МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2018 MATHEMATICS AND EDUCATION IN MATHEMATICS, 2018 Proceedings of the Forty-seventh Spring Conference of the Union of Bulgarian Matheaticians Borovets, April 2 6, 2018 ON THE RELATIVE RANK OF THE SEMIGROUP OF ORIENTATION-PRESERVING TRANSFORMATIONS WITH RESTRICTED RANGE * Ilinka Diitrova, Jörg Koppitz, Kittisak Tinpun In this paper, we deterine the relative rank of the seigroup OP(X,Y) of all orientation-preserving transforations on a finite chain X with restricted range Y X odulo the seigroup O(X,Y) of all order-preserving transforations on X with restricted range Y. Let S be a seigroup. The rank of S (denoted by ranks) is defined to be the inial nuber of eleents of a generating set of S. The ranks of various well known seigroups have been calculated [4, 5, 6, 7]. For a set A S, the relative rank of S odulo A, denoted by rank(s : A), is the inial cardinality of a set B S such that A B generates S. The relative rank of a seigroup odulo a suitable set was first introduced by Ruškuc [10] in order to describe the generating sets of seigroups with infinite rank. But also if the rank is finite, the relative rank gives inforation about the generating sets. In the present paper, we will deterine the relative rank for a particular class of transforation seigroups. Let X be a finite chain, say X = {1 < 2 < < n} and denote by T (X) the onoid (under coposition) of all full transforations on X. A transforation α T (X) is called order-preserving if x y iplies xα yα, for all x,y X. We denote by O(X) the subonoid of T (X) of all order-preserving full transforations on X. We say that a transforation α T (X) is orientation-preserving if there are subsets X 1,X 2 X with X 1 < X 2, (i.e. x 1 < x 2 for x 1 X 1 and x 2 X 2 ), X = X 1 X 2, and xα yα, whenever either (x,y) X 2 1 X2 2 with x < y or (x,y) X 2 X 1. Note that X 2 = provides α O(X). We denote by OP(X) the subonoid of T (X) of all orientationpreserving full transforations on X. An equivalent notion of an orientation-preserving transforation was first introduced by McAlister in [9] and, independently, by Catarino and Higgins in [1]. It is interesting to note that the relative rank of OP(X) odulo O(X) as well as the relative rank of T (X) odulo OP(X) is one (see [1, 8]). Let Y = {a 1 < a 2 < < a } be a nonepty subset of X, and denote by T (X,Y) the subseigroup {α T (X) Xα Y} of T (X) of all eleents with range (iage) restricted to Y. In 1975, Syons [11] introduced and studied the seigroup T (X,Y), which is called seigroup of transforations with restricted range. In [2], Fernandes, * 2010 Matheatics Subject Classification: 20M20. Key words: transforation seigroups with restricted range, orientation-preserving transforations, relative rank. 109
Honya, Quinteiro, and Singha deterine the rank of the order-preserving counterpart O(X,Y) of T (X,Y). Recently, the regularity, the Green s relations, and the rank of the seigroup OP(X, Y) of all orientation-preserving transforations in T (X, Y) were studied by the sae authors in [3]. Recall, the rank of T (X,Y) is the Sterling nuber ( S(n,) ) of second kind with X = n and Y =. On the other hand, ranko(x,y) = + Y #, where Y # denotes the set of all y Y with one of the following 1 properties: (i) y has no successor in X; (ii) y is a no successor of any eleent in X; (iii) both the successor of Y and the eleent whose ( successor ) is y belong to Y. Moreover in n [3], Fernandes et al. show thatrankop(x,y) =. In [12], Tinpun and Koppitz show ( ) that rank(t (X,Y) : O(X,Y)) = S(n,) +a, where a {0,1} depending on 1 the set Y. In this paper, we deterine the relative rank of OP(X,Y) odulo O(X,Y). Letα OP(X,Y). The kernel ofαis the equivalence relationkerα with (x,y) kerα if xα = yα. It corresponds uniquely to a partition on X. This justifies to regard kerα as partition on X. We will call a block of this partition a kerα-class. In particular, xα 1 := {y X : yα = x}, for x Xα, are kerα-classes. We say that a partition P is a subpartition of a partition Q of X if for all p P there is a q Q with p q. A set T X with T xα 1 = 1 for all x Xα, is called a transversal of kerα. Let A X. Then α A : A Y denotes the restriction of α to A and A will be called convex if x < y < z with x,z A iplies y A. Let l {1,...,}. We denote by P l the set of all partitions {A 1,...,A l } of X such that A 2 < A 3 < < A l are convex sets (if l > 1) and A 1 is the union of two convex sets with 1,n A 1. For P P