ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. Let $\rho $ be a nontrivial reflexive binary relation on a finite set $A$ with $|A|\ge3$. We consider clones on $A$ that consist of functions preserving $\rho $ and contain all unary functions with this property. We prove that if $\rho $ is either transitive or strongly intransitive, or symmetric then there exist $2^{\aleph_0}$ such clones provided $\rho $ is not a linear order. We show that, for a linear order on a three-element set, there are only 7 such clones.

AMS Subject Classification (1991): 08A40, 03B50

Received February 22, 2000, and in revised form August 31, 2000. (Registered under 2796/2009.)

Abstract. The dual discriminator function was introduced in connection with weakly associative lattices, a structure where the transitivity of the ``partial ordering'' is forgotten, but least upper bounds and greatest lower bounds exist. In this paper we investigate a more general structure where ``join'' is just one of the upper bounds and ``meet'' is just one of the lower bounds in a directed graph. Our goal is to find a dual discriminator term in this larger variety.

AMS Subject Classification (1991): 08A40, 05C75

Keyword(s): dual discriminator, congruence distributivity, varieties, generalizations of distributive lattices, directed graph

Received July 19, 2000, and in revised form January 25, 2001. (Registered under 2797/2009.)

Abstract. In this paper, we deduce from one common source a class of congruences modulo odd primes $p$ most of which are new, e.g. $$\eqalign{\sum_{k=1}^{[p/6]}{ {(-1)^k}\over k} &\equiv\sum_{k=1}^{(p-1)/2}{{3^k}\over k} ({\rm{mod}} p),\cr\sum_{k=1}^{[p/8]}{1\over k} &\equiv{5\over2}\sum_{k=1}^{(p-1)/2}{1\over k}+\sum_{k=1}^{(p-1)/2}{1\over{k\cdot2^k}} ({\rm{mod}} p).}$$

AMS Subject Classification (1991): 11A07, 11A41

Received May 22, 2000. (Registered under 2798/2009.)

Abstract. We prove that the identity $(1+2+\cdots +n)^2=1^3+2^3+\cdots +n^3$ is not a coincidence but the first and simplest one of a family of multiplicative formulas involving the sums of powers of integers.

AMS Subject Classification (1991): 11B75, 11B68

Keyword(s): 11B75, 11B68

Received July 18, 2000. (Registered under 2799/2009.)

Abstract. For an irrational number $x$, let $k_n(x)$ be the number of partial quotients in the continued fraction expansion given by the first $n$-th decimals of $x$. A basic result of G. Lochs states that $$\lim_{n\to\infty }{k_n(x)\over n}={6\log2\log10\over\pi ^2}\approx0.9702$$ for almost all $x$ in the sense of Lebesgue. In this paper we give a survey of properties of $k_n$. New results are also proved. First we give a bound for the number of decimals to have a prescribed number of partial quotients. We prove also that if $x$ has a Lévy constant $\beta(x)$ then ${k_n(x)\over n}$ converges to the limit $\log10\over2\beta(x)$ if a condition on the growth of partial quotients is satisfied. This result improves the theorem of Lochs and has an important application to the case of quadratic numbers. We study also the possible positive limits of the sequence ${k_n(x)\over n}$ for all irrationals $x$. Finally a condition which ensures that ${k_n(x)\over n}$ converges to 0 is given. For example this condition holds for $x=e$, thus $e$ doesn't satisfy Loch's result.

AMS Subject Classification (1991): 11K50, 11Y65, 11A55

Keyword(s): Continued fraction expansion, decimal expansion, Lévy constants

Received July 6, 2000, and in revised form March 2, 2001. (Registered under 2800/2009.)

Abstract. Bases of full radix representations of algebraic integers are characterised in case their minimal polynomial is a trinomial of a particularly simple form.

AMS Subject Classification (1991): 11R04, 11R16, 11R21

Received May 10, 2000, and in revised form August 7, 2000. (Registered under 2801/2009.)

