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 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.     

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.     

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)     

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.     

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 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).     

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.     

On the semigroup BetaS

Let S be an infinite, discrete, cancellative semigroup and let BetaS be the Stone-Cech compactification of S. Then BetaS is a semigroup with an operation which extends that of S and which is continous only in one variable. We generalize some algebraic properties known to hold for the additive semigroup of the integers.     

Symmetric inverse topological semigroups of finite rank ≤ <Emphasis Type="Italic">n</Emphasis>

We establish topological properties of the symmetric inverse topological semigroup of finite transformations of the rank ≤ n. We show that the topological inverse semigroup is algebraically h -closed in the class of topological inverse semigroups. Also we prove that a topological semigroup S with countably compact square S×S does not contain the semigroup for infinite cardinal λ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations of the rank ≤ n is the trivial semigroup.     

Fixed point set of semigroups of non-expansive mappings and amenability

In this paper, we study the fixed point set of a strongly continuous non-expansive semigroup of a semi-topological semigroup S for which CB(S) is n-extremely left amenable. Also, we study the fixed point set of a strongly continuous semigroup of mappings when S is a semigroup which is a sub-semigroup of a locally convex topological vector space with addition. Some applications to harmonic analysis are also provided.     

A Cantor-Bernstein type theorem for effect algebras

We prove that if E¹ and E² are &sgr;-complete effect algebras such that E¹ is a factor of E² and E² is a factor of E¹, then E¹ and E² are isomorphic.     

Beck, H and Leech Coextensions

J. Leech studied certain coextensions ES , of a monoid S and Ch. Wells studied the relationship between Leech coextensions and those studied in a general categorical setting by J. Beck. In this paper it is completed some of the Wells results and it is given a classification of the Leech coextensions. Specifically, Wells shows that a Leech coextension is a Beck coextension if its corresponding functor is "centralizing" (a generalization of abelian-valued). Our Theorem 1.1 provides a technical condition under which Beck coextensions are actually abelian. This technical condition implies other properties as well. Corollary 1.8 provides three natural conditions under any of which the conditions of 1.1 are satisfied. Also it is provided an example of a non-abelian Beck coextension. Table 1 shows the general classification and Table 2 shows the reduced set of possibilities when the semigroup is a regular monoid. Stronger results are provided for coextensions with S an inverse monoid, and for extensions of groups by monoids.     

Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

We study algebraic and topological properties of the convolution semigroup of probability measures on a topological groups and show that a compact Clifford topological semigroup S embeds into the convolution semigroup P(G) over some topological group G if and only if S embeds into the semigroup exp(G)\exp(G) of compact subsets of G if and only if S is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup S embeds into the functor-semigroup F(G) over a suitable compact topological group G for each weakly normal monadic functor F in the category of compacta such that F(G) contains a G-invariant element (which is an analogue of the Haar measure on G).     

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.     

On Embeddability of Idempotent Separating Extensions of Inverse Semigroups

Let S be an inverse semigroup and rho an idempotent separating congruence on S. It is proved that S can be embedded into a lambda-semidirect product of a group F by S/rho where F belongs to the variety generated by the idempotent classes of rho.     

Finitely Presented, Coherent, and Ultrasimplicial Ordered Abelian Groups

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only if G is finitely generated as a group, G⁺ is well-founded as a partially ordered set, and the set of minimal elements of G⁺\ {0} is finite. (ii) Torison-free, finitely presented partially ordered abelian groups can be represented as subgroups of some Zⁿ, with a finitely generated submonoid of (Z⁺)ⁿ as positive cone. (iii) Every unperforated, finitely presented partially ordered abelian group is Archimedean. Further, we establish connections with interpolation. In particular, we prove that a divisible dimension group G is a directed union of simplicial subgroups if and only if every finite subset of G is contained into a finitely presented ordered subgroup.     

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.     

Biordered Sets of Regular Semigroups with Inverse Transversals1

In a regular semigroup S, an inverse subsemigroup S° of S is called an inverse transversal of S if S° contains a unique inverse x° of each element x of S. An inverse transversal S° of S is called a Q-inverse transversal of S if S° is a quasi-ideal of S.If S is a regular semigroup with set of idempotents E then E is a biordered set. T.E. Hall obtained a fundamental regular semigroup T_E from the subsemigroup E which is generated by the set of idempotents of a regular semigroup. K.S.S. Nambooripad constructed a fundamental regular semigroup by a regular biordered set abstractly. In this paper, we discuss the properties of the biordered sets of regular semigroups with inverse transversals. This kind of regular biordered sets is called IT-biordered sets. We also describe the fundamental regular semigroup T_E when E is an IT-biordered set. In the sequel, we give the construction of an IT-biordered set by a left regular IT-biordered set and a right regular IT-biordered set.This project has been supported by the Provincial Natural Science Foundation of Guangdong Province, PR China