators which satisfy a Korovkin type sequential approximation property.

AMS Subject Classification (1991): 41A35, 46J10, 47B38

Received January 16, 1996. (Registered under 5716/2009.)

Abstract. Explicit expressions of the Cotes numbers of the generalized Gaussian Kronrod--Turán quadrature formulas for the Chebyshev weight and their asymptotic behavior are given.

AMS Subject Classification (1991): 41A55; 65D32

Keyword(s): Kronrod--Turán quadrature formula, Cotes numbers, Chebyshev weight

Received January 3, 1996. (Registered under 5717/2009.)

Abstract. Let $\rho\in C^\infty({\msbm R}^d\setminus\{0\} )$ be a distance function, homogeneous with respect to a dilation group $(\exp t\log P)_{t>0}$. For $f\in L^1({\msbm R}^d)$ we consider the pointwise behavior of generalized Riesz means for the Fourier integral, defined by $\widehat{S^t_{\rho,\lambda } f}=(1-\rho(\xi )/t)^\lambda_+\widehat f$. Let ${\eufm M}_{\rho,\lambda }$ be the associated maximal operator. If $P$ is a multiple of the identity then it is known that $\eufm M_{\rho,\lambda }$ is of weak type $(1,1)$ if $\lambda >(d-1)/2$. We show that if $P$ is not a multiple of the identity then for suitable $\rho $ the weak type inequality may fail to hold for $\lambda < d/2$; moreover $L^p$ boundedness of $\eufm M_{\rho,\lambda }$ may fail to hold if $\lambda\le d(1/p-1/2)$. Sharp results are discussed for the case $d=2$, under an additional finite type assumption.

AMS Subject Classification (1991): 42B25, 42B15

Keyword(s): Quasiradial multipliers, Riesz means, pointwise convergence, maximal operators

Received December 11, 1995. (Registered under 5718/2009.)

Abstract. Nonlinear equations in a lattice normed space are considered. Solvability conditions are obtained. They generalize and make somewhat more precise the Ostrowski result for equations with differentiable mappings in a Banach space. Besides, perturbations of differentiable mappings are investigated. Moreover, existence of positive solutions of differentiable maps and an estimate from below for them are established.

AMS Subject Classification (1991): 46B42, 47H15

Keyword(s): Banach lattices, nonlinear equations, solvability

Received April 6, 1995. (Registered under 5719/2009.)

Abstract. For this concept moduli of smoothness for weighted $L^p$ spaces on $[-1,1]$ are based on algebraic additions on $[-1,1]$ where the additions depend on the weight functions. We first investigate algebraic and differential properties of the additions which will be needed to establish some properties of the moduli. Steklov means and equivalent K-functionals are given.

AMS Subject Classification (1991): 46E35, 46B70, 41A10

Received January 26, 1996 and in revised form March 22, 1996. (Registered under 5720/2009.)

Abstract. The notion of a generalized sequence of Colombeau's generalized functions is introduced and studied. A weakly convergent sequence of Schwartz's distributions which is not an almost constant one does not determine a convergent sequence of Colombeau's generalized functions but it determines a weakly or h-weakly convergent generalized sequence of Colombeau's generalized functions.

AMS Subject Classification (1991): 46F05, 03H05

Keyword(s): Generalized functions, convergence, non-standard analysis

Received May 5, 1995 and in revised form October 11, 1995. (Registered under 5721/2009.)

Abstract. This note is the counterpart of [1] in which it was given a new contruction of the smallest positive self-adjoint extension of a positive linear map. In this note we apply this method to the largest positive self-adjoint extension, and we prove a sufficient condition for a pair of bounded operators to be interwined by this extension.

AMS Subject Classification (1991): 47A20, 47B25

Keyword(s): Friedrichs extension, commutation

Received October 12, 1995. (Registered under 5722/2009.)

