on maximal stable orders on an inverse semigroup of finite rank with

Ukrainian Mathematical Journal, Vol. 60, No. 8, 2008

ON MAXIMAL STABLE ORDERS ON AN INVERSE SEMIGROUP OF FINITE RANK WITH ZERO V. D. Derech

UDC 512.534.5

We consider maximal stable orders on semigroups that belong to a certain class of inverse semigroups of finite rank.

Introduction Let A be an arbitrary set and let S be an arbitrary semigroup of binary relations on A with ordinary operation of composition. It is easy to verify that the inclusion relation on the semigroup S is a stable order, i.e., an order consistent with the operation of composition. Based on this result, Lyapin [1] proved the following statement: Let W ( A ) be the semigroup of all partial transformations of an arbitrary set A. Let K denote the semia group of all transformations of the form ⎛ ⎞ , where a ∈ A and b ∈ A, including the empty transformation. ⎝ b⎠ If the semigroup of partial transformations S is such that K ⊆ S ⊆ W ( A ), then, for any stable order ρ, on the semigroup S one has ρ ⊆ Ω or ρ ⊆ Ω−1 , where Ω is the inclusion relation between transformations. A more general result was obtained in [2]. We formulate it below. Let ( Ai )i ∈I be a family of equipotent sets containing the common element 0 and such that Ai ∩ Aj = 0 if i ≠ j. Let I denote the inverse semigroup of partial one-to-one transformations of the set A = ∪ Ai that possess the following properties: (a) for any f ∈ I, one has af = 0 if and only if a = 0; (b) if f ∈ I , then f −1 ∈ I ; (c) if f ∈ I , then either dom ( f ) and ran ( f ) ( dom ( f ) and ran ( f ) are, respectively, the domain of definition and the range of values of the transformation f ) belong to the family 0 f = ; 0

()

( Ai )i ∈I or

(d) if the sets Ak and Am belong to the family ( Ai )i ∈I , then there exists a transformation ϕ ∈ I such that dom ( ϕ ) = Ak and ran ( ϕ ) = Am . Vinnytsya National Technical University, Vinnytsya, Ukraine. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1035–1041, August, 2008. Original article submitted September 20, 2006. 1210

0041–5995/08/6008–1210

© 2008

Springer Science+Business Media, Inc.

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The semigroup is an inverse Brandt semigroup, and any Brandt semigroup can be represented in this form up to isomorphism. We now formulate the main result of [2]. Theorem [2, p. 15]. Let S be a semigroup of binary relations on the set A = ∪ Ai such that the Brandt semigroup I defined above is a biideal in it. Also assume that the stable orders of the structural group of the semigroup I are exhausted by the trivial order. If ρ is a stable order relation on the semigroup S, then ρ ⊆ Ω o r ρ ⊆ Ω−1 , where Ω denotes the inclusion relation. In the present paper, we continue the investigation of this problem. The main result of the present paper is Theorem 3. 1. Main Terminology and Notation A semilattice E is called a semilattice of finite length if there exists a natural number n such that the length of any chain from E does not exceed n. It is obvious that a semilattice of finite length has the least element (zero). Let S be an arbitrary semigroup and let N0 be the set of all nonnegative integers. The function rank : S → N 0 is called the rank function on the semigroup S if, for any elements a and b ∈ S, the following inequality is true: rank ( ab ) ≤ min { rank ( a ), rank ( b ) }. The number rank ( a ) is called the rank of the element a. Let S be an inverse semigroup whose semilattice of idempotents has finite length. The function rank ( a ) = h (aa−1) , where h (aa−1) is the height of the idempotent aa−1 in the semilattice of idempotents of the semigroup S, is a rank function (see [3], p. 470). We say that an inverse semigroup is an inverse semigroup of finite rank if the semilattice of its idempotents has finite length. A semigroup is called a Δ-semigroup if its congruences form a chain with respect to inclusion. A semigroup is called permutable if any two congruences of it commute with respect to the ordinary operation of composition of binary relations. A nontrivial inverse semigroup with zero is called primitive if any nonzero idempotent of it is primitive. For other necessary definitions, see [4]. 2. Homomorphism of an Inverse Semigroup of Finite Rank with Zero into the Global Supersemigroup of a Primitive Inverse Semigroup Let S be an inverse semigroup of finite rank with zero. It is easy to verify that I1 = { x ∈ S rank ( x ) ≤ 1} is a primitive inverse semigroup. Let P( I1) denote the global supersemigroup of the semigroup I1, i.e., the semigroup of all nonempty subsets of the set I1 with respect to the ordinary operation of global multiplication. Further, let b ∈ S be an arbitrary element of the semigroup S. Denote the set { x ∈ S x ≤ b ∧ rank ( x ) ≤ 1 } by R1(b).

