The Semigroup of Hall Matrices over Distributive Lattices

In this paper, the semigroup Hn(L) of Hall matrices over a complete and completely distributive lattice L is studied. A Hall matrix is a matrix which is greater (for the order associated with the lattice structure) than an invertible matrix. Some necessary and sufficient conditions for a Hall matrix to be regular in the semigroup Hn(L) are given and Green's relations of the semigroup Hn(L) are described. Also, the sandwich semigroup of Hall matrices over the lattice L is studied.     

The Classification of Certain Four-Weight Spin Models

In this paper we show that if one of the matrices {Wi, 1 h i h 4} of a four-weight spin model (X, W1, W2, W3, W4; D) is equivalent to the matrix of a Potts model or a cyclic model as type II matrix and |X| S 5, then the spin model is gauge equivalent to a Potts model or a cyclic model up to simultaneous permutations on rows and columns. Using this fact and Nomura's result [12] we show that every four-weight spin model of size |X| = 5 is gauge equivalent to either a Potts model or a cyclic model up to simultaneous permutations on rows and columns.     

Minimal Presentations and Efficiency of Semigroups

A finite semigroup S is said to be efficient if it can be defined by a presentation (A | R) with |R| -|A|=rank(H2(S)). In this paper we demonstrate certain infinite classes of both efficient and inefficient semigroups. Thus, finite abelian groups, dihedral groups D2n with n even, and finite rectangular bands are efficient semigroups. By way of contrast we show that finite zero semigroups and free semilattices are never efficient. These results are compared with some well-known results on the efficiency of groups.     

Centralizers and Central Idempotents of Semigroup Rings

. Let A be a nonempty subset of an associative ring R . Call the subring CR(A)={r] R\mid ra=ar \quadfor all\quad a] A} of R the centralizer of A in R . Let S be a semigroup. Then the subsemigroup S'= {s] S\mid sa=sb \quador\quad as=bs \quadimplies\quad a=b \quadfor all a,b] S} of S is called the C -subsemigroup. In this paper, the centralizer CR[S](R[M]) for the semigroup ring R[S] will be described, where M is any nonempty subset of S' . An non-zero idempotent e is called the central idempotent of R[S] if e lies in the center of R[S] . Assume that S\backslash S' is a commutative ideal of S and Annl(R)=0 . Then we show that the supporting subsemigroup of any central idempotent of R[S] must be finite.     

The Representation Type of the Full Transforamtion Semigroup T 4

Let Tn be the semigroup of all transformations of a set of n elements and k a field of characteristic 0. According to Ponizovskii, the semigroup algebra kTn is of finite representation type if n h 3. According to Putcha, kTn is of infinite representation type if n S 5. Here, we deal with the remaining case n=4 and show that kT4 is also of finite representation type. Note that the quiver of kT4 already has been exhibited by Putcha, here we determine the relations. It turns out that kT4 is a string algebra and its global dimension is 3.     

Minimal and Maximal Semigroups Related to Causal Symmetric Spaces

Let G/H be an irreducible globally hyperbolic semisimple symmetric space, and let S ³ G be a subsemigroup containing H not isolated in S. We show that if S^o p 0 then there are H-invariant minimal and maximal cones Cmin ³ Cmax in the tangent space at the origin such that H exp Cmin ³ S ³ HZK(a)expCmax. A double coset decomposition of the group G in terms of Cartan subspaces and the group H is proved. We also discuss the case where G/H is of Cayley type.     

Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.

     

Topology of the Semigroup of Singular Endomorphisms

The topological interpretations of some of the algebraic properties of the semigroup Sn of singular endomorphisms of an n-dimensional vector space over K are discussed here. Since Sn is known to be an idempotent generated regular semigroup, we pay more attention to the topological properties of the set En of idempotents in Sn. The local structure of En is shown to be that of a C^infinity-manifold and of a finite-dimensional vector bundle over the Grassmann manifolds. The topology of the biorder relations and sandwich sets are also discussed.     

Invariant characters and coprime actions on finite nilpotent groups

Abstract. Let S be a subgroup of SLn(R), where R is a commutative ring with identity and n \geqq 3n \geqq 3. The order of S, o(S), is the R-ideal generated by x_ij, x_ii - x_jj (i 1 j)x_{ij},\ x_{ii} - x_{jj}\ (i \neq j), where (x_ij) ? S(x_{ij}) \in S. Let En(R) be the subgroup of SLn(R) generated by the elementary matrices. The level of S, l(S), is the largest R-ideal \frak q\frak {q} with the property that S contains all the \frak q\frak {q}-elementary matrices and all conjugates of these by elements of En(R). It is clear that l(S) \leqq o(S)l(S) \leqq o(S). Vaserstein has proved that, for all R and for all n \geqq 3n \geqq 3, the subgroup S is normalized by En(R) if and only if l(S) = o(S)     

The semigroup of singular endomorphisms

Let V be a vector space of dimension n over a field K. Here we denote by Sn the set of all singular endomorphisms of V. Erdos [5], Dawlings [4] and Thomas J. Laffey [6] have shown that Sn is an idempotent generated regular semigroup. In this paper we apply the theory of inductive groupoids, in particular the construction of the idempotent generated regular semigroup given in §6 of [8] to detemine some combinatorial properties of the semigroup Sn.     