Abstract. Finite groups factored to the product of normal subsets are considered. If all factors of composite order are close to groups belonging to some Fitting class then the same holds for the product. In certain cases we can assure that for suitable ordering of the factors the partial products form an ascending chain of normal subgroups.

AMS Subject Classification (1991): 20K01, 20D60

Received September 5, 2000, and in revised form October 25, 2000. (Registered under 2802/2009.)

Abstract. A characterization of subdirectly irreducible bands equipped with an involution is given and their properties are investigated. Also, subdirectly irreducibles in the varieties of semilattices with involution and some other interesting classes are discussed in more detail.

AMS Subject Classification (1991): 20M10, 08B26

Received September 5, 2000, and in revised form February 12, 2001. (Registered under 2803/2009.)

Abstract. A semigroup $S$ is called $E$-($0$-)inversive if for every $a\in S(a\not=0)$ there exists $x\in S$ such that $(ax) ^2=ax(\not=0)$. (For example, every finite and every regular semigroup is $E$-inversive.) Imposing different restrictions on the ordering of the idempotents of such a semigroup $S$ the impact of these conditions on the structure of all of $S$ is investigated. First, those $E$-($0$-)inversive semigroups are characterized which admit a unique idempotent $(\not=0)$. Next, the case when there exists a least idempotent $(\not=0 )$ is described. Also primitive $E$-($0$-)inversive semigroups, that is, all of whose idempotents $(\not=0)$ are incomparable in the natural order, are dealt with. Finally, all $E$-($0$-)inversive semigroups whose (nonzero) idempotents form an $\omega $-chain, that is, which are ordered dually to the natural numbers, are characterized.

AMS Subject Classification (1991): 20M10

Received May 8, 2000, and in final form March 22, 2001. (Registered under 2804/2009.)

Abstract. All fully invariant congruences on free commutative semigroups are constructed. This sharpens results of Kisielewicz.

AMS Subject Classification (1991): 20M14, 20M07

Received March 28, 2000. (Registered under 2805/2009.)

Abstract. This article defines complete and subcomplete commutative semigroups and gives general properties of these semigroups including Ponizovsky families and extended Schützenberger functors. Subcomplete semigroups include finitely generated commutative semigroups.

AMS Subject Classification (1991): 20M14

Keyword(s): complete, completion, subcomplete, finitely generated commutative semigroup, Ponizovsky family, {\cal K}, -class, Schützenberger monoid, extended Schützenberger functor

Received August 15, 2000. (Registered under 2806/2009.)

Abstract. This article constructs all congruences on a free commutative semigroup such that the quotient semigroup is subcomplete; this includes all congruences on a finitely generated free commutative semigroup.

AMS Subject Classification (1991): 20M14

Keyword(s): and phrases: congruence, free commutative semigroup, subelementary, subcomplete, extent cell, strand

Received August 15, 2000, and in revised form July 17, 2001. (Registered under 2807/2009.)

Abstract. This article corrects errors in two previous papers.

AMS Subject Classification (1991): 20M14

Keyword(s): complete, completion, subcomplete, finitely generated commutative semigroup, Ponizovsky family, {\cal K}, -class, Schützenberger monoid, extended Schützenberger functor

Received July 20, 2001. (Registered under 2808/2009.)

Abstract. In his paper [6], S. Doro constructed a partial relationship between Moufang loops and groups with triality. We extend this relationship by showing that the following concepts are equivalent: Groups with triality and trivial centre, Moufang $3$-nets, Latin square designs in which every point is the centre of an automorphism, isotopy classes of Moufang loops. Using this new approach, we also give a simple proof to a theorem of Doro.

AMS Subject Classification (1991): 20N05

Received April 19, 1999, and in final form June 18, 2001. (Registered under 2809/2009.)

Abstract. In this paper we consider the Burnside problems for the class of Moufang and Bol loops. We show that a free finitely generated Moufang loop of exponent 3 is finite and that the orders of the finite 2-generated Bol loops of exponent 2 are not bounded.

