RIGHT SIMPLE SUBSEMIGROUPS AND RIGHT SUBGROUPS OF COMPACT CONVERGENCE SEMIGROUPS

Clifford and Preston (1961) showed several important characterizations of right groups. It was shown in Roy and So (1998) that, among topological semigroups, compact right simple or left cancellative semigroups are in fact right groups, and the closure of a right simple subsemigroup of a compact semigroup is always a right subgroup. In this paper, it is shown that such results can be generalized in convergence semigroups. In the discussion of maximal right simple subsemigroups and maximal right subgroups of semigroups, generalization of the results that no twomaximal right simple subsemigroups and maximal right subgroups of a convergence semigroup intersect, is also established.


1.
Introduction. Discussion of convergence spaces, compactification of convergence spaces, and compact convergence semigroups can be found in [2,3,4,5]; however, a brief summary of essential results will be repeated here. Definition 1.1. A convergence semigroup is a convergence space S together with a continuous function m : S × S → S such that S is Hausdorff and m is associative.
The following notations are useful in the discussion of convergence semigroups: (i) For a, b ∈ S, ab = m(a, b). (ii) For A, B ⊆ S, AB = m(A×B) = {ab | a ∈ A and b ∈ B}. In particular, A{b} will be denoted Ab.
(iii) Ᏺ × Ᏻ is the filter on S × S with {F × G | F ∈ Ᏺ and G ∈ Ᏻ} as its base.
(iv) Ᏺ · Ᏻ is the filter on S with m(Ᏺ × Ᏻ) as its base.
Lemma 1.2. If Ᏺ and Ᏻ are filters on a convergence semigroup S such that Ᏺ → x and Ᏻ → y, then Ᏺ · Ᏻ → xy. Lemma 1.3. If S is a compact convergence semigroup, then S contains an idempotent.
2. Main results. Let S be a semigroup. Then S is left cancellative provided that zx = zy implies x = y for all x, y, z ∈ S; S is right simple if it contains no proper right ideal or aS = S for all a ∈ S; S is a right group if S is both left cancellative and right simple; S is a right zero semigroup if xy = y for all x, y ∈ S.
In [1], Clifford and Preston showed that a semigroup S is a right group if and only if S is right simple and contains an idempotent.
Using this result and Lemma 1.3, the next four results in compact convergence semigroups can be obtained in exactly the same way as in the topological setting. The example in [6] indicates that right subgroups of compact topological semigroups are closely related to their right simple subsemigroups, but not left cancellative subsemigroups. Thus the following discussion focuses only on the relationship between right simple subsemigroups and right subgroups of compact convergence semigroups.
In [6], it is shown that the closure of a right simple subgroup of a compact topological semigroup is always a right group. The next two theorems show that similar results can be obtained in compact convergence semigroups.
Theorem 2.5. If S is a compact convergence semigroup and R is a right simple subsemigroup. Then Cl S R, the closure of R, is also a right simple subsemigroup of S.
Since R is right simple, for F ∈ Ᏺ and G ∈ Ᏻ, let X FG = {x ∈ R : g = xf , f ∈ F, g ∈ G} and let χ be the filter with Ꮾ as base where Ꮾ = {X FG : F ∈ Ᏺ, G ∈ Ᏻ}. Then χ ·Ᏺ is the filter on S with m(χ × Ᏺ) as its base.
Since S is compact, there exists an ultrafilter ᐅ ≥ χ such that ᐅ → y where y ∈ Cl S R. Thus χ · Ᏺ ≤ ᐅ · Ᏺ and ᐅ · Ᏺ → ya. On the other hand, for each F ∈ Ᏺ, G ⊂ X FG · F for all G ∈ Ᏻ. It follows that X FG · F ∈ Ᏻ and χ · Ᏺ ≤ Ᏻ. Let ᐁ be an ultrafilter containing χ ·Ᏺ. Then ᐁ → ya and ᐁ → b. Thus b = ya and it follows that Cl S R is a right simple subsemigroup of S. Let x, y ∈ M. Then x ∈ T and y ∈ T * for some T ,T * ∈ Ꮿ. Let T = max{T ,T * }. Then x, y ∈ T and T being right simple implies y ∈ xT ⊂ xM. Therefore, M is a right simple subsemigroup of S.
Suppose M * is a proper right simple subsemigroup of such that M ⊂ M * . Then M * ∈ Ꮿ so Ꮿ ⊂ Ꮿ∪ᏹ * , which contracts the fact that Ꮿ the maximal chain of . Therefore, M is the maximal right simple subsemigroup of S containing R. Since S is compact, by Theorems 2.1 and 2.5, M is a compact maximal right subgroup of S containing R. Therefore, the following theorem is proved.
Similarly, the following corollaries concerning convergence semigroups can be obtained.
Clifford and Preston [1] showed that a semigroup S is a right group if and only if S is isomorphic to the direct product of G × E where G is a group and E is a right zero semigroup, denoted by S G ×E. In fact, E is the set of all idempotent of S and G = Se for some e ∈ E. This result suggests a different way of analyzing compact right simple convergence semigroups.
Let S be a compact right simple or left cancellative convergence semigroup. It follows from Theorem 2.1 that S is a right group. Since S is compact and G = Sg for some g ∈ E, G is compact. Since S is a right group, ef = f for e, f ∈ E. Thus E is a right zero semigroup. Since E is a closed subset of S, E is compact.
Let Z be a right zero subsemigroup of a compact semigroup S. By Theorem 2.5, Cl S Z is a subsemigroup of S. Let x, y ∈ Cl S Z. Then there exist filters Ᏺ and Ᏻ such that Z ∈ Ᏺ ∩ Ᏻ, Ᏺ → x, and Ᏻ → y.
On the other hand, for H ∈ Ᏼ * and K ∈ * , there exists F ∈ Ᏺ such that (Z ∩F )(Z ∩ G) = Z ∩ F ⊂ HK. Thus Ᏼ * · * ≤ Ᏺ and Ᏺ → xy. It follows from xy = y that Cl S Z is a right zero subsemigroups.
The next two lemmas follow from the above discussion.  It is a well-known result that no two maximal subgroups of a semigroup intersect and the following are its generalizations. In Theorem 2.15, we generalize it for maximal right subgroups and the proof can be found in [6]. Theorem 2.17 is a partial generalization of Theorem 2.15 in commutative semigroups. In fact, the following argument shows that M 1 · M 2 is a right simple subsemigroup of S containing both M 1 and M 2 . For a, b ∈ M 1 · M 2 , a = a 1 a 2 and b = b 1

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009