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Theorem 1. Let S be an inverse semigroup of finite rank with zero. The function F : b  R1(b) is a homomorphism of the semigroup into the global supersemigroup P( I1) . Proof. For any b and c ∈ S, it is necessary to prove the equality R1(b ⋅ c) = R1(b) ⋅ R1(c) . First, we show that R1(b ⋅ c) ⊆ R1(b) ⋅ R1(c) . Let a ∈ R1(b ⋅ c). If a = 0, then there is nothing to prove. Now let a ≠ 0, i.e., rank ( a ) = 1 and a ≤ bc. Multiplying the last inequality from the left by aa−1 , we obtain a ≤ aa−1 bc . Multiplying the obtained inequality from the

right by a−1a , we get

a ≤ aa−1 bca−1a .

(1)

Further, 1 = rank ( a ) ≤ rank (aa−1 bca−1a) ≤ rank (aa−1) = 1. Thus, rank (aa−1 bca−1a) = 1. Using inequality (1), we obtain a = aa−1 bca−1a .

(2)

Further, rank (aa−1b) ≤ rank (aa−1) = 1. Assuming that rank (aa−1b) = 0, we get rank ( a ) = 0, which is a contradiction. Thus, rank (aa−1b) = 1. By analogy, we establish that rank (ca−1a) = 1. Further, since aa−1b ≤ b and ca−1a ≤ c, taking equality (2) into account we conclude that a ∈ R1(b) ⋅ R1(c), i.e., R1(b ⋅ c) ⊆ R1(b) ⋅ R1(c) . We now show that R1(b) ⋅ R1(c) ⊆ R1(b ⋅ c) . Let a ∈ R1(b) ⋅ R1(c). Then a = a1 ⋅ a2 for certain a1 ∈ R1(b) and a2 ∈ R1(c). Since a1 ≤ b and a2 ≤ c, we have a1 ⋅ a2 ≤ b ⋅ c . Therefore, a ∈ R1(b ⋅ c). Thus, R1(b ⋅ c) = R1(b) ⋅ R1(c) . In other words, the function F : b  R1(b) is a homomorphism from the inverse semigroup S into the global supersemigroup P( I1) . The following question naturally arises: Under what conditions is the homomorphism F : b  R1(b) injective? Prior to the formulation of a theorem that answers this question, we recall the definition of dense ideal (see [5, p. 48]). An ideal I of a semigroup S is called dense if any homomorphism of S that is injective on I is injective on S. Theorem 2. Let S be an inverse semigroup of finite rank with zero. The homomorphism F : b  R1(b) is an isomorphism if and only if the ideal I1 = { x ∈ S rank ( x ) ≤ 1} is dense.

ON MAXIMAL STABLE ORDERS ON AN INVERSE SEMIGROUP OF FINITE RANK

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Proof. Let the ideal I1 be dense. It is obvious that the homomorphism F is injective on I1. Therefore, it follows from the definition of denseness that F is an injective homomorphism. Now assume that the homomorphism F is injective. Let us show that the ideal I1 is dense. Let Φ be a homomorphism of the semigroup S injective on I1. We show that Φ is injective on S. Assume that Φ ( b ) = Φ ( c ). It is necessary to prove that b = c. We choose an arbitrary element a ∈ R1(b). Then Φ (aa−1) Φ(b) = Φ (aa−1) Φ(c) , whence Φ (aa−1b) = Φ (aa−1c) . Since a ∈ R1(b), we have a ≤ b and, there-

fore, aa−1b = a. Thus, Φ ( a ) = Φ (aa−1c) . Since a ∈ I1 and aa−1c ∈ I1, we have aa−1c = a. Hence, a−1c = a−1a , i.e., a ≤ c and, consequently, a ≤ c. Thus, R1(b) ⊆ R1(c) . By analogy, one can show that R1(c) ⊆ R1(b). Therefore, R1(c) = R1(b). Since, by assumption, the homomorphism F : b  R1(b) is injective, we conclude that b = c. The theorem is proved.

Lemma 1. Let S be an inverse semigroup of finite rank with zero. If the ideal is dense, then the equivalence R1(b) ⊆ R1(c) ⇔ b ≤ c is true.

