A new central polynomial for 3 × 3 matrices

Abstract

Orthodox semigroups have been studied by many authors, in particular by Hall, Yamada and Petrich. In this paper, we give the standard representation of orthodox semigroups and investigate various e-varieties of orthodox semigroups which are determined by the standard representations.     

Single-electron tunneling and Coulomb blockade in carbon-based quantum dots

Single-electron tunneling (SET) and Coulomb blockade (CB) phenomena have been widely observed in nanoscaled electronics and have received intense attention around the world. In the past few years, we have studied SET in carbon nanotube fragments and fullerenes by applying the so-called “Orthodox” theory [28]. As outlined in this review article, we investigated the single-electron charging and discharging process via current-voltage characteristics, gate effect, and electronic structure-related factors. Because the investigated geometric structures are three-dimensionally confined, resulting in a discrete spectrum of energy levels resembling the property of quantum dots, we evidenced the CB and Coulomb staircases in these structures. These nanostructures are sufficiently small that introducing even a single electron is sufficient to dramatically change the transport properties as a result of the charging energy associated with this extra electron. We found that the Coulomb staircases occur in the I–V characteristics only when the width of the left barrier junction is smaller than that of the right barrier junction. In this case, the transmission coefficient of the emitter junction is larger than that of the collector junction; also, occupied levels enter the bias window, thereby enhancing the tunneling extensively.      

On Generalized Orthodox Transversals

Generalized orthodox transversals are introduced in this article, and some characterizations associated with them are obtained. The related results of Chen on quasi-ideal orthodox transversals obtained in 1999 and Zhang and Wang on generalized inverse transversals obtained in 2008 are generalized and amplified. A structure theorem of a regular semigroup with a perfect generalized orthodox transversal is also established.     

Semiprimitivity of Orthodox Semigroup Algebras

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R₀[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.     

Retractions of split graphs and End-orthodox split graphs

A retraction f of a graph G is an edge-preserving mapping of G with f(v)=v for all v∈V(H), where H is the subgraph induced by the range of f. A graph G is called End-orthodox (End-regular) if its endomorphism monoid End X is orthodox (regular) in the semigroup sense. It is known that a graph is End-orthodox if it is End-regular and the composition of any two retractions is also a retraction. The retractions of split graphs are given and End-orthodox split graphs are characterized.     

On the Construction of Regular Orthocryptogroups

The aim of this paper is to study regular orthocryptogroups. After obtaining some characterizations of such semigroups, we establish the construction theorem of regular orthocryptogroups. As an application, we give the construction theorem of right quasi-normal orthocryptogroups and study homomorphisms between two regular orthocryptogroups. Received September 24, 1999, Accepted May 9, 2001