Green relations Dominique Perrin 23 novembre 2015 Dominique Perrin - - PowerPoint PPT Presentation

Green relations

Dominique Perrin 23 novembre 2015

Dominique Perrin Green relations

Monoids

A semigroup is a set equipped with an associative binary operation. The operation is usually written multiplicatively. A monoid is a semigroup which, in addition, has a neutral element. The neutral element of a monoid M is unique and is denoted by 1M or simply by 1. For any monoid M, the set P(M) is given a monoid structure by defining, for X, Y ⊂ M, XY = {xy | x ∈ X, y ∈ Y } . The neutral element is {1}.

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A submonoid of M is a subset N which is stable under the

• peration and which contains the neutral element of M, that is

1M ∈ N and NN ⊂ N . (1) Note that a subset N of M satisfying (1) does not always satisfy 1M = 1N and therefore may be a monoid without being a submonoid of M. A morphism from a monoid M into a monoid N is a function ϕ : M → N which satisfies, for all m, m′ ∈ M, ϕ(mm′) = ϕ(m)ϕ(m′) , and furthermore ϕ(1M) = 1N . The notions of subsemigroup and semigroup morphism are then defined in the same way as the corresponding notions for monoids. A congruence on a monoid M is an equivalence relation θ on M such that, for all m, m′ ∈ M, u, v ∈ M m ≡ m′ mod θ ⇒ umv ≡ um′v mod θ .

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Cyclic monoids

A cyclic monoid is a monoid with just one generator, that is, M = {an | n ∈ N} with a0 = 1. If M is infinite, it is isomorphic to the additive monoid N of nonnegative integers. If M is finite, the index of M is the smallest integer i ≥ 0 such that there exists an integer r ≥ 1 with ai+r = ai. (2) The smallest integer r such that (2) holds is called the period of

• M. The pair composed of index i and period p determines a

monoid having i + p elements, Mi,p = {1, a, a2, . . . , ai−1, ai, . . . , ai+p−1} .

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Its multiplication is conveniently represented in Figure 1.

1 a a2 · · · ai ai+1 ai+p−1 a a a a a a a a Figure: The monoid Mi,p.

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The monoid Mi,p contains two idempotents (provided i ≥ 1). Indeed, assume that aj = a2j. Then either j = 0 or j ≥ i and j and 2j have the same residue mod p, hence j ≡ 0 mod p. Conversely, if j ≥ i and j ≡ 0 mod p, then aj = a2j. Consequently, the unique idempotent e = 1 in Mi,p is e = aj, where j is the unique integer in {i, i + 1, . . . , i + p − 1} which is a multiple of p.

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Idempotents in compact monoids

Since a finite cyclic semigroup contains an idempotent, any finite semigroup contains an idempotent. This property extends to compact semigroups. Proposition (Namakura) Any compact semigroup contains an idempotent. Let S be a compact semigroup. The family of closed nonempty subsemigroups of S is closed under intersection since S is compact. Thus there is a minimal nonempty closed nonempty subsemigroup T of S. For any x ∈ T, we have Tx = T since Tx is a closed subsemigroup included in T. Let T ′ = {y ∈ T | yx = x}. Since T ′ is a closed nonempty subsemigroup of T, we have T ′ = T. Thus x2 = x.

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Ideals in a monoid

Let M be a monoid. A right ideal of M is a nonempty subset R of M such that RM ⊂ R

• r equivalently such that for all r ∈ R and all m ∈ M, we have

rm ∈ R. Since M is a monoid, we then have RM = R because M contains a neutral element. A left ideal of M is a nonempty subset L of M such that ML ⊂ L. A two-sided ideal (also called an ideal) is a nonempty subset I of M such that MIM ⊂ I . A two-sided ideal is therefore both a left and a right ideal. In particular, M itself is an ideal of M. If M contains a zero, the set {0} is a two-sided ideal which is contained in any ideal of M.

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Minimal ideals

An ideal I (resp. a left, right ideal) is called minimal if for any ideal J (resp. left, right ideal) J ⊂ I ⇒ J = I . If M contains a minimal two-sided ideal, it is unique because any nonempty intersection of ideals is again an ideal. If M contains a 0, the set {0} is the minimal two-sided ideal of M. An ideal I = 0 (resp. a left, right ideal) is then called 0-minimalif for any ideal J (resp. left, right ideal) J ⊂ I ⇒ J = 0 or J = I . For any m ∈ M, the set R = mM is a right ideal. It is the smallest right ideal containing m. In the same way, the set L = Mm is the smallest left ideal containing m and the set I = MmM is the smallest two-sided ideal containing m.

