The Lattice of Compatible Quasiorders of Acyclic Monounary Algebras

Acyclic monounary algebras are characterized by the property that any compatible partial order r can be extended to a compatible linear order. In the case of rooted monounary algebras A=(A,f){\cal A}=(A,f) we characterize the intersection of compatible linear extensions of r by several equivalent conditions and generalize these results to compatible quasiorders of A{\cal A}. We show that the lattice QuordA{\rm{Quord}}{\cal A} of compatible quasiorders is a disjoint union of semi-intervals whose maximal elements equal the intersection of their compatible quasilinear extensions. We also investigate algebraic properties of the lattices QuordA{\rm{Quord}}{\cal A} and ConA{\rm{Con}}{\cal A}.     

The lattice of idempotent binary relations

Order-theoretic properties of the complete latticeE(A) of indempotent binary relations ρ=ρ² on the given setA are investigated. The elements ρ ofE(A) are classified according to theirfixed field I(ρ)={a∈A|(a, a)∈ρ} as being either offinite type, dense, or ofmixed type. When |A|>1E(A) is a non-atomic, non-coatomic lattice in which each element is a meet of meet-irreducible elements. The elements ofE(A) which are joins of join-irreducible elements form a compactly generated complete latticeF(A) which is a join-sublattice ofE(A) consisting of all elements having finite type. The setsD(A), M(A) of elements ρ ofE(A) which are dense (i.e., satisfy ρ≠ϕ andI(ρ)=ϕ) or of mixed type (i.e., are neither dense nor of finite type) resp. are non-empty only when |A| is infinite.D(A) is a partial meet-subsemilattice ofE(A) admitting no minimal elements. The group of order automorphisms of the latticeE(A) is isomorphic toS _A ×Z ₂ and each order automorphism ofE(A) preserves inverses. Presented by B. M. Schein.     

Theory of Completeness for Logical Spaces

A logical space is a pair (A, B){(A, {\mathcal{B}})} of a non-empty set A and a subset B{{\mathcal{B}}} of P A{{\mathcal{P}} A} . Since P A{{\mathcal{P}} A} is identified with {0, 1}^A and {0, 1} is a typical lattice, a pair (A, F){(A, {\mathcal{F}})} of a non-empty set A and a subset F{{\mathcal{F}}} of \mathbbB^A{{\mathbb{B}}^A} for a certain lattice \mathbbB{{\mathbb{B}}} is also called a \mathbbB{{\mathbb{B}}} -valued functional logical space. A deduction system on A is a pair (R, D) of a subset D of A and a relation R between A* and A. In terms of these simplest concepts, a general framework for studying the logical completeness is constructed.     

Almost Invariant Half-spaces of Algebras of Operators

Given a Banach space X and a bounded linear operator T on X, a subspace Y of X is almost invariant under T if TY í Y+F{TY\subseteq Y+F} for some finite-dimensional “error” F. In this paper, we study subspaces that are almost invariant under every operator in an algebra \mathfrak A{\mathfrak A} of operators acting on X. We show that if \mathfrak A{\mathfrak A} is norm closed then the dimensions of “errors” corresponding to operators in \mathfrak A{\mathfrak A} must be uniformly bounded. Also, if \mathfrak A{\mathfrak A} is generated by a finite number of commuting operators and has an almost invariant half-space (that is, a subspace with both infinite dimension and infinite codimension) then \mathfrak A{\mathfrak A} has an invariant half-space.     

Some properties of retract lattices of monounary algebras

Necessary and sufficient conditions for a connected monounary algebra (A, f), under which the lattice R ^∅(A, f) of all retracts of (A, f) (together with ∅) is algebraic, are proved. Simultaneously, all connected monounary algebras in which each retract is a union of completely join-irreducible elements of R ^∅(A, f) are characterized. Further, there are described all connected monounary algebras (A, f) such that the lattice R ^∅(A, f) is complemented. In this case R ^∅(A, f) forms a boolean lattice.     

On the structure of inner ideals of nest algebras

IfA is a nest algebra andA _s=A ∩ A* , whereA* is the set of the adjoints of the operators lying inA, then the pair (A, A _s) forms a partial Jordan *-triple. Important tools when investigating the structure of a partial Jordan *-triple are its tripotents. In particular, given an orthogonal family of tripotents of the partial Jordan *-triple (A, A _s), the nest algebraA splits into a direct sum of subspaces known as the Peirce decomposition relative to that family. In this paper, the Peirce decomposition relative to an orthogonal family of minimal tripotents is used to investigate the structure of the inner ideals of (A, A _s), whereA is a nest algebra associated with an atomic nest. A property enjoyed by inner ideals of the partial Jordan *-triple (A, A _s) is presented as the main theorem. This result is then applied in the final part of the paper to provide examples of inner ideals. A characterization of the minimal tripotents as a certain class of rank one operators is also obtained as a means to deduce the principal theorem.     