I1 of the semigroup S

Proof. The implication b ≤ c ⇒ R1(b) ⊆ R1(c) is obvious. We now prove that the inverse implication is true, i.e., R1(b) ⊆ R1(c) ⇒ b ≤ c. The following statement was proved in [6] (Proposition 2.18): For an ideal I of an inverse semigroup S, the following properties are equivalent: (i) I is a dense ideal; (ii) I is a ∨-basis ideal; (iii) I is a reductive ideal. Property (ii) means that every element b ∈ S can be represented in the form b = sup A, where A ⊆ I. We now pass the proof of the lemma. Since, by assumption, the ideal I1 is dense, we conclude that b = sup A for a certain set A contained in I1. Thus, A ⊆ R1(b). This implies that b = sup A ≤ sup R1(b) ≤ b. Therefore, sup R1(b) = b. Thus, if R1(b) ⊆ R1(c) , then b = sup R1(b) ≤ sup R1(c) = c, i.e., b ≤ c. The lemma is proved. 3. Stable Orders on an Inverse Semigroup of Finite Rank with Zero Let S be an inverse semigroup of finite rank with zero whose ideals are linearly ordered by inclusion and let Σ be a stable quasiorder on S . It is easy to verify that I l = { x ∈ S 〈 x, 0 〉 ∈ Σ } and I r = { x ∈ S 〈 0, x 〉 ∈ Σ } are ideals of the semigroup S. Thus, the stable quasiorder Σ is associated with the ordered pair of ideals 〈 I l , I r 〉 . It is easy to show that I l × I r ⊆ Σ . By virtue of Theorem 2 (see [3]), every ideal of the semigroup S is a rank ideal, and, hence, there exist nonnegative integers k and m such that I l = { x ∈ S rank ( x ) ≤ k }

and

I r = { x ∈ S rank ( x ) ≤ m } .

The pair of numbers 〈 k, m 〉 is called the index of the stable quasiorder Σ and is denoted by ind ( Σ ) .

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Lemma 2. Let S be an inverse semigroup of finite rank with zero whose ideals are linearly ordered by inclusion. If τ is a stable order on the semigroup S and ind ( τ ) = 〈 k, m 〉 , then k = 0 or m = 0. Proof. Assume that the converse statement is true, i.e., k ≠ 0 and m ≠ 0. For definiteness, let k ≤ m. Then I l ⊆ I r . Since k ≠ 0, there exists an element a ∈ I l such that a ≠ 0 and 〈 a, 0 〉 ∈ τ . Since I l ⊆ I r , we have a ∈ I r , i.e., 〈 0, a 〉 ∈ τ . Since τ is an antisymmetric binary relation, we get a = 0, which is a contradiction. The lemma is proved. We now formulate and prove the main theorem of the present paper. Theorem 3. Let S be an inverse semigroup of finite rank with zero whose ideals are linearly ordered by inclusion. Assume that the ideal I1 = { x ∈ S rank ( x ) ≤ 1} is a Brandt semigroup and that the stable orders of the structural group of the ideal I1 are exhausted by the trivial order. Furthermore, the ideal I1 i s dense. If the binary relation Σ is a stable order on the semigroup S and ind ( Σ ) = 〈 0, m 〉 , then Σ ⊆ ω, where ω is the canonical order on the inverse semigroup S. Proof. Let 〈 b, c 〉 ∈ Σ . We show that b ≤ c, where ≤ is another notation for the canonical order ω. By virtue of Lemma 1, we must prove that R1(b) ⊆ R1(c) . Assume that the converse statement is true, i.e., there exists an element a such that a ∈ R1(b) and a ∉ R1(c). Then a ≠ 0 and, consequently, rank ( a ) = 1. Since a ∈ R1(b), we have a ≤ b. Hence, aa−1b = a. Since 〈 b, c 〉 ∈ Σ , we have 〈 aa−1b, aa−1c 〉 ∈ Σ , i.e., 〈 a, aa−1c 〉 ∈ Σ . The last relation yields

〈 a, aa−1ca−1a〉 ∈ Σ.

(3)

Consider the element aa−1ca−1a . The following two cases are possible: (i) aa−1ca−1a = 0; (ii) rank (aa−1ca−1a) = 1. If aa−1ca−1a = 0, then 〈 a, 0 〉 ∈ Σ . Furthermore, as indicated above, we have a ≠ 0. However, by as-

sumption, ind ( Σ ) = 〈 0, m 〉 , i.e., we arrive at a contradiction. If rank (aa−1ca−1a) = 1, then

(

)

rank aa−1ca−1a (aa−1ca−1a)−1 = 1. Furthermore, aa−1ca−1a (aa−1ca−1a)−1 ≤ aa−1 .

