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 1 M 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 } . Dominique Perrin Green relations
A submonoid of M is a subset N which is stable under the operation and which contains the neutral element of M , that is 1 M ∈ N and NN ⊂ N . (1) Note that a subset N of M satisfying (1) does not always satisfy 1 M = 1 N 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 (1 M ) = 1 N . 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 . Dominique Perrin Green relations
Cyclic monoids A cyclic monoid is a monoid with just one generator, that is, M = { a n | n ∈ N } with a 0 = 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 a i + r = a i . (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, M i , p = { 1 , a , a 2 , . . . , a i − 1 , a i , . . . , a i + p − 1 } . Dominique Perrin Green relations
Its multiplication is conveniently represented in Figure 1. a a i +1 a a a a a a 2 · · · a i 1 a a a i + p − 1 a Figure : The monoid M i , p . Dominique Perrin Green relations
The monoid M i , p contains two idempotents (provided i ≥ 1). Indeed, assume that a j = a 2 j . Then either j = 0 or j ≥ i and j and 2 j have the same residue mod p , hence j ≡ 0 mod p . Conversely, if j ≥ i and j ≡ 0 mod p , then a j = a 2 j . Consequently, the unique idempotent e � = 1 in M i , p is e = a j , where j is the unique integer in { i , i + 1 , . . . , i + p − 1 } which is a multiple of p . Dominique Perrin Green relations
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 x 2 = x . Dominique Perrin Green relations
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 or 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 . Dominique Perrin Green relations
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- minimal if 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 . Dominique Perrin Green relations
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 x i nR ′ 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 . Dominique Perrin Green relations
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 . Dominique Perrin Green relations
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. Dominique Perrin Green relations
Green’s relations We now define in a monoid M four equivalence relations L , R , J and H as m R m ′ mM = m ′ M , ⇐ ⇒ m L m ′ Mm = Mm ′ , ⇐ ⇒ m J m ′ MmM = Mm ′ M , ⇐ ⇒ m H m ′ mM = m ′ M and Mm = Mm ′ . ⇐ ⇒ Therefore, we have for instance, m R m ′ if and only if there exist u , u ′ ∈ M such that m ′ = mu , m = m ′ u ′ . Dominique Perrin Green relations
We have R ⊂ J , L ⊂ J , and H = R ∩ L . u p m v v ′ q n u ′ Figure : The relation RL = LR . Dominique Perrin Green relations
Proposition The two equivalences R and L commute : RL = LR . Let m , n ∈ M be such that m RL n . There exists p ∈ M such that m R p , p L n (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 q R n . Furthermore, we have m = pu ′ = ( v ′ n ) u ′ = v ′ ( nu ′ ) = v ′ q . Since q = vm by the definition of q , we obtain m L q . Therefore m L q R n and consequently m LR n . This proves the inclusion RL ⊂ LR . The proof of the converse inclusion is symmetrical. Dominique Perrin Green relations
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. · · · L 1 L 2 R 1 R 2 R 3 . . . 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. Dominique Perrin Green relations
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 ) . Dominique Perrin Green relations
Proposition In a compact monoid M, D = J . Assume that x J y , that is uxv = y and wyt = x . Then wuxvt = x and thus ( wu ) n x ( 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 ux L x . Symmetrically, y R ux . Thus x D y . Dominique Perrin Green relations
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