Families of Metrized Graphs with Small Tau Constants

Baker and Rumely’s tau lower bound conjecture claims that if the tau constant of a metrized graph is divided by its total length, this ratio must be bounded below by a positive constant for all metrized graphs. We construct several families of metrized graphs having small tau constants. In addition to numerical computations, we prove that the tau constants of the metrized graphs in one of these families, the hexagonal nets around a torus, asymptotically approach to \({\frac{1}{108}}\) which is our conjectural lower bound.     

Generalized Bicyclic Semigroups and Jones Semigroups

In this paper, we consider the generalized bicyclic semigroups B_n = a, b | aⁿb = 1 and the Jones semigroups A_n = a, b | aⁿ⁺¹b = a. They are the generalizations of the bicyclic semigroup B = a, b | ab = 1 and its analogous semigroup A = a, b | a²b = a discovered by P.R., Jones in 1987. The word problem for these kinds of semigroups is solved. It is proved that, for n 2, B_n are bisimple right inverse but not inverse semigroups and that the semigroup C = a, b | a²b = a, ab² = b is the smallest idempotent-free homomorphic image of A_n. Moreover, we also prove that A_n and A_m are mutually embeddable but not isomorphic with each other if n m. As a consequence, different kind of -nontrivial [0-]simple semigroups without idempotents are discussed.AMS 1991 Subject Classification: primary 20M10 secondary 20M05.Supported by NNSF of China (19671063) and KSRF of Sichuan Education Committee ([1999]127).     

广义双循环半群和Jones半群

本文刻画了广义双循环半群Ｂｎ＝〈ａ,ｂ｜ａ＾ｎｂ＝１〉和Ｊｏｎｅｓ半群Ａｎ＝〈ａ,ｂ｜ａ＾ｎ＋１ｂ＝ａ〉（ｎ≥１）的结构;证明了每个Ａｎ都具有Ｐ．Ｒ．Ｊｏｎｅｓ所发现的半群Ａ＝〈ａ,ｂ｜ａ＾２ｂ＝ａ〉的所有重要性质,特别地,证明了Ａｎ,Ａｍ可互相嵌入,从而得到：第三个Ｄ－非平凡的无幂等元「０－」单半群若不含Ｃ＝〈ａ,ｂ｜ａ＾２ｂ＝ａ,ａｂＴ＾２＝ｂ〉,则必含每个Ａｎ或它们的对偶,作为推论,每人广义     

On Finite Semigroups Embeddable in Inverse Semigroups

No Abstract. October 29, 1998     

右Clifford左商半群

本文讨论了右Ｃｌｉｆｆｏｒｄ左商半群．利用所谓的可适对刻划了右Ｃｌｉｆｆｏｒｄ半群中的左次半群,因此扩展了商半群的研究范围．     

Semigroups with I-Quasi Length and Syntactic Semigroups

A module M is said to be extending (𝒢-extending) if for each submodule X of M there exists a direct summand D of M such that X is essential in D (X ∩ D is essential in both X and D). It is known that for a nonsingular module the concepts of 𝒢-extending and extending coincide. However, in the not nonsingular case, they are distinct. In this article, we obtain a characterization of the right 𝒢-extending generalized triangular matrix rings. This result and its corollaries improve and generalize the existing results on right extending generalized triangular matrix rings. It is well known that the ring of n-by-n triangular matrices over a right selfinjective ring is not, in general, right extending. One application of our characterization shows that such rings are right 𝒢-extending. Connections to Operator Theory and a characterization of the class of right extending right SI-rings are also obtained. Examples are given to illustrate and delimit the theory.     

Power Semigroups of a Class of Cliford Semigroups

Let S =∪(Gα : α∈ E) be a semilattice of groups(i.e., a Cliford semigroup) and n a natural number. E is called an n-element chain of groups if it is an n-element chain. Denote by Cn the set of all n-element chains of groups. In this paper we shall show that for any natural number n, the class of semigroups Cn satisfies the strong isomorphism property.     

