Indecomposable matrices over a distributive lattice

In this paper, the concepts of indecomposable matrices and fully indecomposable matrices over a distributive lattice L are introduced, and some algebraic properties of them are obtained. Also, some characterizations of the set F _n (L) of all n × n fully indecomposable matrices as a subsemigroup of the semigroup H _n (L) of all n × n Hall matrices over the lattice L are given.     

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.     

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.     

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.     

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.     

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 Circulant Matrices Over a Lattice

Let C_n(L) denote the set of all n × n circulant matrices over a distributive lattice L. Then C_n(L) forms a semigroup under the usual matrix product. In this paper, we shall characterize all idempotents in C_n(L), and also estabish the Euler-Fermat theorem for the semigroup C_n(L).AMS Subject Classification (2000): 20MSupported by the Educational Committee of Fujian, China.     

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.     

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.     

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.     

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.     

Some Conditions for Matrices over an Incline To Be Invertible and General Linear Group on an Incline

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn（L） = PLn （L） for any n （n 〉 2）, in which GLn（L） is the group of all n × n invertible matrices over L and PLn（L） is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.     

Wavelets with Frame Multiresolution Analysis

A frame multiresolution (FMRA for short) orthogonalwavelet is a single-function orthogonal wavelet such that theassociated scaling space V0 admits a normalized tight frame(under translations). In this article, we prove that for anyexpansive matrix A with integer entries, there existA-dilation FMRA orthogonal wavelets. FMRA orthogonal waveletsfor some other expansive matrix with non integer entries are also discussed.     

K-Loops in The Minkowski World Over an Ordered Field

Let K be a commutative ordered field and L =K(i) the quadratic extension of K with i² = ?1. Let H be the set of all Hermitian 2 × 2 matrices over the field extension (L,K) and let H^(2),+ ? {A ∈ H ¦ det A ∈ K⁽²⁾, Tr A > 0}. Then we prove that (H⁽²⁾,+,⊕) is a K-loop with respect to the operation $$ {\rm A}\ \oplus \ {\rm B}= {1 \over {\rm TrA} + 2{\sqrt {\rm det A}}} ({\sqrt {\rm det A}}\ E +A){\rm B} ({\sqrt {\rm det A}}\ E +A) $$ where E is the identity matrix.     

Matrix semigroup homomorphisms into higher dimensions

We study non-degenerate irreducible homomorphisms from the multiplicative semigroup of all n-by-n matrices over an algebraically closed field of characteristic zero to the semigroup of m-by-m matrices over the same field. We prove that every non-degenerate homomorphism from the multiplicative semigroup of all n-by-n matrices to the semigroup of (n + 1)-by-(n + 1) matrices when n ? 3 is reducible and that every non-degenerate homomorphism from the multiplicative semigroup of all 3-by-3 matrices to the semigroup of 5-by-5 matrices is reducible.     

On spectral theory in discrete programming: References added in proof

In different parts of discrete programming so-called (H,A,L_C)-problems are studied, where one must find an h_o?H (H is a set of permutation), for which L_C(h_o) = min_h?HL_C(h), L_C(h) = c_1h(1) +...+ c_nh(n), C = | c_ij | is an n ×n-matrix over A, and A is a totally ordered commutative semigroup (for example, semigroup of positive real numbers or a finite commutative totally ordered semigroup). We are dealing with the full spectrum of values of the function L_C and not only with the solutions of a (H,A,L_C)-problem. Equivalence theorems for different classes of these problems are proved. Realizability of spectra in some classes of (H,A,L_C)-problems is studied.     

Representation of Finite Lattices by Annihilators of Completely 0-Simple Semigroups

L the explicit construction of a 0-simple Rees matrix semigroup is suggested such that the lattice of left annihilators of this semigroup is isomorphic to L.     

The group-theoretic complexity of subsemigroups of boolean matrices

We find the group-theoretic complexity of many subsemigroups of the semigroup B_n of n × n Boolean matrices, including Hall matrices, reflexive matrices, fully indecomposable matrices, upper triangular matrices, row-rank-n matrices, and others.     

Algebraic characterization of the isometries of the hyperbolic 5-space

Let \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} be the group of invertible 2 × 2 matrices over the division algebra \mathbbH{\mathbb{H}} of quaternions. \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} acts on the hyperbolic 5-space as the group of orientation-preserving isometries. Using this action we give an algebraic characterization of the orientation-preserving isometries of the hyperbolic 5-space. Along the way we also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} .     

Invariants of free Lie algebras

Let k be a principal ideal domain with identity and characteristic zero. For a positive integer n, with n \geqq 2n \geqq 2, let H(n) be the group of all n x n matrices having determinant ±1\pm 1. Further, we write SL(n) for the special linear group. Let L be a free Lie algebra (over k) of finite rank n. We prove that the algebra of invariants L^B(n) of B(n), with B(n) ? { H(n), SL(n)}B(n) \in \{ H(n), {\rm SL}(n)\} , is not a finitely generated free Lie algebra. Let us assume that k is a field of characteristic zero and let áSem(n) ?\langle {\rm Sem}(n) \rangle be the Lie subalgebra of L generated by the semi-invariants (or Lie invariants) Sem(n). We prove that áSem(n) ?\langle {\rm Sem}(n) \rangle is not a finitely generated free Lie algebra which gives a positive answer to a question posed by M. Burrow [4].