International Mathematical Forum, 5, 2010, no. 48, 2381 - 2385

Regular Elements of Some Semigroups of Order-Preserving Partial Transformations 1

W. Mora and 2 Y. Kemprasit

Department of Mathematics, Faculty of Science Chulalongkorn University, Bangkok 10330, Thailand 1 [email protected], 2 [email protected] Abstract Let X be a chain, OP (X) the order-preserving partial transformation semigroup on X and OI(X) the order-preserving 1–1 partial transformation semigroup on X. It is known that both OP (X) and OI(X) are regular semigroups. We extend these results by characterizing the regular elements of the semigroups OP (X, Y ), OI(X, Y ), OP (X, Y ) and OI(X, Y ) where ∅ = Y ⊆ X, OP (X, Y ) = {α ∈ OP (X) | ran α ⊆ Y }, OP (X, Y ) = {α ∈ OP (X) | (dom α ∩ Y )α ⊆ Y }, OI(X, Y ) and OI(X, Y ) are defined analogously. The semigroups OP (X, Y ) and OP (X, Y )[OI(X, Y ), OI(X, Y )] may be counted as generalizations of OP (X)[OI(X)]. In addition, it is shown that each of these semigroups becomes a regular semigroup only the case that Y = X.

Mathematics Subject Classification: 20M20, 20M17 Keywords: Order-preserving partial transformation semigroup, regular element of a semigroup

1

Introduction

An element x of a semigroup S is called regular if x = xyx for some y ∈ S and S is called a regular semigroup if every element of S is regular. The set of all regular elements of S is denoted by Reg(S). For a nonempty set X, let T (X), P (X) and I(X) be the full transformation semigroup on X, the partial transformation semigroup on X and the 1-1 partial transformation semigroup (the symmetric inverse semigroup) on X, respectively. The domain and the range (image) of α ∈ P (X) are denoted by dom α and ran α, respectively. For α ∈ P (X) and x ∈ dom α, the image of x under α is written as xα. Note that T (X) = {α ∈ P (X) | dom α = X},

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I(X) = {α ∈ P (X) | α is 1-1} and T (X) and I(X) are subsemigroups of P (X). Recall that for α, β ∈ P (X), dom(αβ) = (ran α ∩ dom β)α−1 ⊆ dom α, ran(αβ) = (ran α ∩ dom β)β ⊆ ran β, for x ∈ X, x ∈ dom(αβ) ⇔ x ∈ dom α and xα ∈ dom β. It is well-known that T (X), P (X) and I(X) are regular semigroups ([1], p.4). In fact, I(X) is an inverse semigroup, i.e., for every α ∈ I(X), there is a unique β ∈ I(X) such that α = αβα and β = βαβ. For a poset X and α ∈ P (X), α is said to be order-preserving if for  x, x ∈ dom α, x ≤ x implies xα ≤ x α. Let OP (X) = {α ∈ P (X) | α is order-preserving} . We define OT (X) and OI(X) analogously. Kemprasit and Changphas [3] studied the regularity of OT (X) for certain chains X. In [5], the authors characterized the regular elements of OT (X) for any chain X and then showed that the results on OT (X) provided in [3] can be obtained as its consequences. Symmon [6] and Magill [4] have studied the subsemigroups T (X, Y ) = {α ∈ T (X) | ran α ⊆ Y } and T (X, Y ) = {α ∈ T (X) | Y α ⊆ Y } of T (X), respectively where Y is a nonempty subset of a set X. Since T (X, X) = T (X, X) = T (X), we can count these semigroups as generalizations of T (X). For a chain X, OT (X, Y ) is defined in [5] analogously, i.e., OT (X, Y ) = {α ∈ OT (X) | ran α ⊆ Y }. Then OT (X, Y ) may be considered as a generalization of OT (X). In [5], the regular elements of OT (X, Y ) were characterized. The regularity of OT (X, Y ) was also determined. It was shown in [3] that for any chain X, OP (X) and OI(X) are regular semigroups. In this paper, the semigroups OP (X, Y ), OI(X, Y ), OP (X, Y ) and OI(X, Y ) are defined similarly where ∅ = Y ⊆ X, i.e., OP (X, Y ) = {α ∈ OP (X) | ran α ⊆ Y }, OI(X, Y ) = {α ∈ OI(X) | ran α ⊆ Y }, OP (X, Y ) = {α ∈ OP (X) | (dom α ∩ Y )α ⊆ Y }, OI(X, Y ) = {α ∈ OI(X) | (dom α ∩ Y )α ⊆ Y }. Then OP (X, X) = OP (X, X) = OP (X) and OI(X, X) = OI(X, X) = OI(X). Hence OP (X, Y ) and OP (X, Y ) generalize OP (X) while OI(X, Y ) and OI(X, Y ) generalize OI(X). Notice that OP (X, Y ) ⊆ OP (X, Y ) and OI(X, Y ) ⊆ OI(X, Y ). We characterize the regular elements of the semigroups OP (X, Y ), OI(X, Y ), OP (X, Y ) and OI(X, Y ) and show that Y = X is necessary for each of these semigroups to be a regular semigroup.

