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...

Sufficient conditions are developed, under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. Recently, one of the authors [O. Johnson, Stoch. Proc. Appl., 2007] used a semigroup approach to show that the Poisson has maximal entropy among all ultra-log-concave distributions with fixed mean. We show via a non-tr...

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 ...

in this paper we study some results on noetherian semigroups. we  show that if  $s_s$ is an  strongly  faithful $s$-act and $s$ is a duo weakly noetherian, then we have the following.

Let M2 be the semigroup of linear endomorphisms of a plane. We show that the space of idempotents in M2 is a hyperboloid of one sheet, the set of semigroup-theoretic inverses of a nonzero singular element in M2 is a hyperbolic paraboloid, and the set of nilpotent elements in M2 is a right circular cone. This is the story of the rediscovery of classical three-dimensional geometry, especially the...

We define the length of a semigroup presentation and related ideas. Theoretical results and bounds are presented, gained while investigating the lengths of groups defined by semigroup presentations. Minimal length semigroup presentations are obtained in certain cases. AMS Mathematics Subject Classification (2000): 20F05, 20M05

Let S be a finite semigroup. In this paper we introduce the functions φs : S ∗ → S, first defined by Rhodes, given by φs([a1, a2, . . . , an]) = [sa1, sa1a2, . . . , sa1a2 · · · an]. We show that if S is a finite aperiodic semigroup, then the semigroup generated by the functions {φs}s∈S is finite and aperiodic.

For strongly continuous semigroups on a Hilbert space, we present a short proof of the fact that the left-inverse of a left-invertible semigroup can be chosen to be a semigroup as well. Furthermore, we show that this semigroup need not to be unique.

In this paper we have introduced fuzzy quasi-ideal and fuzzy left(right, two-sided) ideals in LA-semigroup. We have proved some results related to fuzzy quasi-ideals and fuzzy left(right, two-sided) ideals of an LA-semigroup. Further we characterize an intra-regular LA-semigroup by the properties of their fuzzy ideals.