In the following text, the main aim is to distinguish some relations between Smarad-
che semigroups and (topological) transformation semigroups areas. We will see that a transformation group is not distal if and only if its enveloping semigroup is a Smarandache semigroup. Moreover we will find a classifying of minimal right ideals of the enveloping semigroup of a transformation semigroup

Ayatollah, Zadeh Shirazi

Hosseini, A.

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Scientia MagnaVol. 3 (2007), No. 3, 1-4Some faces of Smarandache semigroups’ conceptin transformation semigroups’ approachF. Ayatollah Zadeh Shirazi† and A. Hosseini‡† Fatemah Shirazi Ayatollah Zadeh Shirazi, Faculty of Mathematics, Statistics, and ComputerScience, College of Science, University of TehranEnghelab Ave., Tehran, Irane-mail: fatemah@khayam.ut.ac.ir‡ Arezoo Hosseini, Department of Mathematics, Faculty of Science, Guilan UniversityManzarieh Ave., Rasht, IranAbstract In the following text, the main aim is to distinguish some relations between Smarad-che semigroups and (topological) transformation semigroups areas. We will see that a transfor-mation group is not distal if and only if its enveloping semigroup is a Smarandache semigroup.Moreover we will find a classifying of minimal right ideals of the enveloping semigroup of atransformation semigroup.Keywords A−minimal set, distal, Smarandache semigroup, transformation semigroup.§1. PreliminariesBy a transformation semigroup (group) (X,S, pi) (or simply (X,S)) we mean a compactHausdorff topological space X, a discrete topological semigroup (group) S with identity e anda continuous map pi : X × S → X (pi(x, s) = xs (∀x ∈ X, ∀s ∈ S)) such that:1) ∀x ∈ X, xe = x,2) ∀x ∈ X, ∀s, t ∈ S, x(st) = (xs)t.In the transformation semigroup (X,S) we have the following definitions:1. For each s ∈ S, define the continuous map pis : X → X by xpis = xs (∀x ∈ X),then E(X,S)(or simply E(X)) is the closure of {pis| s ∈ S} in XX with pointwise convergence,moreover it is called the enveloping semigroup (or Ellis semigroup) of (X,S). We used to writes instead of pis (s ∈ S). E(X,S) has a semigroup structure [1, Chapter 3], a nonempty subset Kof E(X,S) is called a right ideal if KE(X,S)⊆ K, and it is called a minimal right ideal if none ofthe right ideals of E(X,S) be a proper subset of K. For each p∈E(X), Lp : E(X)→ E(X) withLp(q) = pq (q ∈ E(X)) is continuous.2. Let a ∈ X, A be a nonempty subset of X and K be a closed right ideal of E(X,S)[2,Definition 1]:1) We say K is an a−minimal set if: aK = aE(X,S), K does not have any proper subsetlike L, such that L is a closed right ideal of E(X,S) and aL = aE(X,S). The set of alla−minimal sets is denoted by M(X,S)(a).2 F. Ayatollah Zadeh Shirazi and A. Hosseini No. 32) We say K is an A−minimal set if: ∀b ∈ A bK = bE(X,S), K does not have any propersubset like L, such that L is a closed right ideal of E(X,S) and bL = bE(X,S) for all b ∈ A.The set of all A−minimal sets is denoted by M(X,S)(A).3) We say K is an A−minimal set if: AK = AE(X,S), K does not have any proper subsetlike L, such that L is a closed right ideal of E(X,S) and AL = AE(X,S).The set of all A − minimal sets is denoted by M(X,S)(A). M(X,S)(A) and M(X,S)(a) arenonempty.3. Let a ∈ X, A be a nonempty subset of X [2, Definition 13]:1) (X,S) is called distal if E(X,S) is a minimal right ideal.2) (X,S) is called a−distal if M(X,S)(a) = {E(X,S)}.3) (X,S) is called A (M)distal if M(X,S)(A) = {E(X,S)}.4) (X,S) is called A (M)distal if M(X,S)(A) = {E(X,S)}.4. Let a ∈ X, A be a nonempty subset of X and C be a nonempty subset of E(X,S), weintroduce the following notations F(a,C) = {p ∈ C|ap = p}, F(A,C) = {p ∈ C|∀b ∈ A bp = p},F(A,C) = {p ∈ C| Ap = A}, J(C) = {p ∈ C| p2 = p} (the set of all idempotents of C),M(X,S) = {∅ 6= B ⊆ X| ∀K ∈ M(X,S)(B) J(F(B,K)) 6= ∅}, and M(X,S) = {∅ 6= B ⊆X|M(X,S)(B) 6= ∅ ∧ (∀K ∈ M(X,S)(B) J(F(B,K)) 6= ∅)}.In the following text in the transformation semigroup (X,S) suppose S acts effective onX, i.e., for each s, t ∈ S if s 6= t, then there exists x ∈ X such that xs 6= xt. Moreover considerS is a Smarandache semigroup if S has a semigroup structure which does not provides a groupstructure on S, although S has at least a proper subset with more than one element, whichcarries a group structure induced by the action considered on S [3, Chapter 4].