AMS Subject Classification (1991): 20N05

Keyword(s): Burnside problem, Moufang loop, Bol loop, triality

Received November 15, 1999, and in revised form October 16, 2000. (Registered under 2810/2009.)

Abstract. It is proved that finitely many continuous functions defined on the real line can be approximated simultaneously by $C^{\infty }({\msbm R}) $-solutions of the single algebraic differential equation $y_1'y_2''-y_2'y_1''=0 $ for any two of the approximating functions. Moreover, this result does no longer hold in the case of analytic solutions. A former result of the author is improved concerning the approximation of Lipschitz continuous functions by $C^{\infty }({\msbm R}) $-solutions of a third-order algebraic differential equation.

AMS Subject Classification (1991): 26A16, 26E10, 34A05

Received September 12, 2000. (Registered under 2811/2009.)

Abstract. This paper characterizes pullback properties linking both Bloch space and Hardy classes via boundedness and compactness of composition operators.

AMS Subject Classification (1991): 30D55, 46E15, 47B38

Keyword(s): Holomorphic self-maps, the unit disc, pullbacks, Bloch space, Hardy classes

Received May 22, 2000. (Registered under 2812/2009.)

Abstract. It is proved that --- under certain conditions --- continuous solutions $f$ of the functional equation $$ f(x)=h(x,y,f(g_1(x,y)),\ldots,f(g_n(x,y))), (x,y)\in D\subset{{\msbm R}^s}\times{\msbm R}^l, $$ are ${{\cal C}^\infty }$, even if $1\le l\le s$. As a tool we introduce new function classes which --- roughly speeking --- interpolate between differentiable and continuous functions.

AMS Subject Classification (1991): 39B05, 26B05

Received October 27, 2000. (Registered under 2813/2009.)

Abstract. We give a description of all 2-local (surjective linear) isometries of $C_0(X)$ where $X$ is a locally compact Hausdorff space. Further, we prove that if $X$ is also first countable and $\sigma $-compact then every 2-local (surjective linear) isometry of $C_0(X)$ is a (surjective linear) isometry. Finally, we show that this result cannot be generalized for arbitrary locally compact Hausdorff spaces.

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

Keyword(s): Reflexivity, isometry, local isometry, 2-local isometry, function algebra

Received May 21, 2001. (Registered under 2814/2009.)

Abstract. Let $A$ be a unital $C^*$-algebra and let $L$ be a $w_{\rho }$ completely bounded map $(1\leq\rho \leq2)$. Then $\left\|L\right\|_{w_{\rho }cb}=\left\|L_{2}\mid_{D_{\rho }\otimes A}\right\|_{w_{2}cb}.$ Moreover, there exist completely positive linear maps $\phi_{i}$ from $A$ to $B(H)$ with $\left\|\phi_{i}\right\|=\left\|L\right\|_{w_{\rho }cb}(i=1,2)$ such that the linear maps $\pmatrix{\phi_{1} & \sqrt{\rho(2-\rho )}L \cr\sqrt {\rho(2-\rho )}L^* & \phi_{2}}$ and $\pmatrix{\phi_{2} & (1-\rho )L \cr(1-\rho )L^* & \phi_{2}}$ from $A\otimes M_{2}$ to $B(H)\otimes M_{2}$ are completely positive. The above properties extend the results [8, Theorem 7.3] and [16, Corollary 3.11]. Let $M$ be a subspace of a unital $C^*$-algebra $A$, $\widehat{L}$ be a linear map from $M$ to $B(H)$ with $\left\|\widehat{L}\right\|_{w_{\varrho }cb}< \infty $ ($1\leq\rho \leq2$), then there exists a linear map $\widetilde{L}$ from $A\otimes M_{2}$ to $B(H)\otimes M_{2}$ such that $\widetilde{L}|_{D_{\rho }\otimes M}=\widehat{L}_{2}|_{D_{\rho }\otimes M}$ and $\left\|\widetilde{L}\right\|_{w_{2}cb}=\left\|\widehat{L}_{2}|_{D_{\rho }\otimes M}\right\|_{w_{2}cb}=\left\|\widehat{L}\right\|_{w_{\varrho }cb}.$ When $\varrho =1$, we have Haagerup, Paulsen, and Wittstock's extension theorem [4,8,18].When $\rho =2,$ then there exists a linear map $\overline{L}$ from $A$ to $B(H)$ such that $\overline{L}\mid_{M}=L$ and $\left\|\overline{L}\right\|_{w_{2}cb}=\left\|L\right\|_{w_{2}cb}.$ Let $M$ be an operator subspace of $A$ and $L$ be a linear map from $M$ to $M_{n}(\bf C)$. We prove that $\left\|L\right\|_{cb}=\left\|L\otimes I_{n}\right\|$ [10] and $\left\|L\right\|_{w_{2}cb}=\left\|L\otimes I_{n}\right\|_{w_{2}}.$ In general, $\left\|L\right\|_{w_{\rho }cb}=$ $\left\|L\otimes I_{2n}\right\|_{w_{\rho }}(1<\rho < 2).$