On quasiorder lattices and topology lattices of algebras

In this paper, it is shown that the dual [(\textQord)\tilde]\mathfrakA \widetilde{\text{Qord}}\mathfrak{A} of the quasiorder lattice of any algebra \mathfrakA \mathfrak{A} is isomorphic to a sublattice of the topology lattice á( \mathfrakA ) \Im \left( \mathfrak{A} \right) . Further, if \mathfrakA \mathfrak{A} is a finite algebra, then [(\textQord)\tilde]\mathfrakA @ á( \mathfrakA ) \widetilde{\text{Qord}}\mathfrak{A} \cong \Im \left( \mathfrak{A} \right) . We give a sufficient condition for the lattices [(\textCon)\tilde]\mathfrakA\text, [(\textQord)\tilde]\mathfrakA \widetilde{\text{Con}}\mathfrak{A}{\text{,}} \widetilde{\text{Qord}}\mathfrak{A} , and á( \mathfrakA ) \Im \left( \mathfrak{A} \right) . to be pairwise isomorphic. These results are applied to investigate topology lattices and quasiorder lattices of unary algebras.     

Lattice of subalgebras of the ring of continuous functions and Hewitt spaces

The latticeA(X) of all possible subalgebras of the ring of all continuous ℝ-valued functions defined on an ℝ-separated spaceX is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line ℝ. The main achievement of the paper is the proof of the fact that any Hewitt spaceX is determined by the latticeA(X). An original technique of minimal and maximal subalgebras is applied. It is shown that the latticeA(X) is regular if and only ifX contains at most two points. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 687–693, November, 1997. Translated by A. I. Shtern     

On a geometric property of perfect graphs

LetG be a graph,VP(G) its vertex packing polytope and letA(G) be obtained by reflectingVP(G) in all Cartersian coordinates. Denoting byA*(G) the set obtained similarly from the fractional vertex packing polytope, we prove that the segment connecting any two non-antipodal vertices ofA(G) is contained in the surface ofA(G) and thatG is perfect if and only ifA*(G) has a similar property.     

A generalization of Mathieu subspaces to modules of associative algebras

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $ \mathcal{A} $ \mathcal{A} , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to $ \mathcal{A} $ \mathcal{A} -modules $ \mathcal{M} $ \mathcal{M} . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable elements and quasi-stable elements, respectively, for all R-subspaces N of $ \mathcal{A} $ \mathcal{A} -modules $ \mathcal{M} $ \mathcal{M} , where R is the base ring of $ \mathcal{A} $ \mathcal{A} . We then prove some general properties of the sets σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.     

Feedback invariants of pairs of matrices and restrictions

If (A,B) εF ^n×n ×F ^×m is a given pair and S is an (A,B)-invariant subspace we investigate the relationship between the feedback invariants of (A, B) and those of its restrictions

to S.     

A Classification of 2-Arc-Transitive Circulants

A graph X is k-arc-transitive if its automorphism group acts transitively on the set of k-arcs of X. A circulant is a Cayley graph of a cyclic group. A classification of 2-arc-transitive circulants is given.     

The Ext-group of unitary equivalence classes of unital extensions

Let A be a unital separable nuclear C*-algebra which belongs to the bootstrap category N and B be a separable stable C*-algebra. In this paper, we consider the group Ext _u (A, B) consisting of the unitary equivalence classes of unital extensions τ: A→Q(B). The relation between Ext _u (A, B) and Ext(A, B) is established. Using this relation, we show the half-exactness of Ext _u (·, B) and the (UCT) for Ext _u (A, B). Furthermore, under certain conditions, we obtain the half-exactness and Bott periodicity of Ext _u (A, ·).     

