Applied Mathematics E-Notes, 1(2001), 111-117 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

L-Classes of Inverse Semigroups ^{∗}

Shigeru Kobayashi and Haruki Shimada^{†}

Received 4 June 2001

Abstract

In this paper, we provide some properties of inverse semigroups from those of their L-classes.

1 Introduction

Let S be a semigroup and e be an idempotent element in S. Then a H-class H_{e} is a subgroup of S. If there exists a morphism θ from S to H_{e}, then we can make N^{0}×S×N^{0} into a semigroup (here and in the sequel, N stands for the set of positive integers and N^{0} the set of nonnegative integers). This is called the Bruck-Reilly extension of S determined by θ. If S is a group and θ is an endomorphism of S, then the Bruck-Reilly extension is a bisimple inverse ω-semigroup. In this case, the H-classes are important in the study of semigroups. Furthermore, let S be a complete regular semigroup, then every H-class is a group, and it is known that the Clifford semigroup is a semilattice [1]. On the other hand, for L-classes and R-classes of an inverse semigroup, fundamental properties were given by several authors [5,6,7]. In this paper, we consider the properties of inverse semigroups which are related to the structure of their L-classes. If every L-class of an inverse semigroup S is a semigroup, then we will show S is a Clifford semigroup. We give a characterization of inverse semigroups whose L-classes contain a semigroup. We shall also show that if every L-class of an inverse semigroup S is not a semigroup, then S has a chain of idempotent elements which is not well ordered.

2 Preliminaries

Let S be a semigroup. Then an equivalence L on S is defined by the rule that aLb if, and only if, S^{1}a = S^{1}b, where S^{1} = Sa ∪ {a}. Similarly we define the equivalence R by the rule that aRb if, and only if, aS^{1} = bS^{1}. It is well known that L is a right congruence and R is a left congruence. The intersection of L and R is denoted by H and the join of L and R is denoted by D. The L-class (resp. R-class, H-class, D-class) containing the element a will be denoted by L_{a} (resp. R_{a}, H_{a}, D_{a}). If e is an idempotent element of S, then H_{e} is a subgroup of S, and no H-class can contain more than one idempotent.

∗Mathematics Subject Classifications: 20M18

†Department of Mathematics, Naruto Universit of Education, Takashima, Naruto, Tokushima 772-8502, Japan

A semigroup S is called an I-semigroup if a unary operation a ↦ a^{−1} is defined on S such that, for all a, b ∈ S,

(a^{−1})^{−1} = a, aa^{−1}a = a.

Completely regular semigroup is specified within an I-semigroup by

aa^{−1} = a^{−1}a,

and that a Clifford semigroup is specified by the properties

aa^{−1} = a^{−1}a, aa^{−1}bb^{−1} = bb^{−1}aa^{−1}.

An inverse semigroup is an I-semigroup S such that for all a, b ∈ S,

aa^{−1}bb^{−1} = bb^{−1}aa^{−1}.

PROPOSITION 2.1. The following statements are equivalent.

(1) S is an inverse semigroup.

(2) S is regular, that is for any element a in S, there exists x in S such that axa = a and its idempotents commute.

(3) Every L-class and every R-class contains exactly one idempotent.

(4) Every element of S has a unique inverse. PROPOSITION 2.2. Let S be an inverse semigroup with semilattice E of idempo-

tents. Then the following hold.

(1) (ab)^{−1} = b^{−1}a^{−1} for every a, b in S.

(2) Both aea^{−1} and a^{−1}ea are idempotent for every a in S and e in E.

(3) aLb if, and only if, a^{−1}a = b^{−1}b; aRb if, and only if, aa^{−1} = bb^{−1}.

(4) For e, f ∈ E, eDf if, and only if, there exists a in S such that aa^{−1} = e and a^{−1}a = f.

The proofs of these two results can be found in [2, Theorem 5.1.1] and [2, Proposition 5.1.2] respectively.