Regular elements of some semigroups

2

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The Semigroups OP (X, Y ) and OI(X, Y )

In this section, Reg(OP (X, Y )), Reg(OI(X, Y )) and the regularity of OP (X, Y ) and OI(X, Y ) are determined. The following result is clearly seen. It will be often used in what follows: Lemma 2.1. Let X be a chain. If α ∈ OP (X) and a, b ∈ ran α are such that a < b, then s < t for all s ∈ aα−1 and t ∈ bα−1 . Theorem 2.2. Let X be a chain and ∅ = Y ⊆ X. Then for α ∈ OP (X, Y ), α ∈ Reg(OP (X, Y )) if and only if ran α = (dom α ∩ Y )α. Proof. Assume that α ∈ Reg(OP (X, Y )). Let β ∈ OP (X, Y ) be such that α = αβα. Then ran(αβ) ⊆ Y , so ran α = ran(αβα) = (ran(αβ) ∩ dom α)α ⊆ (Y ∩ dom α)α ⊆ ran α which implies that ran α = (dom α ∩ Y )α. Conversely, assume that ran α = (dom α ∩ Y )α. Then xα−1 ∩ Y = ∅ for all x ∈ ran α. For each x ∈ ran α, choose dx ∈ xα−1 ∩ Y. Then dx α = x for all x ∈ ran α. Define β : ran α → Y by   x β= . dx x ∈ ran α Since α ∈ OP (X), by Lemma 2.1, β is order-preserving. Then β ∈ OP (X, Y ). Since for x ∈ dom α, xα ∈ dom β and xαβ ∈ dom α, it follows that dom(αβα) = dom α. If x ∈ dom α, then xαβα = (xα)βα = dxα α = xα. Therefore α = αβα, so α ∈ Reg(OP (X, Y )), as desired. Notice that β defined in the proof of Theorem 2.2 is 1-1. Then β ∈ OI(X, Y ). Theorem 2.3. Let X be a chain and ∅ = Y ⊆ X. Then for α ∈ OI(X, Y ), α ∈ Reg(OI(X, Y )) if and only if dom α ⊆ Y . Proof. Assume that α ∈ Reg(OI(X, Y )). Since OI(X, Y ) is clearly a subsemigroup of OP (X, Y ), it follows that α ∈ Reg(OP (X, Y )). By Theorem 2.2, ran α = (dom α∩Y )α. Then (dom α)α = (dom α∩Y )α, so dom α = dom α∩Y since α is 1-1. Hence dom α ⊆ Y . Conversely, assume that dom α ⊆ Y . Then dom α = dom α ∩ Y, so ran α = (dom α)α = (dom α ∩ Y )α. From the proof of Theorem 2.2, there is a β ∈ OI(X, Y ) such that α = αβα. Hence α ∈ Reg(OI(X, Y )). Corollary 2.4. Let X be a chain, ∅ = Y ⊆ X and let OS(X, Y ) be OP (X, Y ) or OI(X, Y ). Then OS(X, Y ) is a regular semigroup if and only if Y = X.

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 Proof. Suppose that Y  X. Let a ∈ X  Y and b ∈ Y . Then α = ab ∈ OI(X, Y ) ⊆ OP (X, Y ). But dom α ∩ Y = ∅, ran α = {b} and dom α = {a}  Y, so by Theorem 2.2 and Theorem 2.3, α ∈ / Reg(OS(X, Y )). If Y = X, then OP (X, Y ) = OP (X) and OI(X, Y ) = OI(X) and both OP (X) and OI(X) are regular semigroups. Hence the corollary is proved.

3

The Semigroups OP (X, Y ) and OI(X, Y )

