ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. Shallon invented a means of deriving algebras from graphs, yielding numerous examples of so-called graph algebras with interesting equational properties. Here we study directed graph algebras, derived from directed graphs in the same way that Shallon's undirected graph algebras are derived from graphs. We classify the finitely based looped directed graph algebras and find the five finitely based varieties generated by them. We show that every looped directed graph algebra is either finitely based or inherently nonfinitely based. We find an equational basis for the variety generated by all directed graph algebras. We also develop a general two-part method for showing that varieties are finitely based; this method is then developed further, into syntactic and semantic components, in the specific case of varieties generated by directed graph algebras.

AMS Subject Classification (1991): 03C05; 08B15, 08B05

Keyword(s): Directed graph algebra, finite equational basis, finite basis problem, lattice of varieties, looped graphs, inherently nonfinitely based

Received August 30, 2004, and in revised form August 7, 2006. (Registered under 5930/2009.)

Abstract. Weakly associative lattices, a generalization of lattices, led to the introduction of dual discriminator. For a further generalization, graph algebras (or weak lattices), the original term did not produce a ``suitable'' term for dual discriminator function. Our goal is to describe why the original term was applicable for weakly associative lattices, and we present a class of algebras where the original term yields the dual discriminator function.

AMS Subject Classification (1991): 06B99, 08B99

Keyword(s): varieties, dual discriminator function, generalizations of lattices, directed graphs, projective planes

Received February 6, 2003, and in final form April 16, 2006. (Registered under 5931/2009.)

Abstract. An EQ-monoid $A$ is a monoid with distinguished subsemilattice $L$ with $1\in L$ and such that any $a,b\in A$ have a largest right equalizer in $L$. The class of all such monoids equipped with a binary operation that identifies this largest right equalizer is a variety. Examples include Heyting algebras, Cartesian products of monoids with zero, as well as monoids of relations and partial maps on sets. The variety is 0-regular (though not ideal determined and hence congruences do not permute), and we describe the normal subobjects in terms of a global semilattice structure. We give representation theorems for several natural subvarieties in terms of Boolean algebras, Cartesian products and partial maps. The case in which the EQ-monoid is assumed to be an inverse semigroup with zero is given particular attention. Finally, we define the derived category associated with a monoid having a distinguished subsemilattice containing the identity (a construction generalising the idea of a monoid category), and show that those monoids for which this derived category has equalizers in the semilattice constitute a variety of EQ-monoids.

AMS Subject Classification (1991): 06F05, 08A99, 08B05, 20M20

Received September 15, 2003, and in final form June 2, 2006. (Registered under 5932/2009.)

Abstract. It is well-known that for lattices both associativities are equivalent to the transitivity of the corresponding partially ordered set. It is also well-known that both distributivity is equivalent to the medial identity. We shall prove that ``in general'' this is not the case. Some related topics will be investigated.

AMS Subject Classification (1991): 08B99, 08B10, 06D99

Keyword(s): lattices, absorption laws, associativities, distributivities

Received February 3, 2005, and in final form June 8, 2006. (Registered under 5933/2009.)

Abstract. Let $\varphi $ denote the Euler function. For a fixed integer $k\not=0$, we study positive integers $n$ for which the largest prime factor of $\varphi(n)$ also divides $\varphi(n+k)$. We obtain an unconditional upper bound on the number of such integers $n\le x$, as well as unconditional lower bounds in each of the cases $k>0$ and $k< 0$. We also obtain some conditional lower bounds, for example, under the Prime $K$-tuplets Conjecture. Our lower bounds are based on explicit constructions.

AMS Subject Classification (1991): 11A25

Keyword(s): Euler function, largest prime factor, shift

Received July 6, 2004, and in revised form April 4, 2006. (Registered under 5934/2009.)