3 Inverse semigroups

In this section, we assume that S is an inverse semigroup. If every L-class, or, every R-class in S is a semigroup, then we have the following.

THEOREM 3.1. If every L-class, or, every R-class in S is a semigroup, then S is a Clifford semigroup.

PROOF. Suppose that (a, b) ∈ L. Then by Proposition 2.2(3), a^{−1}a = b^{−1}b. Since every L-class is a semigroup, we have aLba. Hence

a^{−1}a = (ba)^{−1}ba = a^{−1}(b^{−1}b)a = a^{−1}(a^{−1}a)a = a^{−2}a^{2}.

Also for each a in S, aa^{−1} and a^{−1}a are idempotent. Since idempotents commute, so we obtain

aa^{−1} = aa^{−1}aa^{−1} = aa^{−2}a^{2}a^{−1} = (aa^{−1}) (a^{−1}a) (aa^{−1})
= (a^{−1}a) (aa^{−1}) (aa^{−1}) (a^{−1}a) = a^{−1}a^{2}a^{−2}a = a^{−1}aa^{−1}a = a^{−1}a.

Thus S is a Clifford semigroup. A similar argument can be applied to the case where each R-class in S is a semigroup. The proof is complete.

Next if there is a L-class L_{a} in S such that L_{a} is a semigroup, then we have the following characterization.

THEOREM 3.2. Let a be an element of S. Then the following conditions are equivalent.

(1) L_{a} is a semigroup.

(2) For every element b in L_{a}, b^{−1}b = b^{−2}b^{2}.

(3) For every element c in R_{a^{−1}}, cc^{−1} = c^{2}c^{−2}.

(4) R_{a^{−1}} is a semigroup.

PROOF. To see that (1) implies (2), let b be an element in L_{a}. Then a^{−1}a = b^{−1}b. Since L_{a} is a semigroup, bLab, thus

b^{−1}b = (ab)^{−1}ab = b^{−1}(a^{−1}a)b = b^{−1}(b^{−1}b)b = b^{−2}b^{2}.

(2) ⇒ (3): Let c be an element in R_{a^{−1}}. Then since a^{−1} ∈ R_{a^{−1}},

a^{−1}a = a^{−1} (a^{−1})^{−1} = cc^{−1} = (c^{−1})^{−1} c^{−1}.

It follows that aLc^{−1}, so c^{−1} ∈ L_{a}. By (2), this shows that

cc^{−1} = (c^{−1})^{−1} c^{−1} = (c^{−1})^{−2} (c^{−1})^{2} = c^{2}c^{−2}.

(3) ⇒ (4): Let g, h ∈ R_{a^{−1}}. Then gRh. Since R is a left congruence, it follows that g^{2}Rgh and by our assumption, gRg^{2} and hence that gRgh. Thus gh ∈ R_{g} = R_{a^{−1}}.

(4) ⇒ (1): Let x, y ∈ L_{a}, then x^{−1}x = a^{−1}a = y^{−1}y. Hence

x^{−1}(x^{−1})^{−1} = a^{−1}(a^{−1})^{−1} = y^{−1}(y^{−1})^{−1}.

This shows that x^{−1}, y^{−1} ∈ R_{a^{−1}}. Since R_{a^{−1}} is a semigroup, we see that y^{−1}x^{−1} = (xy)^{−1} ∈ R_{a^{−1}}. Thus

(xy)^{−1}xy = (xy)^{−1}((xy)^{−1})^{−1} = a^{−1}(a^{−1})^{−1} = a^{−1}a

It follows that xy ∈ L_{a}, so L_{a} is a semigroup.

THEOREM 3.3. Suppose that a is an element in S such that L_{a} is a semigroup. Then the following hold.

(1) a^{−1}a is the largest idempotent in the D-class D_{a}. This shows that each D-class contains at most one L-class which is semigroup.