Abstract. We construct a (non-integrable) function $f$ and two measure preserving, ergodic transformations ${\bf S}$ and ${\bf T}$ on a measure space $({\cal X}, {\cal A},\mu )$, $\mu({\cal X})=1$, in such a way that the ergodic means $\lim_{n\to\infty }{1\over n} \sum_{k=1}^n f({\bf S}^k x)$ and $\lim_{n\to\infty } {1\over n}\sum_{k=1}^n f({\bf T}^k x)$ exist for almost all $x$, they are finite constants not depending on $x$, but these constants differ when we are averaging with respect to the operators ${\bf S}$ and ${\bf T}$. This means that in the case of a non-integrable function $f$ and an ergodic transformation ${\bf T}$ the ergodic mean depends not only on the function $f$, but also on the transformation ${\bf T}$. The construction applies some probabilistic arguments.

AMS Subject Classification (1991): 47A35

Received July 26, 1995 and in revised form December 14, 1995. (Registered under 5723/2009.)

Abstract. Let $A$ and $B$ be selfadjoint operators on a Hilbert space with spectral families $E(\lambda )$ and $F(\lambda )$, respectively. $A\preceq B$ means $E(\lambda )\geq F(\lambda )$ for every $\lambda $, and this order is called the \it spectral order. \rm A selfadjoint operator valued real analytic function which is defined on $\bf R$ is called a \it real analytic wave \rm if it is piecewise monotone in the spectral order sense. We show that $ A\preceq A +s B$ for every $s>0$ implies $AB=BA$, and that a finite set of selfadjoint operators is commutative if and only if these operators are connected by a real analytic wave.

AMS Subject Classification (1991): 47B15

Received November 20, 1995. (Registered under 5724/2009.)

Abstract. Using a KKM theorem in this paper a theorem on best approximations in a class of not necessarily locally convex topological vector spaces is proved.

AMS Subject Classification (1991): 47H10

Received June 26, 1995. (Registered under 5725/2009.)

Abstract. A common fixed point theorem is proved for a multifunction, not necessarily convex valued, with open fibres and a family of commuting affine continuous mappings each of which commutes strongly with the multifunction. As a consequence of this result, a Browder's fixed point theorem for convex valued multifunction with open fibres is obtained. Also, a common fixed point theorem due to Itoh and Takahashi is generalized to a common fixed point theorem for a Kakutani factorizable multifunction and a family of commuting continuous affine mappings each of which commutes with all the factors of the factorizable multifunction.

AMS Subject Classification (1991): 47H10; 54H25

Received July 19, 1995 and in revised form March 7, 1996. (Registered under 5726/2009.)

Abstract. In the affine plane $AG(2,q)$ over the Galois field $GF(q)$, $q=p^h$ and $p$ an odd prime, let the parabola ${\cal P}$ be given in its canonical form $y=x^2$. For a positive integer $v< h$, put $d=p^v$ and denote by $K_d$ the set of size $q/d$ consisting of all points $(x^d-x,(x^d-x)^2)$ with $x$ ranging over $GF(q)$. The set $K_d$ lies on ${\cal P}$, and hence $K_d$ may be viewed as a $q/d$-gon in ${\cal A}$, because no three points of $K_d$ are collinear. Szőnyi [7] proved that if $q/d$ is sufficiently large, then the chords of $K_d$ cover every affin point apart from those in ${\cal P} \setminus K_d$. In this paper we give an upper bound for the number $n_P(d)$ of chords of $K_d$ through a given point $P$ outside of ${\cal P}$.

AMS Subject Classification (1991): 51E20, 51E21

Received October 24, 1995 and in revised form March 5, 1996. (Registered under 5727/2009.)

Abstract. In a Sasakian manifold, we discuss the special projective Killing $p$-form with constant $k$. Moreover, we seek some conditions for a Sasakian manifold to be isometric with a unit sphere.

AMS Subject Classification (1991): 53C65, 53C25

Received November 18, 1994. (Registered under 5728/2009.)

Abstract. We investigate the relations between metric completeness and order completeness and obtain characterizations of metric completeness using fixed point theorems and the existence of maximal elements, respectively.

AMS Subject Classification (1991): 54E50, 54H25

Keyword(s): Completeness, fixed point, stationary point, maximal element

Received July 19, 1995. (Registered under 5729/2009.)