Abstract. We say that a semigroup $S$ is a permutable semigroup if, for every congruences $\alpha $ and $\beta $ of $S$, $\alpha\circ \beta = \beta\circ \alpha $. In [4], A. Nagy showed that every permutable semigroup satisfying an arbitrary non-trivial permutation identity is medial or an ideal extension of a rectangular band by a non-trivial commutative nil semigroup. The author raised the following problem: Is every permutable semigroup satisfying a non-trivial permutation identity medial? In the present paper we give a positive answer for this problem.

AMS Subject Classification (1991): 20M10

Received January 13, 2006, and in revised form June 7, 2006. (Registered under 5935/2009.)

Abstract. We characterize those positive measures on a Boolean $\sigma $-algebra $A$ which can be represented as the variation of a measure on $A$ with values in an Abelian normed group $G$. We also show that if there exists such a representation, then there is one in which $G$ is an $F^*$-lattice.

AMS Subject Classification (1991): 28B10, 28B05, 28A12, 28A60

Received October 10, 2005, and in revised form July 6, 2006. (Registered under 5936/2009.)

Abstract. We give sufficient conditions for the convergence of the symmetric as well as unsymmetric rectangular partial sums of the series conjugate to double Fourier series of a complex-valued function $f\in L^1({\msbm T}^2)$ at a given point $(x_0, y_0) \in{\msbm T}^2$. It turns out that this convergence essentially depends on the convergence behavior of the series conjugate to the single Fourier series of the so-called marginal functions $f(x, y_0)$, $x\in{\msbm T}$, and $f(x_0, y)$, $y\in{\msbm T}$, at $x:= x_0$ and $y:=y_0$, respectively. Our theorems apply to functions in the multiplicative Lipschitz or Zygmund classes in two variables.

AMS Subject Classification (1991): 42A50, 42B05

Keyword(s): Pringsheim test, double Fourier series, conjugate series, symmetric and unsymmetric rectangular partial sums, pointwise convergence, Riemann--Lebesgue lemma, multiplicative Lipschitz and Zygmund classes in two variables

Received January 26, 2006. (Registered under 5937/2009.)

Abstract. In this paper we inverstigate approximation properties of partial sums of Walsh-Fourier series of functions of generalized bounded oscilation.

AMS Subject Classification (1991): 42C10

Keyword(s): Walsh function, Uniform convergence, Bounded variation

Received April 21, 2006. (Registered under 5938/2009.)

Abstract. In this paper two independent and unitarily invariant projection matrices $P(N)$ and $Q(N)$ are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size $N$ converges to infinity. The result is formulated on the tracial state space $TS({\cal A})$ of the universal $C^*$-algebra ${\cal A}$ generated by two selfadjoint projections. The random pair $(P(N),Q(N))$ determines a random tracial state $\tau_N \in TS({\cal A})$ and $\tau_N$ satisfies the large deviation. The rate function is in close connection with Voiculescu's free entropy defined for pairs of projections.

AMS Subject Classification (1991): 15A52, 60F10, 46L54

Keyword(s): Eigenvalue density, large deviation, random matrices, free entropy, C^*, universal-algebra, tracial state space

Received October 18, 2005. (Registered under 5939/2009.)

Abstract. Given a Hilbert space $({\cal H}, \langle, \rangle )$ and a bounded selfadjoint operator $B$ consider the sesquilinear form over ${\cal H}$ induced by $B$, $$\langle x,y\rangle_B=\langle Bx,y\rangle, x,y\in{\cal H}. $$ A bounded operator $T$ is $B$-selfadjoint if it is selfadjoint with respect to this sesquilinear form. We study the set ${\cal P}(B,{\cal S})$ of $B$-selfadjoint projections with range ${\cal S}$, where ${\cal S}$ is a closed subspace of ${\cal H}$. We state several conditions which characterize the existence of $B$-selfadjoint projections with a given range; among them certain decompositions of ${\cal H}$, $R(|B|)$ and $R(|B|^{1/2})$. We also show that every $B$-selfadjoint projection can be factorized as the product of a $B$-contractive, a $B$-expansive and a $B$-isometric projection. Finally two different formulas for $B$-selfadjoint projections are given.