§2. Proof of the theoremsTheorem 1. Transformation group (X,G) (with |G| > 1) is not distal if and only ifE(X,G) is a Smarandache semigroup.Proof. If E(X,G) is a Smarandache semigroup if and only if it is not a group (sinceG ⊆ E(X,G)), and by [1, Proposition 5.3], equivalently (X,G) is not distal.Theorem 2. In the transformation semigroup (X,S), if a ∈ A ⊆ X, and S is a Smaran-dache semigroup, then we have:1. E(X,S) is a Smarandache semigroup if and only if (X,S) is not distal;2. If F(a,E(X)) is a Smarandache semigroup, then (X,S) is not a−distal;3. If A ∈ M(X) and F(A,E(X)) is a Smarandache semigroup, then (X,S) is not AMdistal;4. If A ∈ M(X) and F(A,E(X)) is a Smarandache semigroup, then (X,S) is not AMdistal.Proof.1. If (X,S) is not distal, then E(X) is not a group (see [1, Proposition 5.3]), S is aSmarandache semigroup thus there exists a group A ⊆ S ⊆ E(X,S), such that |A| > 1.2, 3, 4. Use [2, Theorem 18] and a similar method described for (1).Lemma 3. In the transformation semigroup (X,S) if I a minimal right ideal of E(X)(resp. βS), then at least one of the following statements occurs:1) I is a Smarandache semigroup;Vol. 3 Some faces of Smarandache semigroups’ concept in transformation semigroups’ approach 32) I is a group (or equivalently |J(I)| = 1);3) I = J(I).Proof. In the transformation semigroup (X,S) if I is a minimal right ideal of E(X) (resp.βS), then {Iv : v ∈ J(I)} is a partition of I to some of its isomorphic subgroups, and for eachv ∈ J(I), Iv is a group with identity v (see [1, Proposition 3.5]).Theorem 4. In the transformation semigroup (X,S), at least one of the following state-ments occurs:1) For each I minimal right ideal of E(X), I is a Smarandache semigroup;2) For each I minimal right ideal of E(X), I is a group (or equivalently |J(I)| = 1);3) For each I minimal right ideal of E(X), I = J(I).Proof. We distinguish the following steps:Step 1. If I and J are minimal right ideals of E(X,S) and v ∈ J(J), then Lv|I : I → Jis bijective, since where exists u ∈ J(I) with uv = u and vu = v ([1, Proposition 3.6], thusLu|J ◦ Lv|I = idI .Step 2. If I and J are minimal right ideals of E(X,S) and I is a Smarandache semigroup,then there exists H ⊆ I such that H is a group with more than one element, suppose u be theidentity element of H, then Iu is a subgroup of I with identity u and Iu 6= u, choose v ∈ J(J)such that uv = u and vu = v, vIu(= Lv(Iu)) is a subgroup of J with identity v, moreover|vIu| = |Iu| ≥ |H| > 1 and since Lv|I : I → J is bijective and Iu 6= I, thus vIu 6= J , thereforeJ is a Smarandache semigroup.Step 3. If there exists minimal right ideal I of E(X) such that I is a group, it means|J(I)| = 1, since for each minimal right ideal J of E(X), we have |J(J)| = |J(I)|, thus J has aunique idemppotent element say v. {Jv} = {Jw : w ∈ J(J)} is a partition of J to some of itsisomorphic subgroups, thus Jv = J is a group.Step 4. If none of minimal right ideals of E(X) satisfies items 1 and 2 in Lemma 3, thenby Lemma 3, all of them will satisfy item 3 in Lemma 3.Using Lemma 3, will complete the proof.Example 5. Let X = { 1n : n ∈ N} ∪ {0} ∪ {−1,−2} with the induced topology of R;S = {ϕiψj : i, j ∈ N ∪ {0}} with discrete topology, where ψ : X → X and ϕ : X → X suchthat:xψ =−1 x = −2−2 x = −10 x ∈ X − {−1,−2}, xϕ = 1n+1 x = 1n , n ∈ Nx x ∈ {0,−1,−2} ,and ψ0 = ϕ0 = idX , then S is a Smarandache semigroup(under the composition of maps) sinceA = {ψ, idX} is a subgroup of S, moreover in the transformation group (X,S), E(X) = S.Example 6. If X = [0, 1] with induced topology of R, and G := {f : [0, 1] → [0, 1] isa homeomorphism} with discrete topology, then E(X,G) is a Smarandache semigroup since itcontains G and it is not a group, note that if:xp := 0 0 ≤ x < 121 12 ≤ x ≤ 1 ,4 F. Ayatollah Zadeh Shirazi and A. Hosseini No. 3References[1] Ellis, R., Lectures on topological dynamics, W. A. Benjamin, New York, 1969.[2] Sabbaghan, M., Ayatollah Zadeh Shirazi, F., A−minimal sets and related topics intransformation semigroups (I), International Journal of Mathematics and Mathematical Sci-ences, 25(2001), No. 10, 637-654.[3] Vasantha Kandasamy, W. B., Smarandache Semigroups, American Research Press,Rehoboth, 2002.

A−minimal sets and related topics in transformation semigroups (I),

Lectures on topological dynamics,

Smarandache Semigroups,

Some faces of Smarandache semigroups' concept in transformation semigroups' approach

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Some faces of Smarandache semigroups' concept in transformation semigroups' approach

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