Unary Iterative Hyperidentities for Semigroups and Inverse Semigroups

Research supported by NSERC.     

一类Clifford半群的幂半群

Let S =∪（Gα ： α ∈ E） be a semilattice of groups（i.e., a Cliford semigroup） and n a natural number. E is called an n-element chain of groups if it is an n-element chain. Denote by Cn the set of all n-element chains of groups. In this paper we shall show that for any natural number n, the class of semigroups Cn satisfies the strong isomorphism property.     

Two-Parameter Semigroups

In this paper we study compact semigroups that can be written as a product of two one-parameter semigroups. We show that such semigroups admit locally a unique factorization and can locally be embedded into a two-dimensional Lie group. The latter fact allows us to determine their local structure. We show how these results provide generalizations of a number of earlier results of this type.     

Permanent Semigroups

A permanent semigroup is a semigroup of n × n matrices on which the permanent function is multiplicative. If the underlying ring is an infinite integral domain with characteristic p > n or characteristic 0 we prove that any permanent semigroup consists of matrices with at most one nonzero diagonal. The same result holds if the ring is a finite field with characteristic p > n and at least n²+n elements. We also consider the Kronecker product of permanent semigroups and show that the Kronecker product of permanent semigroups is a permanent semigroup if and only if the pennanental analogue of the formula for the determinant of a Kronecker product of two matrices holds. This latter result holds even when the matrix entries are from a commutative ring with unity.     

Derived Rees Matrix Semigroups as Semigroups of Transformations

An ordered pair (e,f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. Previously [1] we have established that there are four distinct types of skew pairs of idempotents. We have also described (as quotient semigroups of certain regular Rees matrix semigroups [2]) the structure of the smallest regular semigroups that contain precisely one skew pair of each of the four types, there being to within isomorphism ten such semigroups. These we call the derived Rees matrix semigroups. In the particular case of full transformation semigroups we proved in [3] that T_X contains all four skew pairs of idempotents if and only if |X| ≥ 6. Here we prove that T_X contains all ten derived Rees matrix semigroups if and only if |X| ≥ 7.     

Constructing Almost Symmetric Numerical Semigroups from Irreducible Numerical Semigroups

Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number.     

A Structure Preserving Embedding of Semigroups Into Regular Semigroups

For any semigroup S a regular semigroup 𝒞(S) that embeds S can be constructed as the direct limit of a sequence of semigroups each of which contains a copy of its predecessor as a subsemigroup whose elements are regular. The construction is modified here to obtain an embedding of S into a regular semigroup R such that the nontrivial maximal subgroups of R are isomorphic to the Schützenberger groups of S and such that the restriction to S of any of Green's relations on R is the corresponding Green's relation on S.     

完全单半群及完全正则半群的逆断面

指出完全单半群Ｓ的任何一个Ｆ－类是逆断面,且为Ｑ－逆断面,而Ｓ的任何一个逆断面必是一个Ｆ－类,因而所有逆断面同构。并且给出完全正则半群的逆断面存在的充要条件。     

One-dimensional Tiling Semigroups

We investigate the structure of Kellendonks tiling semigroup in the case of 1-dimensional tilings.     

Power Centralized Semigroups

A semigroup is said to be power centralized if for every pair of elements x and y there exists a power of x commuting with y. The structure of power centralized groups and semigroups is investigated. In particular, we characterize 0-simple power centralized semigroups and describe subdirectly irreducible power centralized semigroups. Connections between periodic semigroups with central idempotents and periodic power commutative semigroups are discussed.     

Semigroups with apartness

Proving a constructive version of the Spectral Mapping Theorem, Bridges and Havea used a constructive semigroup with inequality in 8 . This motivated us to achieve a little progress in that direction. The starting point is the structure called a semigroup with apartness. Our primary objective is to prove isomorphism theorems for such constructive semigroups. In doing so our main ideas and notions come from 10 .