(2) For any element b ∈ L_{a}, each b^{n}b^{−n} (n ∈ N) is idempotent and

bb^{−1} ≥ b^{2}b^{−2} ≥ ... ≥ b^{n}b^{−n} ≥ ...

PROOF. To see that (1) holds, let f be an idempotent in D_{a}. Then there exists an element b in S such that b^{−1}b = a^{−1}a and bb^{−1} = f. This shows that b is contained in L_{a}, so by Theorem 3.2(2), b^{−1}b = b^{−2}b^{2}. Since idempotents commute,

f = bb^{−1} = bb^{−1}bb^{−1} = bb^{−2}b^{−2}b^{−1} = f (b^{−1}b) f = (b^{−1}b) f = (a^{−1}a) f.

This implies that a^{−1}a ≥ f. Therefore a^{−1}a is the largest idempotent in D_{a}. Next we assume that there exists L-class L_{b} in the D-class D_{a} which is a semigroup. Then a^{−1}a and b^{−1}b are the largest idempotent elements in D_{a}, hence a^{−1}a = b^{−1}b and L_{a} = L_{b}.

To see that (2) holds, suppose that b ∈ L_{a}. Then b^{n}b^{−n}, n ∈ N, are idempotent elements and by Theorem 3.2(2),

a^{−1}a = b^{−1}b = b^{−2}b^{2} = ... = b^{−n}b^{n}.

It follows that b^{−1}b = a^{−1}a ≥ bb^{−1} by (1). Therefore we obtain bb^{−1} = (bb^{−1}) (b^{−1}b) and

b^{2}b^{−2} = b (bb^{−1}) b^{−1} = b (bb^{−1}) (b^{−1}b) b^{−1} = (b^{2}b^{−2}) (bb^{−1}) .

This implies that bb^{−1} ≥ b^{2}b^{−2}. Next we assume that b^{k}b^{−k} ≥ b^{k+1}b^{−(k+1)} for k ≥ 1. Then

b^{k+2}b^{−(k+2)} = b (b^{k+1}b^{−(k+1)}) b^{−1} = b (b^{k+1}b^{−(k+1)}) (b^{k}b^{−k}) b^{−1}
= b^{k+2}b^{−1} (b^{−k}b^{k}) b^{−(k+1)} = b^{k+2}b^{−1} (b^{−(k+1)}b^{k+1}) b^{−(k+1)}
= (b^{k+2}b^{−(k+2)}) (b^{k+1}b^{−(k+1)})

This shows that b^{k+1}b^{−(k+1)} ≥ b^{k+2}b^{−(k+2)}, so an inductive argument leads to bb^{−1} ≥ b^{2}b^{−2} ≥ ... ≥ b^{n}b^{−n} ≥ ... . The proof is complete.

COROLLARY 3.4. Let a be an element in a periodic inverse semigroup S and assume that the L-class L_{a} is a semigroup. Then L_{a} is group.

PROOF. Since S is periodic, L_{a} becomes a periodic semigroup. Let b be an element in L_{a}. Then there exists a positive integer n such that b^{n} is idempotent. It follows that b^{n} = a^{−1}a since a^{−1}a is the largest idempotent in D_{a} and b^{n} ∈ L_{a}, hence b^{n}b^{−n} = a^{−1}a. Now Theorem 3.3 (2) shows that

a^{−1}a ≥ bb^{−1} ≥ b^{2}b^{−2} ≥ ⋯ ≥ b^{n}b^{−n}.

Therefore we have that b ∈ H_{a^{−1}a}, and so L_{a} ⊆ H_{a^{−1}a} ⊆ L_{a}. Thus L_{a} = H_{a^{−1}a} is a group. The proof is complete.

Corollary 3.4 shows that there is no finite inverse semigroup with a L-class which is a semigroup but not a group.

