(Generalized)Green's Relations of GV-Semigroups

Green's relations and generalized Green's relations play a fundamental role in the study of semigroups.GV-semigroups are the generalizations of completely regular semigroups in the range of π-regular semigroups.In this paper,Green's relations and generalized Green's relations on GV-semigroups are considered by the structure of GV-semigroups.D=j and D C D* on GV-semigroups will be proved.

Green＇s Relations on Semigroups of Transformations Preserving Two Equivalence Relations

Let Tx be the full transformation semigroup on a set X. For a non-trivial equivalence F on X, letTF（X） = {f ∈ Tx ： arbieary （x, y） ∈ F, （f（x）,f（y）） ∈ F}.Then TF（X） is a subsemigroup of Tx. Let E be another equivalence on X and TFE（X） = TF（X） ∩ TE（X）. In this paper, under the assumption that the two equivalences F and E are comparable and E lohtain in F, we describe the regular elements and characterize Green＇s relations for the semigroup TFE（X）.

On Some Subsemigroups of the 4-cyclic Semigroup-Ⅱ

In this paper, we study subsemigroups Bd,o of the 4-cyclic semigroup Bω2 and characterize the Green＇s relations of Bd,o. We show that, up to isomorphism, the only fundamental simple inverse w2-semigroups such that D│E（s） =Md are the semigroupsBd,o（d = 1, 2, 3,... ）.

Some Characterizations of Cryptogroups

In this paper, we give some characterizations of（regular, normal） cryptogroups with Green＇s relations, left（right） translations and homomorphisms.

On Green’s Relations, 2~0-regularity and Quasi-ideals in Γ-semigroups

In this paper, we introduce the definition of （m, n）0-regularity in Г-semigroups. we in- vestigate and characterize the 20-regular class of F-semigroups using Green＇s relations. Extending and generalizing the Croisot＇s Theory of Decomposition for F-semigroups, we introduce and study the absorbent and regular absorbent Г-semigroups. We approach this problem by examining quasi-ideals using Green＇s relations.

Transformation Semigroup of Alternating Nonnegative Integers

Set of integers, Zn is split into even-odd parts. The even part is arranged in ways, while the odd part fixes one point at a time to compliment the even part thereby forming the semigroup, AZn. Thus, -spaces are filled choosing maximum of two even points at a time. Green’s relations have formed important structures that enhance the algebraic study of transformation semigroups. The semigroup of Alternating Nonnegative Integers for n-even (AZn-even) is shown to have only two D-classes, and there are -classes for n≥4. The cardinality of L-classes is constant. Certain cardinalities and some other properties were derived. The coefficients of the zigzag triples obtained are 1, and . The second and third coefficients can be obtained by zigzag addition.

Green Economy, Green Capitalism and the Imperial Mode of Living： Limits to a Prominent Strategy, Contours of a Possible New Capitalist Formation

Since some years, the strategy of a Green Economy as a current form of ecological modernization was proposed. The paper highlights the core issues of the concept and its structural constraints. Several constraints of a far-reaching realization of a Green Economy are presented, and one constraint is highlighted, i.e., the imperial mode of （production and） living. The latter gives emphasis to the continuous and largely unquestioned access to products produced by cheap labor and under environmentally problematic conditions. This mode of living, the paper argues, is also attractive among the upper and middle classed in countries with emerging economies like China. Moreover, it compares it with the concept of sustainable development which emerged some 15 years ago. Like sustainable development, proposals for a Green Economy might become ＂a tranquilizing dis- positive＂ in order to silence doubt and criticisms. However, it is argued that, despite the improbability of realizing the ambitious objectives, the Green Economy strategy might contribute to further capitalist development. Environmental issues might be integrated into the mode of production and living. However, given capitalist and imperial structures and dynamics, this will occur in highly selective and partial ways. The author calls this emerging constellation and possible new mode of development in some countries or regions of the world Green Capitalism.

On the Ranks of Semigroups of Transformations on a Finite Set with Restricted Range

Let ΡΥ（X） be the semigroup of all partial transformations on X, Υ（X） and Ι（X） be the subsemigroups of ΡΥ（X） of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let ΡΥ（X, Y） = {α∈ ΡΥ（X） ： Xα Y}, Υ（X, Y） = ΡΥ（X, Y） ∩Υ（X） and Ι（X, Y） = ΡΥ（X, Y） ∩ Ι（X）. In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green＇s relations of Υ（X,Y）. In this paper, we present analogous results for bothΡ Υ（X, Y） and Ι（X, Y）. For a finite set X with ｜x｜≥ 3, the ranks of ΡΥ（X） = ΡΥ（X, X）, Υ（X） = Υ（X, X） and.Ι（X） = Ι（X, X） are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of ΡΥ（X,Y）, Υ（X, Y） and Ι（X, Y） for any proper non-empty subset Y of X.

Inverse Semigroups of Matrices.

We discuss some fundamental properties of inverse semigroups of matrices, and prove that the idempotents of such a semigroup constitute a subsemilattice of a finite Boolean lattice, and that the inverse semigroups of some matrices with the same rank are groups. At last, we determine completely the construction of the inverse semigroups of some 2 × 2 matrices： such a semigroup is isomorphic to a linear group of dimension 2 or a null-adjoined group, or is a finite semilattice of Abelian linear groups of finite dimension, or satisfies some other properties. The necessary and sufficient conditions are given that the sets consisting of some 2 ×2 matrices become inverse semigroups.