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Theorem (Suschkewitsch) If a semigroup S has a minimal right ideal, then the union of all minimal right ideals is the minmal ideal of S. If R is a minimal right ideal and L a minimal left ideal, then R ∩ L is a maximal subgroup of S. Let R be a minimal right ideal. Then for any a ∈ S, aR is a right

• ideal. Let us show that it is minimal. Let R′ be a right ideal

contained in aR. The set {y ∈ R | ay ∈ R′} is nonempty since for every xinR′ there is y ∈ R such that x = ay. It is also a right ideal. By minimality of R, it is equal to R. This shows that aR ⊂ R′, and thus aR = R′. Thus aR is minimal and moreover we obtain that the union I of all minimal right ideals is a left ideal. If J is an ideal of M, we have RJ ⊂ R ∩ J ⊂ R and thus R ∩ J = R which implies R ⊂ J. Thus I is the minimal ideal of M.

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For any minimal left ideal L, consider the set G = RL, which is contained in R ∩ L. For every a ∈ G, we have aR = R and La = L by mimimality of R and L. It follows that aG = Ga = G. Thus G is a group. The identity e of G is an idempotent of M in R ∩ L. From eR = R and Le = L we see that every element x ∈ R ∩ L is such that x = ex = xe = exe = (ex)e ∈ RL. Therefore G = RL = R ∩ L. If G ′ is a subgroup of M admitting e as identity, then G ′ = eG ′ = G ′e imply G ′ ⊂ R ∩ L = G proving the maximality of G.

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Proposition A compact monoid M has minimal right and left ideals and a minimal two-sided ideal. The assertion follows from the obervation that for any m ∈ M the sets mM, Mm and MmM are closed. Thus, for instance, a minimal element R of the family of closed right ideals is equal to each rM for r ∈ R and is therefore a minimal right ideal.

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Green’s relations

We now define in a monoid M four equivalence relations L, R, J and H as mRm′ ⇐ ⇒ mM = m′M, mLm′ ⇐ ⇒ Mm = Mm′, mJ m′ ⇐ ⇒ MmM = Mm′M, mHm′ ⇐ ⇒ mM = m′M and Mm = Mm′. Therefore, we have for instance, mRm′ if and only if there exist u, u′ ∈ M such that m′ = mu, m = m′u′ .

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We have R ⊂ J , L ⊂ J , and H = R ∩ L.

q m n p u v u′ v ′ Figure: The relation RL = LR.

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Proposition The two equivalences R and L commute : RL = LR. Let m, n ∈ M be such that mRLn. There exists p ∈ M such that mRp, pLn (see Figure 2). There exist by the definitions, u, u′, v, v ′ ∈ M such that p = mu, m = pu′, n = vp, p = v ′n. Set q = vm. We then have q = vm = v(pu′) = (vp)u′ = nu′ , n = vp = v(mu) = (vm)u = qu . This shows that qRn. Furthermore, we have m = pu′ = (v ′n)u′ = v ′(nu′) = v ′q . Since q = vm by the definition of q, we obtain mLq. Therefore mLqRn and consequently mLRn. This proves the inclusion RL ⊂ LR. The proof of the converse inclusion is symmetrical.

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Since R and L commute, the relation D defined by D = RL = LR is an equivalence relation. We have the inclusions H ⊂ R, L ⊂ D ⊂ J . The classes of the relation D, called D-classes, can be represented by a schema called an ”egg-box” as in Figure 16.

R1 R2 R3 . . . L1 L2 · · ·

The R-classes are represented by rows and the L-classes by

• columns. The squares at the intersection of an R-class and an

L-class are the H-classes.

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We denote by L(m), R(m), D(m), H(m), respectively, the L, R, D, and H-class of an element m ∈ M. We have H(m) = R(m) ∩ L(m) and R(m), L(m) ⊂ D(m) .

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Proposition In a compact monoid M, D = J . Assume that xJ y, that is uxv = y and wyt = x. Then wuxvt = x and thus (wu)nx(vt)n = x for any n ≥ 0. Since M is compact, there are idempotents e, f which are limit of powers of wu and vt. Then exf = x by continuity and thus ex = xf = x. Since wue belongs to the H-class of e, it has an inverse z. Then zwuex = ex = x, showing that uxLx. Symmetrically, yRux. Thus xDy.

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Green’s lemma

Proposition Let M be a monoid. Let m, m′ ∈ M be R-equivalent. Let u, u′ ∈ M be such that m = m′u′, and m′ = mu. The mappings ρu : q → qu, ρu′ : q′ → q′u′ are bijections from L(m) onto L(m′) inverse to each other which map an R-class onto itself. We first verify that ρu maps L(m) into L(m′). If q ∈ L(m), then Mq = Mm and therefore Mqu = Mmu = Mm′. Hence qu = ρu(q) is in L(m′). Analogously, ρu′ maps L(m′) into L(m). Let q ∈ L(m) and compute ρu′ρu(q). Since qLm, there exist v, v ′ ∈ M such that q = vm, m = v ′q (see Figure 3). Since muu′ = m′u′ = m, we have ρu′ρu(q) = quu′ = vmuu′ = vm = q .

