Elasticity in certain block monoids via the Euclidean table

This paper continues the study begun in [GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤ _n , S) where S = (hereafter denoted ℬ _a (n)). We introduce in Section 2 the notion of a Euclidean table and show in Theorem 2.8 how it can be used to identify the irreducible elements of ℬ _a (n). In Section 3 we use the Euclidean table to compute the elasticity of ℬ _a (n) (Theorem 3.4). Section 4 considers the problem, for a fixed value of n, of computing the complete set of elasticities of the ℬ _a (n) monoids. When n = p is a prime integer, Proposition 4.12 computes the three smallest possible elasticities of the ℬ _a (p). Part of this work was completed while the second author was on an Academic Leave granted by the Trinity University Faculty Development Committee.     

右消去幺半群、左正则带和左正则型A幺半群

本文利用右消去幺半群,左正则带建立了真左正则型A幺半群．在证明了任一左正则型A幺半群均有P－覆盖后,给出P－覆盖的结构．     

The clone of operations preserving a cycle with loops

We consider all the binary relations on a k-element set which, when viewed as directed graphs, consist of a k-cycle and some loops. If k ≥ 3 and the relation has at least two loops, we show that it is only preserved by essentially unary operations. In all other cases, the relation is preserved by operations that depend on a greater number of variables. Received March 9, 2006; accepted in final form September 14, 2007.     

Generalized symmetry of graphs — A survey

Symmetry of graphs has been extensively studied over the past fifty years by using automorphisms of graphs and group theory which have played and still play an important role for graph theory, and promising and interesting results have been obtained, see for examples, [L.W. Beineke, R.J. Wilson, Topics in Algebraic Graph Theory, Cambridge University Press, London, 2004; N. Biggs, Algebraic Graph Theory, Cambridge University Press, London, 1993; C. Godsil, C. Royle, Algebraic graph theory, Springer-Verlag, London, 2001; G. Hahn, G. Sabidussi, Graph Symmetry: Algebraic Methods and Application, in: NATO ASI Series C, vol. 497, Kluwer Academic Publishers, Dordrecht, 1997]. We introduced generalized symmetry of graphs and investigated it by using endomorphisms of graphs and semigroup theory. In this paper, we will survey some results we have achieved in recent years. The paper consists of the following sections.

1. Introduction

2. End-regular graphs

3. End-transitive graphs

4. Unretractive graphs

5. Graphs and their endomorphism monoids.

由n次幂等矩阵确定的交换幺半群

设R是含幺结合环,n≥2为自然数.对所有的k≥1,本文给出了n次幂等矩阵集Pk^n（R）={P｜P^n=P∈Mk（R）}上的一种等价关系,证明了P^n（R）=∪k=1^∞Pk^n（R）中的等价类在给定的加法运算下构成一个交换幺半群.     

图的P-正则自同态幺半群

刻划了具有Ｐ－正则自同态幺半群的二分图,讨论了字典序积图的自同态幺半群的Ｐ－正则性。     

On Primitive Idempotents of the Strong Endomorphism Monoid of a Graph

51.IntroductionandPreIiminariesThemonoidofendomorphismsofagraph,inparticular,thatofstrongendomorphismsofagraph,hasbeentheobjectofresearchesinthetheoryofsemigroupsforquitesometime(cf.Llj-Lloj).LlJandL2jcanserveasasurvey.Theaimoftheseresearchesistocon-tributetothealgebraicanalysisofgraphs.Thesubjectyieldssomeinterestsincetheresultsoftheseresearchesmayopenvastpossibilitiesforapplicationsofsemigrouptheorytographtheo-ry.InL1]andL3j,ithasbeenprovedthatforfinitegraphG,sEnd(G),themonoidofstrongen…     

Hsu-Riordan 阵/partial monoid

本文首先对Shapiro的Riordan群进行了推广。给出了Hsu-Riordan partial monoid的概念，然后在此框架内，对徐利治先生的两类扩展型广义Stirling数偶进行了统一处理；建立了Hsu-Wang转换定理。Brown-Sprugnoli转换公式，以及广义Brown转换引理-它揭示了一些不同类型的Hsu-Riordan阵之间转换的方法。由此可产生大量的恒等式。     

Construction of lattice orders on the semigroup ring of a positive rooted monoid

Lattice orders on the semigroup ring of a positive rooted monoid are constructed, and it is shown how to make the monoid ring into a lattice-ordered ring with squares positive in various ways. It is proved that under certain conditions these are all of the lattice orders that make the monoid ring into a lattice-ordered ring. In particular, all of the partial orders on the polynomial ring A[x] in one positive variable are determined for which the ring is not totally ordered but is a lattice-ordered ring with the property that the square of every element is positive. In the last section some basic properties of d-elements are considered, and they are used to characterize lattice-ordered division rings that are quadratic extensions of totally ordered division rings.