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Since

(

)

rank aa−1ca−1a (aa−1ca−1a)−1 = rank (aa−1) = rank (a) = 1, we have aa−1ca−1a (aa−1ca−1a)−1 = aa−1 .

(4)

(aa−1ca−1a)−1 aa−1ca−1a = a−1a .

(5)

By analogy, we obtain

Using relations (3) – (5) and the condition that the structural group of the ideal −1

−1

−1

−1

I1 admits only the trivial stable

order, we get aa ca a = a. However, aa ca a ≤ c , i.e., a ≤ c. Thus, a ∈ R1(c), which is a contradiction. Thus, R1(b) ⊆ R1(c) . Using the last inclusion and Lemma 1, we establish that b ≤ c, i.e., 〈 b, c 〉 ∈ω . The theorem is proved. 4. Corollaries and Examples In this section, we formulate several corollaries of Theorem 3. Corollary 1. Maximal stable orders of an inverse semigroup S that satisfies the conditions of Theorem 3 are exhausted by the relations ω and ω −1, where ω is the canonical order on the inverse semigroup S. Corollary 2. Let S be a finite inverse Δ-semigroup (for the definition, see Sec. 1) with zero. Then, for any stable order Σ on S, one has Σ ⊆ ω or Σ ⊆ ω −1 , where ω is the canonical order. Proof. It is clear that the ideals of a Δ-semigroup are ordered by inclusion. Moreover, by assumption, the semigroup S is finite. Using these two remarks, we easily establish that the ideal I1 = { x ∈ S rank ( x ) ≤ 1} is a Brandt semigroup. It is easy to show (which was indicated, e.g., in [7]) that any nonzero ideal of the semigroup S is dense. Furthermore, it is known (see, e.g., [8, p. 297]) that only the trivial stable order exists on a finite group. Therefore, by virtue of Theorem 3 (and Lemma 2), we can conclude that Σ ⊆ ω or Σ ⊆ ω −1 . The corollary is proved. Further, let E be a semilattice of finite length and let T E denote the Mann semigroup, i.e., the semigroup of all isomorphisms between the principal ideals of the semilattice E with respect to the ordinary operation of superposition. Corollary 3. If Σ is a stable order on the permutable Mann semigroup TE , then Σ ⊆ ω o r Σ ⊆ ω −1 , where ω is the canonical order. Proof. It is obvious that the inverse semigroup TE contains zero. Since, by assumption, it is permutable, its ideals form a chain with respect to inclusion (see [9], Theorem 4). Moreover, any idempotent of the ideal

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I1 = { x ∈ S rank ( x ) ≤ 1} is primitive. Using these remarks, we easily establish that the ideal I1 is a Brandt semigroup. It was proved in [10] that any nonzero ideal of the semigroup T E is dense. Furthermore, it is clear that the structural group of the ideal I1 is a singleton. Thus, all conditions of Theorem 3 are satisfied for the semigroup TE . Therefore, Σ ⊆ ω or Σ ⊆ ω −1 . The corollary is proved. A stable order ρ on a semigroup S is called fundamental (see [11]) if the ordered semigroup 〈 S, ρ〉 is O-isomorphic to a certain semigroup of transformations that is ordered by the inclusion relation. If ρ is a fundamental stable order on a semigroup S, then ρ−1 is called an antifundamental stable order. It is known (see [12, p. 303]) that a stable order relation τ on an inverse semigroup is fundamental if and only if τ ⊆ ω, where ω is the canonical order. Corollary 4. Let S be an inverse semigroup that satisfies the conditions of Theorem 3. Then any stable order on S is fundamental or antifundamental. Let V be a finite-dimensional vector space over a finite field and let Aut p (V ) denote the inverse semigroup of all partial automorphisms between subspaces of the vector space V. It is easy to verify that all conditions of Theorem 3 are satisfied for the inverse semigroup Aut p (V ) . Corollary 5. For any stable order Σ on the inverse semigroup Aut p (V ), one has either Σ ⊆ ω o r Σ ⊆ ω −1 , where ω is the canonical order. Note that this corollary also follows from the main theorem in [2]. 5. Density of Ideal and Permutability of Congruences As noted above, any nonzero ideal of a Δ-semigroup is dense. It is obvious that a Δ-semigroup is permutable. It has been noted in the proof of Corollary 2 that any nonzero ideal of the permutable inverse Mann semigroup TE , where E is a semilattice of finite length, is dense. Finite symmetric inverse semigroups and semigroups of partial automorphisms of a finite-dimensional vector space also belong to permutable inverse semigroups in which any nonzero ideal is dense. The following question naturally arises: Is every nontrivial ideal of a permutable inverse semigroup S of finite rank with zero dense? The answer is negative. We illustrate this by the example presented below. Example. On the set { 1, 2, 3, 4 }, we consider the set of transformations ⎧ ⎛1⎞ ⎛ 1⎞ ⎛ 2⎞ ⎛ 2⎞ ⎛1234⎞ ⎛1234⎞ ⎫ S = ⎨ ∅, ⎜ ⎟ , ⎜ ⎟ , ⎜ ⎟ , ⎜ ⎟ , ⎜ ⎟, ⎜ ⎟ ⎬. ⎩ ⎝1⎠ ⎝ 2⎠ ⎝ 2⎠ ⎝ 1⎠ ⎝1234⎠ ⎝1243⎠ ⎭ It is easy to verify that S is an inverse semigroup that has exactly three linearly ordered ideals, namely, 1 ∅ ⊂ S ⎛ ⎞ S ⊂ S. ⎝ 1⎠