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This proves that ρu′ρu is the identity on L(m). One shows in the same way that ρuρu′ is the identity on L(m′). Finally, since quu′ = q for all q ∈ L(m), the elements q and ρu(q) are in the same R-class.

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The last Proposition has the following consequence which justifies the regular shape of D-classes.

q m q′ m′ u v u′ v ′ Figure: The reciprocal bijections.

Proposition Any two H-classes contained in the same D-class have the same cardinality.

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We now address the problem of locating the idempotents in an

• ideal. The first result describes the H-class of an idempotent.

Proposition Let M be a monoid and let e ∈ M be an idempotent. The H-class

• f e is the group of units of the monoid eMe.

Let m ∈ H(e). Then, we have for some u, u′, v, v ′ ∈ M e = mu , m = eu′ , e = vm , m = v ′e . Therefore em = e(eu′) = eu′ = m and in the same way me = m. This shows that m ∈ eMe. Since m(eue) = mue = e , (eve)m = evm = e , the element m is both right and left invertible in M. Hence, m belongs to the group of units of eMe. Conversely, if m ∈ eMe is right and left invertible, we have mu = vm = e for some u, v ∈ eMe. Since m = em = me, we obtain mHe.

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Proposition An H-class of a monoid M is a group if and only if it contains an idempotent. Let H be an H-class of M. If H contains an idempotent e, then H = H(e) is a group by Proposition 7. The converse is obvious.

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Proposition (Clifford, Miller) Let M be a monoid and m, n ∈ M. Then mn is in R(m) ∩ L(n) if and only if R(n) ∩ L(m) contains an idempotent. If R(n) ∩ L(m) contains an idempotent e, then e = nu , n = eu′ , e = vm , m = v ′e for some u, u′, v, v ′ ∈ M. Hence mnu = m(nu) = me = (v ′e)e = v ′e = m , so that mnRm. We show in the same way that mnLn. Thus mn ∈ R(m) ∩ L(n). Conversely, if mn ∈ R(m) ∩ L(n), then mnRm and nLmn. By Proposition 5 the multiplication on the right by n is a bijection from L(m) onto L(mn). Since n ∈ L(mn), this implies the existence of e ∈ L(m) such that en = n. Since the multiplication by n preserves R-classes, we have additionally e ∈ R(n). Hence there exists u ∈ M such that e = nu. Consequently nunu = enu = nu and e = nu is an idempotent in R(n) ∩ L(m).

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Proposition Let M be a monoid and let D be a D-class of M. The following conditions are equivalent. (i) D contains an idempotent. (ii) Each R-class of D contains an idempotent. (iii) Each L-class of D contains an idempotent. Obviously, only (i) implies (ii) requires a proof. Let e ∈ D be an

• idempotent. Let R be an R-class of D. The H-class H = L(e) ∩ R

is nonempty. Let n be an element of H (See Figure 4). Since nLe, there exist v, v ′ ∈ M such that n = ve , e = v ′n . Let m = ev ′. Then mn = e because mn = (ev ′)n = e(v ′n) = ee = e . Moreover, we have mRe since mn = e and m = ev ′. Therefore, e = mn is in R(m) ∩ L(n). This implies, by Proposition 9, that R = R(n) contains an idempotent.

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R n e m v v ′ Figure: Finding an idempotent in R.

A D-class satisfying one of the conditions of the last Proposition is called regular

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Proposition Let M be a monoid and let H be an H-class of M. The two following conditions are equivalent. (i) There exist h, h′ ∈ H such that hh′ ∈ H. (ii) H is a group. (i) = ⇒ (ii). If hh′ ∈ H, then by Proposition 9 H contains an

• idempotent. By Proposition 8, it is a group. The implication

(ii) = ⇒ (i) is obvious.

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The Sch¨ utzenberger group of a D-class

Proposition Any two maximal subgroups contained in the same D-class of a monoid M are isomorphic. Let e, f be two idempotents in the same D-class. Let a ∈ R(e) ∩ L(f ). We have ea = a and a′a = f for some a′ ∈ M. Green’s Lemma implies that the composite of ρa and λa′ is a bijection from H(e) onto H(f ) mapping e onto a′ea = a′a = f . Note that aa′a = af = a which shows that aa′ is an idempotent in R(a). It follows that for z ∈ R(a) we have aa′Rz and thus aa′z = z. In particular for y ∈ H(e), aa′y = y. Consequently, for every x, y ∈ H(e), a′xya = a′x(aa′)ya = (a′xa)(a′ya). Thus x → a′xa is an isomorphism from H(e) onto H(f ).

Dominique Perrin Green relations