On Free Inverse Semigroups

Using techniques of Rewriting Theory, we present a new proof of the known theorem of Munn that FIX , the free inverse semigroup on X, is isomorphic to birooted word-trees on X.     

Rational realization of the minimum ranks of nonnegative sign pattern matrices

A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set {+,?, 0} ({+, 0}, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of A. Using a correspondence between sign patterns with minimum rank r ≥ 2 and point-hyperplane configurations in R^r?1 and Steinitz’s theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every d-polytope determines a nonnegative sign pattern with minimum rank d + 1 that has a (d + 1) × (d + 1) triangular submatrix with all diagonal entries positive. It is also shown that there are at most min{3m, 3n} zero entries in any condensed nonnegative m × n sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established.     

The Theory of Multiresolution Analysis Frames and Applications to Filter Banks

The notion of a frame multiresolution analysis (FMRA) is formulated. An FMRA is a natural extension to affine frames of the classical notion of a multiresolution analysis (MRA). The associated theory of FMRAs is more complex than that of MRAs. A basic result of the theory is a characterization of frames of integer translates of a function φ in terms of the discontinuities and zero sets of a computable periodization of the Fourier transform of φ. There are subband coding filter banks associated with each FMRA. Mathematically, these filter banks can be used to construct new frames for finite energy signals. As with MRAs, the FMRA filter banks provide perfect reconstruction of all finite energy signals in any one of the successive approximation subspacesV_jdefining the FMRA. In contrast with MRAs, the perfect reconstruction filter bank associated with an FMRA can be narrow band. Because of this feature, in signal processing FMRA filter banks achieve quantization noise reduction simultaneously with reconstruction of a given narrow-band signal.     

The Monoid of the Random Graph

We investigate properties of the endomorphism monoid of the countable random graph R. We show that End(R) is not regular and is not generated by its idempotents. The Rees order on the idempotents of End(R) has 2^N0 many minimal elements. We also prove that the order type of Q is embeddable in the Rees order of End(R).     

Spectral Inclusions for Semigroups of Closed Operators

We show that several spectral inclusions known for C0-semigroups fail for semigroups of closed operators, even if they can be regularized. We introduce the notion of spectral completeness for the regularizing operator C which implies equality of the spectrum and the C-spectrum of the generator. We prove spectral inclusions under this additional assumption. We give a series of examples in which the regularizing operator is spectrally complete including generators of integrated semigroups, of distribution semigroups, and of some semigroups that are strongly continuous for t > 0.     

A polynomial algorithm for integer programming covering problems satisfying the integer round-up property

LetA be a non-negative matrix with integer entries and no zero column. The integer round-up property holds forA if for every integral vectorw the optimum objective value of the generalized covering problem min{1y: yA w, y 0 integer} is obtained by rounding up to the nearest integer the optimum objective value of the corresponding linear program. A polynomial time algorithm is presented that does the following: given any generalized covering problem with constraint matrixA and right hand side vectorw, the algorithm either finds an optimum solution vector for the covering problem or else it reveals that matrixA does not have the integer round-up property.     

Possible numbers of positive entries of imprimitive nonnegative matrices

Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n,?k) and m(n,?k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d?≤?M(n,k), there exists an n?×?n irreducible nonnegative matrix A with imprimitivity index k such that?σ?(A)=d.     

Parseval frame wavelets with -dilations

We study Parseval frame wavelets in with matrix dilations of the form , where A is an arbitrary expanding n×n matrix with integer coefficients, such that |detA|=2. We show that each A-MRA admits either Parseval frame wavelets, or Parseval frame bi-wavelets. The minimal number of generators for a Parseval frame associated with an A-MRA (i.e. 1 or 2) is determined in terms of a scaling function. All Parseval frame (bi)wavelets associated with A-MRA's are described. We then introduce new classes of filter induced wavelets and bi-wavelets. It is proved that these new classes strictly contain the classes of all A-MRA Parseval frame wavelets and bi-wavelets, respectively. Finally, we demonstrate a method of constructing all filter induced Parseval frame (bi)wavelets from generalized low-pass filters.     

When does the inverse have the same sign pattern as the transpose?

By a sign pattern (matrix) we mean an array whose entries are from the set {+, –, 0}. The sign patterns A for which every real matrix with sign pattern A has the property that its inverse has sign pattern A ^T are characterized. Sign patterns A for which some real matrix with sign pattern A has that property are investigated. Some fundamental results as well as constructions concerning such sign pattern matrices are provided. The relation between these sign patterns and the sign patterns of orthogonal matrices is examined.