Krylov sequences of maximal length and convergence of GMRES

In most practical cases, the convergence of the GMRES method applied to a linear algebraic systemAx=b is determined by the distribution of eigenvalues ofA. In theory, however, the information about the eigenvalues alone is not sufficient for determining the convergence. In this paper the previous work of Greenbaum et al. is extended in the following direction. It is given a complete parametrization of the set of all pairs {A, b} for which GMRES(A, b) generates the prescribed convergence curve while the matrixA has the prescribed eigenvalues. Moreover, a characterization of the right hand sidesb for which the GMRES(A, b) converges exactly inm steps, wherem is the degree of the minimal polynomial ofA, is given. This work was supported by AS CR Grant A2030706. Part of the work was performed while the third author visited Instituto di Analisi Numerica (IAN CNR).     

A note on the rank of semigroups

The rank of a semigroup $\mathcal{A}The rank of a semigroup A\mathcal{A} of functions from a finite set X to X is the minimum of |f(X)| over f ? Af\in \mathcal{A}. Given a finite set X and a subset Y of X, we show that if A\mathcal{A} is a semigroup of functions from X to X and ℬ a transitive semigroup of functions from Y to Y, then the rank of A\mathcal{A} divides that of ℬ provided that f(X)⊆Y for some f ? Af\in \mathcal{A} and that each function in ℬ is the restriction of a function in A\mathcal{A} to Y. To prove this, we generalize a result of Friedman which says that one can partition Y into q subsets of equal weight where q is the rank of ℬ. When one extends a transitive automaton by adding new states and letters, a similar condition guarantees that the rank of the extension divides the original rank.     

Extensions of the Hadamard determinant theorem

Denote byH _n the set ofn byn, positive definite hermitian matrices. Hadamard proved thath(A)≧det(A) for allA∈H _n, whereh(A) is the product of the main diagonal elements ofA. Subsequently, M. Marcus showed that per(A)≧h(A) for allA∈H _n. This article contains a result for all generalized matrix functions from which it follows thath(A)≧(per(A^1/n )) ⁿ ,A∈H _n.     

Derivations into duals of ideals of Banach algebras

We introduce two notions of amenability for a Banach algebra A. LetI be a closed two-sided ideal inA, we sayA is I-weakly amenable if the first cohomology group ofA with coefficients in the dual space I* is zero; i.e.,H ¹(A, I*) = {0}, and,A is ideally amenable ifA isI-weakly amenable for every closed two-sided idealI inA. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study theI-weak amenability of a Banach algebraA for some special closed two-sided idealI.     

The number of nearest and farthest points to a compactum in Euclidean space

A typical (in the sense of Baire category) compactA inE, whereE is either the Euclidean spaceE ⁸,s≧2, or the separable Hilbert space ℍ, generates a dense subsetC ^n,m(A) of the underlying space, such that everyx∈C ^n,m(A) has exactlyn nearest andm farthest points fromA, whenevern andm are positive integers satisfyingn+m≦ dimE+2. Research of this author is in part supported by Consiglio Nazionale delle Ricerche, G.N.A.F.A., Italy.     

A property of the character table for a finite group

A character table X of a finite group is broken up into four squares: A, B, C, and D. We establish relations via which ranks of the matrices inX are connected. In particular, ifX is an l × l-matrix, A is an s × t-matrix, and, moreover, the squares A and C are opposite, thenr(A)=r(C) + s + t − l; here.r(M) is the rank of a matrix M. Associated with such each block ofX is some integral nonnegative parameter m, and we have m=0 iff A, B, C, and D are active fragments ofX. Supported by RFFR grant No. 96-01-00488. Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 273–279, May–June, 2000.     

Regularity properties of solutions of some abstract parabolic equations

Consider the equation (i) (da/dt)—A(t)u(t)=f(t) where fort ∈ [a, b],A(t) is a densely defined and closed linear operator in a Banach spaceX. Assume the existence of bounded projectionsE _i(t),i=1, 2, such thatA(t) E ₁(t) and —A(t)E ₂(t) are infinitesimal generators of analytic semigroups andA(t) is completely reduced by the direct sum decompositionX = Σ _i ^{b = 1/2} ⊕E _i (t)X. We show that any solutionu(t) of (i) is inC ^∞(a, b) and satisfies the inequalities (1.2) provided thatf(t) andA(t) are infinitely differentiable in [a, b] in a suitable sense. In caseA(t) andf(t) are in a Gevrey class determined by the constants {M _n} we have (1. 3). Applications are given to the study of solution of (i) where fort ∈ [a, b]A(t) is the unbounded operator inH ^0,p (G) associated with an elliptic boundary value problem that satisfies Agmon’s conditions on the rays λ=±iτ, τ > 0. Research partially supported by an N.S.F. grant at Brandeis University.