AMS Subject Classification (1991): 47A07, 46C20, 46C50

Keyword(s): Indefinite metric, Krein space, oblique projections, selfadjoint operators

Received November 16, 2005, and in revised form July 10, 2006. (Registered under 5940/2009.)

Abstract. We consider the classification, up to unitary equivalence, of commuting $n$-tuples $(V_{1},V_{2},\ldots,V_{n})$ of isometries on a Hilbert space. As in earlier work by Berger, Coburn, and Lebow, we start by analyzing the Wold decomposition of $V=V_{1}V_{2}\cdots V_{n}$, but unlike their work, we pay special attention to the case when $\ker V^*$ is of finite dimension. We give a complete classification of $n$-tuples for which $V$ is a pure isometry of multiplicity $n$. It is hoped that deeper analysis will provide a classification whenever $V$ has finite multiplicity. Further, we identify a pivotal operator in the case $n=2$ which captures many of the properties of a bi-isometry.

AMS Subject Classification (1991): 47A13, 47A45

Received June 15, 2006, and in revised form September 7, 2006. (Registered under 5941/2009.)

Abstract. In a sequence of four recent papers (cf. below), it was eventually shown that the hyperinvariant-subspace lattice $\mathop{\rm Hlat}(T)$ of an arbitrary nonalgebraic operator $T$ on Hilbert space is lattice-isomorphic to $\mathop{\rm Hlat}(A)$ for some $A$ in a special class $({\cal A}_{\theta })$ of operators defined below. In this note, which might be regarded as a first step in an attempt to better understand the structure of the class $({\cal A}_{\theta })$, we construct and study a certain subclass ${\cal(S}_{\theta }{\cal )}$ of this collection consisting of some operator--weighted bilateral shifts.

AMS Subject Classification (1991): 47A15, 47A45

Received May 15, 2006, and in revised form May 24, 2006. (Registered under 5942/2009.)

Abstract. In this paper we show the following extensions of the results by Yamazaki and Furuta--Yanagida: If $T$ is a $p$-hyponormal operator for $p\in(0,1]$, then $(T^{n+1^{\ast }}T^{n+1})^{n+p\over n+1}\geq(T^{n^{\ast }}T^n)^{n+p\over n}$ and $(T^nT^{n^{\ast }})^{n+p\over n}\geq(T^{n+1}T^{n+1^{\ast }})^{n+p\over n+1}$ hold for all positive integer $n$. And if $T$ is a $p$-hyponormal operator for $p>1$, then $T^{n+1^{\ast }}T^{n+1}\geq(T^{n^{\ast }}T^n)^{n+1\over n}$ and $(T^nT^{n^{\ast }})^{n+1\over n}\geq T^{n+1}T^{n+1^{\ast }}$ hold for all positive integer $n$. And we also discuss the best possibility of our results.

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

Received January 3, 2005, and in final form May 30, 2006. (Registered under 5943/2009.)

Abstract. In this note we describe the set of {\it extended eigenvalues} of a certain class of weighted Toeplitz operators.

AMS Subject Classification (1991): 47B35, 47B37

Keyword(s): \alpha, -commutant, Topeplitz operators

Received November 28, 2005, and in revised form April 12, 2006. (Registered under 5944/2009.)

Abstract. The inner derivation $\delta_{A}$ implemented by an element $A$ of the algebra $L(H)$ of all bounded linear operators on the separable complex Hilbert space $H$ into itself is the map $X\mapstochar\longrightarrow AX-XA$ for $X\in L(H)$. In this paper, we are interested in the class of operators $A\in L(H)$ which satisfy the following property $AT=TA$ implies $A^{\ast }T=TA^{\ast }$ for all $T\in C_{1}(H)$(trace class operators). Such operators are termed P-symmetric. We establish some properties of this class. We also turn our attention to commutants and derivation ranges. Hence, we obtain new results concerning the intersection of the kernel and the closure of the range of an inner derivation.