EXAMPLE 3.5. Let I_{X} be a symmetric inverse semigroup, that is, the set of all partial one to one maps of a set X. Let X = {1, 2}. Then the following two L-classes

\{( \begin{matrix}1 & 2 \\ 1 & 2 \end{matrix} ) , ( \begin{matrix}1 & 2 \\ 2 & 1 \end{matrix} )\} and \{∅\}

are groups.

EXAMPLE 3.6. Let

σ = ( \begin{matrix}0 & 1 & 2 & ⋯ & n & ⋯ \\ 1 & 2 & 3 & ⋯ & n + 1 & ⋯ \end{matrix} )

and let S = {σ^{−m}σ^{n} | m, n ∈ N^{0}}. Then S has a L-class which is a semigroup but not a group. Note that S is an inverse semigroup and isomorphic to bicyclic semigroup N^{0} ×N^{0}. Thus L-class L_{σσ^{−1}} ≅ L_{(0,0)} = {(m, 0) ∈ N^{0} × N^{0} | m ∈ N^{0}} is a semigroup but not a group.

For any subset T in an inverse semigroup S, we define R(T) = {x ∈ S | T x ⊆ T }. Clearly R(T) is a semigroup. Let a be an element of S. Then since a^{−1}a is a right identity in R(L_{a}), a^{−1}a ∈ R(L_{a})

THEOREM 3.7. The following statements hold.

(1) For any y ∈ R(L_{a}), y^{−1}y ∈ R(L_{a}).

(2) a^{−1}a is the smallest idempotent element in R(L_{a}).

(3) There is an unique idempotent element in R(L_{a}) if, and only if, R(L_{a}) ⊆ L_{a}.

(4) L_{a} is a semigroup if, and only if, L_{a} ⊆ R(L_{a}).

(5) L_{a} is a group if, and only if, D_{a} ⊆ R(L_{a}).

PROOF. (1): Let y ∈ R(L_{a}). Then since L_{a}y ⊆ L_{a}, (by)^{−1}(by) = a^{−1}a = b^{−1}b for any element b ∈ L_{a}. Hence for any b ∈ L_{a},

(by^{−1}y)^{−1} by^{−1}y = y^{−1}yb^{−1}by^{−1}y
= y^{−1}y (y^{−1}b^{−1}by) y^{−1}y = y^{−1}b^{−1}by = b^{−1}b = a^{−1}a.

It follows that L_{a}y^{−1}y ⊆ L_{a}. Thus y^{−1}y ∈ R(L_{a}).

(2): Let f be an idempotent element in R(L_{a}). Then since af ∈ L_{a}, a^{−1}a = (af)^{−1}af = f (a^{−1}a). This implies that a^{−1}a ≤ f, so a^{−1}a is the smallest idempotent element in R(L_{a}).

(3): Assume that there is an unique idempotent element in R(L_{a}). Then by (2), a^{−1}a is the unique idempotent in R(L_{a}). Let y ∈ R(L_{a}). Then by (1), y^{−1}y ∈ R(L_{a}). Since y^{−1}y is an idempotent element, we have that y^{−1}y = a^{−1}a, so y ∈ L_{a}. The converse is clear since L_{a} contains the unique idempotent element.

(4): Assume that L_{a} is a semigroup. Then L_{a}L_{a} ⊆ L_{a}, so L_{a} ⊆ R(L_{a}). Conversely since L_{a} ⊆ R(L_{a}), L_{a}L_{a} ⊆ L_{a}, hence L_{a} is a semigroup.

(5): Assume that L_{a} is a group. Then L_{a} = R_{a} = D_{a}. Thus by (4), D_{a} = L_{a} ⊆ R(L_{a}). Conversely since L_{a}L_{a} ⊆ L_{a}D_{a} ⊆ L_{a}R(L_{a}) ⊆ L_{a}, L_{a} become a semigroup. Further, by Theorem 3.3(1), D_{a} contains an unique idempotent element. This shows that D_{a} = R_{a} = L_{a} = H_{a}. Hence L_{a} is a group.