AMS Subject Classification (1991): 46L05, 46L10

Received July 19, 2000, and in revised form March 14, 2001. (Registered under 2815/2009.)

Abstract. In this paper we consider the reverse order rule of the form $(AB)_{K,L}^{(2)} =B_{T,S}^{(2)} A_{M,N}^{(2)}$ for outer generalized inverses with prescribed range and kernel. As corollaries, we get generalizations of the well-known results of Bouldin (SIAM J. Appl. Math. {\bf25} (1973), 489--495; ``Recent Applications of Generalized Inverses'', Pitman Ser. Res. Notes in Math. vol {\bf66}, (1982), 233--248) and Izumino (Tohoku Math. J. {\bf34} (1982), 43--52) for the ordinary Moore--Penrose inverse, and Sun and Wei (SIAM J. Matrix Anal. Appl. {\bf19} (1998), 772--775) for the weighted Moore--Penrose inverse. Results of Bouldin (the second paper mentioned above) for the reverse order rule for the Drazin inverse are improved. Finally, necessary and suficient conditions such that the reverse order rule holds for the group inverse are introduced.

AMS Subject Classification (1991): 47A05, 15A09

Received June 20, 2000, and in revised form January 15, 2001. (Registered under 2816/2009.)

Abstract. A new construction for the form sum of positive, selfadjoint operators is given in this paper. The situation is a bit more general, because our aim is to add positive, symmetric operators. With the help of the used method, some commutation properties of the form sum extension are observed.

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

Received July 18, 2000, and in revised form June 25, 2001. (Registered under 2817/2009.)

Abstract. This paper concerns the single valued extension property of a bounded operator $T\in L(X)$ on a Banach space $X$. We shall be mostly interested in a local version of this property, the single valued extension property at a point $\lambda_o \in\bf C$, in the case that $T$ admits a generalized Kato decomposition property.

AMS Subject Classification (1991): 47A53, 47A55

Keyword(s): Single valued extension property, Fredholm theory

Received July 6, 2000, and in final form October 2, 2000. (Registered under 2818/2009.)

Abstract. In this paper, we shall define $*$-Aluthge transformation (i.e., $\widetilde{T}^{(*)}=|T^*|^{1/2}U|T^*|^{1/2}$), and show some properties of it. Next, we shall show parallelisms between Aluthge transformation and powers of $p$-hyponormal operators, $\log $-hyponormal operators and $w$-hyponormal operators by using $*$-Aluthge transformation.

AMS Subject Classification (1991): 47B20, 47A63

Received September 12, 2000. (Registered under 2819/2009.)