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Furthermore, it is obvious that condition 2 of Theorem 1 (see [13]) is satisfied. Therefore, by virtue of Theo1 rem 1 (see [13]), the semigroup S is permutable. We now show that the ideal S ⎛ ⎞ S is not dense. Let G ⎝ 1⎠ denote the group of invertible elements of the semigroup S. It is clear that ⎧ ⎛1234⎞ ⎛1234⎞ ⎫ G = ⎨⎜ ⎟, ⎜ ⎟ ⎬. ⎩ ⎝1234⎠ ⎝1243⎠ ⎭ Since ⎛1⎞ ⎧ S ⎜ ⎟ S = ⎨ ∅, ⎝1⎠ ⎩

⎛1⎞ ⎜ ⎟, ⎝1⎠

⎛ 1⎞ ⎜ ⎟, ⎝ 2⎠

⎛ 2⎞ ⎜ ⎟, ⎝ 2⎠

⎛ 2⎞ ⎫ ⎜ ⎟ ⎬, ⎝ 1⎠ ⎭

we have ⎛1⎞ S ⎜ ⎟ S = { x ∈ S rank ( x ) ≤ 1} . ⎝1⎠ As above, we denote the last ideal by I1. Consider the binary relation Σ = G × G ∪ Δ I1 , where Δ I1 is the identity transformation on the ideal I1. It is easy to verify that Σ is a congruence, and, furthermore, it is an identity congruence on I1. However, it is not an identity congruence on the entire semigroup S. Thus, the ideal I1 is not dense. REFERENCES 1. E. S. Lyapin, “On maximal two-sided stable orders in semigroups,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 34, No. 3, 88–94 (1963). 2. V. D. Derech, “On maximal stable orders on certain biideal extensions of the Brandt semigroup,” in: Semigroups and Their Homomorphisms [in Russian], Leningrad (1991), pp. 12–18. 3. V. D. Derech, “Congruences of a permutable inverse semigroup of finite rank,” Ukr. Mat. Zh., 57, No. 4, 469–473 (2005). 4. A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. 1, American Mathematical Society, Providence, RI (1964). 5. M. Petrich, Inverse Semigroups, Wiley, New York (1984). 6. B. M. Schein, “Completions, translational hulls and ideal extensions of inverse semigroups,” Czech. Math. J., 23, 575–610 (1973). 7. A. Nagy and P. R. Jones, “Permutative semigroups whose congruences form a chain,” Semigroup Forum, 69, No. 3, 446–456 (2004). 8. A. G. Kurosh, Lectures on General Algebra [in Russian], Nauka, Moscow (1973). 9. H. Hamilton, “Permutability of congruences on commutative semigroups,” Semigroup Forum, 10, No. 1, 55–66 (1975). 10. V. D. Derech, “Structure of a permutable Mann semigroup of finite rank,” Ukr. Mat. Zh., 58, No. 6, 742–746 (2006). 11. B. M. Schein, “Representation of ordered semigroups,” Mat. Sb., 65, No. 2, 188–197 (1964). 12. S. M. Goberstein, “Fundamental order relations on inverse semigroups and on their generalizations,” Semigroup Forum, 21, 285–328 (1980). 13. V. Derech, “On permutable inverse semigroups of finite rank,” in: Abstracts of the Fifth International Algebraic Conference in Ukraine, Odessa (2005), p. 57.