COROLLARY 3.8. Let a be an element in the inverse semigroup S. Then the following statements hold.

(1) There exists an unique idempotent element in R(L_{a}) if, and only if, a^{−1}a is a right identity element of R(L_{a}).

(2) L_{a} is a semigroup if, and only if, L_{a} = R(L_{a}) ∩ D_{a}.

PROOF. (1): Let y be an element in R(L_{a}). Then since R(L_{a}) ⊆ L_{a}, ya^{−1}a = yy^{−1}y = y. Thus a^{−1}a is a right identity element of R(L_{a}). Conversely let f be an idempotent element in R(L_{a}), then a^{−1}a ≤ f. Further since a^{−1}a is a right identity element in R(L_{a}), fa^{−1}a = f. It follows that f = a^{−1}a.

(2): Let z ∈ R(L_{a}) ∩ D_{a}, then a^{−1}a ≤ z^{−1}z and z^{−1}z ≤ a^{−1}a by Theorem 3.3(1) and Theorem 3.6(2). Hence z^{−1}z = a^{−1}a which implies z ∈ L_{a}, that is, we have equality. The converse is clear from Theorem 3.6(4).

Finally, we consider an inverse semigroup S which satisfies the condition that each L-class is not a semigroup. We have the following.

THEOREM 3.9. Let S be an inverse semigroup such that each L-class is not a semigroup. Then there is a chain of idempotents in S which is not well ordered.

PROOF. Let e_{λ} be an idempotent element in S. Then since L_{e_{λ}} is not a semigroup, there is an idempotent e_{μ} such that L_{e_{λ}}L_{e_{μ}} ∩L_{e_{μ}} ≠ ∅ and e_{λ} ≠ e_{μ}. Let c be an element in L_{e_{λ}}L_{e_{μ}} ∩ L_{e_{μ}}. Then there exist a, b ∈ L_{e_{λ}} such that c = ab. Since e_{λ} is a right identity of L_{e_{λ}}, it follows that b = be_{λ}. Hence we have

Se_{λ} = Sc = (Sa)b = (Se_{λ})be_{λ} = (Se_{λ}be_{λ})e_{λ} = Se_{μ}e_{λ}.

This shows that e_{λ}e_{μ} is an element in L_{e_{λ}}, so Proposition 2.1(3) implies that e_{μ}e_{λ} = e_{μ}, which means that e_{λ} > e_{μ}. We can apply this process repeatedly to obtain a chain of idempotents in S. We claim that this chain is not well-ordered. Indeed, if there is a minimal element in the chain, then the L-class which contains the minimal element must become a semigroup, but this is contrary to our assumption.

EXAMPLE 3.10. Let S be the inverse semigroup generated by the following set

\{( \begin{matrix}ℓ & ℓ + 1 & ⋯ & ℓ + n & ⋯ \\ ℓ + 1 & ℓ + 2 & ⋯ & ℓ + n + 1 & ⋯ \end{matrix} ) | ℓ ∈ Z, n ∈ N^{0}\}

and its inverse. Then each L-class is

L_{m} = {σ ∈ S | Imσ = (m, m + 1, m + 2, ⋯)} .

Clearly L_{m} is not semigroup.

References

[1] A. H. Clifford, Semigroups admitting relative inverses, Ann. of Math., 42(1941), 1037–1049.

[2] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. II. Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I. 1967.

[3] J. M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, 1995.

[4] J. M. Howie and B. M. Schein, Anti-uniform semilattice, Bull. Australian Math. Soc., 1(1969), 263–268.

[5] A. E. Liber, On the theory of generalized groups (in Russian), Doklady Akad. Nauk. SSSR, 97(1954), 25–28.

[6] B. M. Schein, Representations of generalized groups, Izv. Uchebn. Zaved. Mathematika, 3(28)(1962), 164–176.

[7] V. V. Vagner, Generalized groups (in Russian), Doklady Akad. Nauk. SSSR, 84(1952), 1119–1122.