Sistema de Generación Eólico

There are various notions of how large a substructure is inside of an algebraic object. We introduce some of these notions as they apply to semigroups.

In groups the notion of how large a substructure is inside a group is called the group index. Let G be a group and let H be a subgroup of G. We say

two elements, x, y, of G lie in the same coset of H in G if xy−1 _∈ H. The index of H in G is defined to be the number of distinct cosets of H in G. Or equivalently, we say H is of index n if n is the least cardinal such that there exists g1, g2, . . . , gn such that for any g ∈ G there exists i such that

gH = giH. We say H is of finite index if n is finite. Many properties are

shared between groups and subgroups of finite index. In particular if G is a finitely generated group andH is a subgroup of finite index thenHis finitely generated.

In semigroups there are many notions of index, for example Rees index, Green index and syntactic index. Here we shall introduce just two and pro- vide theorems relating to finite generation and behaviour of subsemigroups of groups.

Let S be a semigroup and T be a subsemigroup of S. We say T is of finite Rees index in S if _|S_\T_| < _∞. We say the Rees index of T in S is

|S_\T_|+ 1. The notion of Rees index has been used to study how finiteness properties, such as finite presentability and solvable word problem, pass to subsemigroups in papers such as [7] and [20].

The following theorem is attributed to Jura in [25], however, a more constructive proof can be found in [7, Corollary 3.2].

finite Rees index. Then S is finitely generated if and only if T is finitely generated.

Lemma 2.4. Let S be an infinite group and let T be a subsemigroup ofS of finite Rees index. Then T =S.

Proof. Let x _∈ S_\T. As S is infinite and S _\T is finite it follows that T is infinite. As S is a group the set xT must be of the same cardinality as T and hence is infinite. This means xT _∩T is also infinite as S_\T is finite. As S _\T is finite there are only finitely many t _∈ T such that xt _∈ T and t−1 _{∈S\T. It follows there exists t∈T such that xt∈T and t}−1 _{∈T. Now} x=xtt−1 but xt_∈T and t−1 _∈T and therefore x_∈T, a contradiction.

Another definition of index is that ofGreen index introduced in [17]. Let S be a semigroup and let T be a subsemigroup of S. We say T is of finite Green index in S if S _\T has finitely many _HT_{-classes. We say the Green}

index of T in S is the number of _HT_{-classes in S\T plus one. The proof}

that this is well-defined is covered in [17].

Theorem 2.5. [6, Corollary 9.2] Let S be a semigroup andT be a subsemi- group of S of finite Green index. Then S is finitely generated if and only if T is finitely generated.

Theorem 2.6. [17, Corollary 34] Let S be a group and let T be a subsemi- group of S of finite Green index. Then T is a subgroup of S of finite group

index.

The notions of Rees index and Green index are quite different. If a subsemigroup T of a semigroup S is of finite Rees index then T is also of finite Green index, however, the values of their respective index may be different. In contrast if a subsemigroup has finite Green index then it may not have finite Rees index.

A related notion to index is that of aquotient. LetS be a semigroup and letρbe an equivalence relation onS. We sayρis aright-congruence if for all (x, y)_∈ρand for alls_∈S we have (xs, ys)_∈ρ. A left-congruence is defined similarly. We call an equivalence relation ρ on a semigroup S a congruence if it is both a left- and a right-congruence. We say a congruence is of finite index if it has finitely many equivalence classes.

LetS be a semigroup and let ρ be a congruence. Let s_ρ be the equiv- alence class of s in ρ. The quotient semigroup of S with respect to ρ is the semigroup with elements S

ρ and multiplication sρ ·tρ = stρ. For details, such as the fact that this product is well-defined, see [9, Section 1.5] A special example of this is the Rees quotient. LetS be a semigroup and let I be an ideal. The Rees quotient, denoted S/_I, is the quotient of S by the congruence I_×I_{∪ {}(s, s) :s _∈S_}.