In this paper we develop an analog of the notion of the con- jugacy graph of  nite groups for the  nite semigroups by considering the Green relations of a  nite semigroup. More precisely, by de ning the new graphs $Gamma_{L}(S)$, $Gamma_{H}(S)$, $Gamma_{J}(S)$ and $Gamma_{D}(S)$ (we name them the Green graphs) related to the Green relations L R J H and D of a  nite semigroup S , we  first atte...

The concepts of ∗-relation of a (Γ-)semigroup and Γ̄-adequate transversal of a (Γ-)abundant semigroup are defined in this note. Then we develop a matrix type theory for abundant semigroups. We give some equivalent conditions of a Rees matrix semigroup being abundant and some equivalent conditions of an abundant Rees matrix semigroup having an adequate transversal. Then we obtain some Rees matrix...

The 3x + 1 semigroup is the multiplicative semigroup S of positive rational numbers generated by { 2k+1 3k+2 : k ≥ 0} together with {2}. This semigroup encodes backwards iteration under the 3x + 1 map, and the 3x + 1 conjecture implies that it contains every positive integer. This semigroup is proved to be the set of positive rationals a b in lowest terms with b 6≡ 0( mod 3), and so contains al...

The main purpose of the present note is to show (Theorem 2) that any regular ©-class of any semigroup is a partial homomorphic image of a Brandt groupoid. It follows from this that a semigroup with zero is a partial homomorphic image of a Brandt semigroup if and only if it is regular and O-bisimple. In the first section, an alternative formulation is given of the determination by H.-J. Hoehnke ...

We define some new algebraic structures, termed coloured Hopf algebras, by combining the coalgebra structures and antipodes of a standard Hopf algebra set H, corresponding to some parameter set Q, with the transformations of an algebra isomorphism group G, herein called colour group. Such transformations are labelled by some colour parameters, taking values in a colour set C. We show that vario...

This paper discusses the enveloping grasp of nraltiple objects unde~ rolling contacts. We first pro7yide a ge71m-al 71~athematical forn~ulatio?~ on the kinematic 7Aztiollshi,p for nudtiple ob,jects el(l~eloped by a 7nultijingered robot ha7id, and then deril!e a col).ditio?). for judgzl).g whether the rollzT~g condition can be satisfied at each contact point. We also show a sufjcient condit ion ...