The purpose of this section is to characterize the elements of Reg(OP (X, Y )) and Reg(OI(X, Y )). The regularity of OP (X, Y ) and OI(X, Y ) is also considered. Theorem 3.1. Let X be a chain and ∅ = Y ⊆ X. Then for α ∈ OP (X, Y ), α ∈ Reg(OP (X, Y )) if and only if ran α ∩ Y = (dom α ∩ Y )α. Proof. Assume that α ∈ Reg(OP (X, Y )). Since (dom α ∩ Y )α ⊆ Y , we have that (dom α ∩ Y )α ⊆ ran α ∩ Y . To show that ran α ∩ Y ⊆ (dom α ∩ Y )α, let β ∈ OP (X, Y ) be such that α = αβα. Let x ∈ ran α∩Y . Then x = aα for some a ∈ dom α. Thus x = aα = aαβα = xβα which implies that x ∈ dom β and xβ ∈ dom α. It follows that x ∈ dom β ∩Y and hence xβ ∈ (dom β ∩Y )β ⊆ Y . We then deduce that xβ ∈ dom α∩Y . Consequently, x = xβα ∈ (dom α∩Y )α. This proves that ran α ∩ Y = (dom α ∩ Y )α. For the converse, assume that ran α∩Y = (dom α∩Y )α. Then xα−1 ∩Y = ∅ for all x ∈ ran α ∩ Y . For each x ∈ ran α ∩ Y, choose dx ∈ xα−1 ∩ Y and for each x ∈ ran α  Y, choose ex ∈ xα−1 . Then dx α = x for all x ∈ ran α ∩ Y and ex α = x for all x ∈ ran α  Y . Define β : ran α → dom α by   x u . β= dx eu x ∈ ran α ∩ Y



u ∈ ran α Y

Then (dom β ∩ Y )β = (ran α ∩ Y )β = {dx | x ∈ ran α ∩ Y } ⊆ Y . Since α ∈ OP (X), it follows from Lemma 2.1 that β is order-preserving. Hence β ∈ OP (X, Y ). Since for x ∈ dom α, xα ∈ dom β and xαβ ∈ dom α, we deduces that dom α = dom(αβα). If x ∈ dom α, then  dxα α = xα if xα ∈ Y, xαβα = exα α = xα if xα ∈ / Y, so α = αβα. Thus α ∈ Reg(OP (X, Y )), as desired. It can be seen that β constructed in the proof of Theorem 3.1 is 1-1. Then β ∈ OI(X, Y ).

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Theorem 3.2. Let X be a chain and ∅ = Y ⊆ X. Then for α ∈ OI(X, Y ), α ∈ Reg(OI(X, Y )) if and only if (ran α ∩ Y )α−1 ⊆ Y . Proof. Assume that α ∈ Reg(OI(X, Y )). Since OI(X, Y ) is a subsemigroup of OP (X, Y ), we have that α ∈ Reg(OP (X, Y )). By Theorem 3.1, ran α ∩ Y = (dom α ∩ Y )α. Then (ran α ∩ Y )α−1 = (dom α ∩ Y )αα−1. Since αα−1 is the identity mapping on dom α, it follows that (dom α ∩ Y )αα−1 = dom α ∩ Y . Hence (ran α ∩ Y )α−1 = dom α ∩ Y ⊆ Y . Conversely, assume that (ran α ∩ Y )α−1 ⊆ Y . But (ran α ∩ Y )α−1 ⊆ dom α, so (ran α ∩ Y )α−1 ⊆ dom α ∩ Y . Thus (ran α ∩ Y )α−1 α ⊆ (dom α ∩ Y )α ⊆ ran α ∩ Y . Since α−1 α is the identity mapping on ran α, we have that (ran α ∩ Y )α−1 α = ran α ∩ Y . This implies that (dom α ∩ Y )α = ran α ∩ Y . From the proof of Theorem 3.1, α = αβα for some β ∈ OI(X, Y ). Hence α ∈ Reg(OI(X, Y )), as desired. Corollary 3.3. Let X be a chain, ∅ = Y ⊆ X and let OS(X, Y ) be OP (X, Y ) or OI(X, Y ). Then OS(X, Y ) is a regular semigroup if and only if Y = X.  Proof. Suppose that Y  X. Let a ∈ X  Y and b ∈ Y . Then α = ab ∈ OS(X, Y ). Since dom α ∩ Y = ∅, ran α ∩ Y = {b} and bα−1 = a ∈ / Y ,by Theorem 3.1 and Theorem 3.2, α ∈ / Reg(OS(X, Y )). If Y = X, then OP (X, Y ) = OP (X) and OI(X, Y ) = OI(X) and both OP (X) and OI(X) are regular semigroups. Therefore the corollary is proved.

References [1] P. M. Higgins, Techniques of Semigroup Theory, New York, Oxford University Press, 1992. [2] J. M. Howie, Fundamentals of Semigroup Theory, Oxford, Clarendon Press, 1995. [3] Y. Kemprasit and T. Changphas, Regular order-preserving transformation semigroups, Bull. Austral. Math. Soc. 62(2000), 511–524. [4] K. D. Magill, Jr., Subsemigroups of S(X), Math. Japonicae 11(1966), 109-115. [5] W. Mora and Y. Kemprasit, Regular elements of some order-preserving transformation semigroups, submitted. [6] J.S.V. Symons, Some results concerning a transformation semigroup, J. Austral. Math. Soc. 19(Series A)(1975), 413-425. Received: April, 2010