Abstract. In this paper, the following result is shown: Let $X$ be an arbitrary real Banach space and $K$ a nonempty closed convex subset of $X$ and $T\colon K\to K$ a Lipschitz $\phi$-hemicontractive operator. Define the sequence $\{x_n\}_{n=0}^\infty$ iteratively by $x_0, u_0, v_0 \in K$, $$\eqalign{ x_{n+1}&= a_nx_n + b_nTy_n+ c_nu_n, \quad n\ge0,\cr y_{n}&= a'_nx_n + b'_nTx_n+ c'_nv_n\quad n\ge0, }$$ where $\{u_n\}_{n=0}^\infty$, $\{v_n\}_{n=0}^\infty$ are arbitrary bounded sequences in $K$; $\{a_n\}$, $\{b_n\}$, $\{c_n\}$, $\{a'_n\}$, $\{b'_n\}$ and $\{c'_n\}$ are real sequences in $[0,1]$ satisfying the following conditions: \item{(i)} $a_n+ b_n+ c_n= a'_n+ b'_n + c'_n=1,$ $n\ge0$; \item{(ii)} $\sum_{n=0}^\infty c_n < \infty, \sum_{n=0}^\infty b_nb'_n < \infty$, $\sum_{n=0}^\infty b_nc'_n < \infty, \sum_{n=0}^\infty b_n^2 < \infty,$ \item{(iii)} $\sum_{n=0}^\infty b_n =+\infty.$ \par\noindent Then the sequence $\{x_n\}_{n=0}^\infty$ converges strongly to the unique fixed point of $T$. Our result extends, improves and unifies the corresponding results in [2]-[8], [10]-[15], [20], [21], [24], [27].

AMS Subject Classification (1991): 47H17, 47H15, 47H05

Keyword(s): The Ishikawa iteration sequence with errors, the Mann iteration sequence with errors, strongly pseudocontractive operator, \phi, \phi, -strongly pseudocontractive operator.-henicontractive operator, real Banach space

Received May 3, 2000, and in revised form March 5, 2001. (Registered under 2820/2009.)

Abstract. We consider a series of overlapping products of the form $X_1X_2+X_2X_3+X_3X_4+\cdots $ where $X_1,X_2,\ldots $ are independent Bernoulli random variables. We compute the exact distribution of every tail section for a particular choice of the $X$'s, thus extending a result of Csörgő and Wu [2]. As a generalization, sums of multiple products are also studied.

AMS Subject Classification (1991): 60E05, 62E15

Keyword(s): Ewens sampling formula, beta mixture of Poisson distribution, generating function

Received March 1, 2001, and in revised form May 24, 2001. (Registered under 2821/2009.)

Abstract. First the turning points $\alpha =1,2$ and $3$ in the asymptotic behavior, as $n\to\infty $, of the number of elements in the random integer-set ${\cal I}_n(\alpha ) = \{\lfloor\lfloor n^\alpha\rfloor U_{n,1}\rfloor,\ldots, \lfloor\lfloor n^\alpha\rfloor U_{n,n}\rfloor\} $ are delineated, where $\{U_{n,1}, \ldots, U_{n,n}\} _{n=1}^{\infty }$ is an array of independent Uniform$(0,1)$ random variables and $\alpha > 0$ is a fixed parameter. Then, proving some conjectures made in [6], for $\alpha >2$ the distribution of the number $N_n(\alpha )$ of elements in the random multiset ${\cal I}_n(\alpha )\cap[ \cup_{j=n+1}^{\infty } {\cal I}_j(\alpha )]$ is approximated in the variation distance by suitable binomial and Poisson distributions, with specified rates of approximation depending on $\alpha $, and an almost sure bound of the order of $n^{3-\alpha }$ is obtained for $N_n(\alpha )$. Finally, these results are used to extend necessary conditions concerning the strong convergence of some randomly rarefied and bootstrap means.

AMS Subject Classification (1991): 60F05, 60F15, 62G09

Received December 20, 2000. (Registered under 2822/2009.)