AMS Subject Classification (1991): 47B47, 47B10; 47A30

Keyword(s): Derivations, P-symmetric operators, subnormal operators, cyclic vector

Received February 2, 2006, and in revised form May 12, 2006. (Registered under 5945/2009.)

Abstract. A canonical differential equation is a system $y'=zJHy$ with a real, nonnegative and locally integrable $2\times2$-matrix valued function $H$. The theory of a canonical system is closely related to the spectral theory of a symmetric operator $T_{\min }(H)$ which acts in a Hilbert space $L^2(H)$, and, moreover, is closely related to the theory of positive definite Nevanlinna functions by means of the Titchmarsh--Weyl coefficient associated to it. In the present paper we define an indefinite analogue of canonical systems, construct an operator model which now acts in a Pontryagin space, and show that the spectral theory of the indefinite model is the perfect analogue of the classical theory of $T_{\min }(H)$.

AMS Subject Classification (1991): 47E05, 46C20; 47B25, 34L05

Keyword(s): canonical system, indefinite inner product, operator model

Received November 18, 2005, and in revised form September 6, 2006. (Registered under 5946/2009.)

Abstract. Here we categorize all of the subloops of the unit octonions and in particular describe all of the finite maximal subloops. Lagrange's Theorem for Moufang loops then follows as a corollary.

AMS Subject Classification (1991): 20N05, 17A75

Received November 4, 2005, and in revised form May 29, 2006. (Registered under 7/2006.)

Abstract. The normal form of a $C^r$-differentiable loop multiplication with respect to a distinguished parametrization is investigated in case the loop is defined on the real line and the group topologically generated by the left translations is locally compact. The normal form is applied to the classification of isomorphism classes of such loops by pairs of $C^r$-differentiable real functions satisfying a differential inequality.

AMS Subject Classification (1991): 20N05, 22A30, 70G65

Keyword(s): loops, PSL_2({\msbm R}), Lie-group and Lie-algebra methods

Received December 13, 2005, and in revised form August 28, 2006. (Registered under 17/2005.)

Abstract. Any point set ${\cal A}$ of the plane defines a simple graph on its elements as follows: let $P$ and $Q$ be adjacent if and only if the slope of their connecting line $\ell_{PQ}$ belongs to a prescribed set ${\cal S}$. A graph $G$ is $k$-slope if there exist a proper ${\cal A}$ and a set ${\cal S}$ of size $k$ realizing $G$. The slope parameter $sl(G)$ is the minimal such $k$. We completely characterize the $2$-slope graphs in terms of excluded induced subgraphs. For any tree $T$, we observe that $sl(T)=\Delta(T)$. We also show that if $G$ is an outerplanar graph with maximum degree at most three, then $sl(G)\leq3$. We also establish some general lower bounds, for instance ${2\over\omega (G)-1}\cdot{|E(G)|\over |V(G)|}\leq sl (G)$, where $\omega(G)$ is the size of the maximal clique of $G$.

AMS Subject Classification (1991): 05C62, 05C75

Received October 3, 2005, and in final form May 5, 2006. (Registered under 5947/2009.)

Abstract. We consider permutations of $\{1,\ldots,n\} $ obtained by $\lfloor\sqrt nt\rfloor $ independent applications of random stirring. In each step the same marked stirring element is transposed with probability $1/n$ with any one of the $n$ elements. Normalizing by $\sqrt n$, we describe the asymptotic distribution of the cycle structure of these permutations, for all $t\ge0$, as $n\to\infty $.

AMS Subject Classification (1991): 60C05

Received July 1, 2005, and in final form May 8, 2006. (Registered under 5948/2009.)

Abstract. The above-titled paper of mine appeared in the Acta Sci. Math. (Szeged), 71 (2005), 159--174. Regrettably, the proof of our main result contains an error which we want to correct here.

AMS Subject Classification (1991): 40C10, 42B10

Received August 15, 2006. (Registered under 5949/2009.)