with the blocks A 1,A 2 < < A, let α P be the transforation on X defined by xα P := a i, whenever x A i for 1 i in the case 1 / Y or n / Y and { ai+1, if x A xα P := i for 1 i < a 1 if x A in the case 1,n Y. Clearly, kerα P = P. With X 1 := {1,...,axA }, X 2 := {axa + 1,...,n} and X 1 = {1,...,axA 1 }, X 2 = {axa 1 + 1,...,n}, respectively, we can easy verify that α P is orientation-preserving. Further, let η T (X,Y) be defined by a i+1 if a i x < a i+1 for 1 i < { 1 if 1 / Y xη := a 1 if x = a with Γ := 2 otherwise, otherwise a Γ in the case 1 / Y or n / Y and a i+1 if a i x < a i+1, 2 i < xη := a 1 if x = a = n a 2 if x < a 2 in the case ( 1,n Y. In fact, η) OP(X,Y) and η Y is a perutation on Y, naely a1... a η Y = 1 a. We will show that A := {α a 2 a a P : P P } {η} is a 1 110
relative generating set of OP(X, Y) odulo O(X, Y). Lea 1. For each α OP(X,Y) with rankα =, there is an α {α P : P P } O(X,Y) with kerα = ker α. Proof. Let α OP(X,Y) and let X 1,X 2 X as in the definition of orientationpreserving transforation. If X 2 = then α O(X,Y). Suppose now that X 2 and let X 1 α = {x 1 < < x r } and X 2 α = {y 1 < < y s } for suitable natural nubers r and s. We observe that X 1 α and X 2 α have at ost one joint eleent (only x 1 = y s could be possible). If x 1 ( y s then kerα = {x 1 α 1 < < x r α 1 < y 1 α 1 < < y s α 1 x1 α 1 x } = ker α with α = r α 1 y 1 α 1 y s α 1 ) O(X,Y). If a 1 a r a r+1 a r+s x 1 = y s then 1,n x 1 α 1 = y s α 1 and kerα = kerα P with P = {x 1 α 1,x 2 α 1 < < x r α 1 < y 1 α 1 < < y s 1 α 1 } P. Lea 2. OP(X,Y) = O(X,Y),A. Proof. Let β OP(X,Y) with rankβ =. Then there is θ {α P : P P } O(X,Y) with kerβ = kerθ by Lea 1. In particular, there is r {0,..., 1} with a 1 θ 1 = a r+1 β 1. Then it is easy to verify that β = θη r, where η 0 := η. Suppose now that i := rankβ < and that kerβ P i, say kerβ = {A 1,A 2 < < A i } with 1,n A 1. Then there is a subpartition P P of kerβ. We put θ := α P, a := inxβ, and lett be a transversal ofkerθ. In particular, we havey = {x(θ T )η k : x T} for all k {1,...,}. Since both appings θ T : T Y and η Y : Y Y are bijections, there is k {1,...,} with a 1 ((θ T )η k ) 1 β = a and a 1 ((θ T )η k+1 ) 1 β a. Moreover, since (θ T )η k is a bijection fro T to Y and both transforations θη k and β are orientation-preserving, it is easy to verify that f := ((θ T )η k ) 1 β can be extended to an orientation-preserving transforation f defined by a 1 f if x < a 1 xf := a i f if a i x < a i+1, 1 i < a f if a x, i.e. f and f coincide on Y. Moreover, a 1 f = a 1 f = a 1 ((θ T )η k ) 1 β = a. In order to show that f is order-preserving, it left to verify that nf a. Assue that nf = a, where n a. Then nf = a f = a f, i.e. (n,a ) kerf and nη = a η = a 1. So, there is x T such that x ((θ T )η k ) = a, i.e. x = a ((θ T )η k ) 1. Now, we have a = nf = a f = a ((θ T )η k ) 1 β = a (η k Y ) 1 (θ T ) 1 β = a 1 (η Y ) 1 (η k Y ) 1 (θ T ) 1 β = a 1 ((θ T )η k+1 ) 1 β a, a contradiction. Finally, we will verify that β = θη k f O(X,Y),A. For this let x X. Then there is x T such that (x, x) kerβ. So, we have xθη k f = xθη k f = xθη k ((θ T )η k ) 1 β = xβ = xβ. Suppose now that kerβ / P i and let Xβ = {b 1,...b i } such that b 1 β 1 < < b i β 1. Then we define a transforation ϕ by xϕ := a j for all x b j 1 β 1 and 2 j i+1. Clearly, ϕ O(X, Y). Further, we define a transforation ν T (X, Y) by { bj if a xν := j < x a j+1, 2 j i otherwise. b i Since β is orientation-preserving, there is k {1,...,i} such that k = i or b 1 < < b k 1 > b k < < b i. Then X 1 := {a 1,...,a k+1 1} and X 2 := {a k+1,...,n} give a partition of X providing that ν is orientation-preserving. Clearly, rank ν = i and 111
1ν = nν = b i. Thus, it is easy to verify that kerν P i. Hence, ν O(X,Y),A by the previous case and it reains to show that β = ϕν O(X,Y),A. For this let x X. Then x b j β 1 for soe j {1,...,i}, i.e. xϕν = a j+1 ν = b j = xβ. The previous lea shows that A is a relative generating set for OP(X,Y) odulo O(X,Y). It reains to show that A is of inial size. Lea 3. Let B OP(X,Y) be a relative generating set of OP(X,Y) odulo O(X,Y). Then P {kerα : α B}. Proof. LetP P. Sinceα P OP(X,Y) = O(X,Y),B, there areθ 1 O(X,Y) B and θ 2 OP(X,Y) with α P = θ 1 θ 2. Because of rankα P =, we obtain kerα P = kerθ 1. Since 1α P = nα P, we conclude that θ 1 / O(X,Y), i.e. θ 1 B with kerθ 1 = kerα P = P. In order to find a forula for the nuber of eleents in P, we have to( copute ) the nuber of possible partitions of X into +1 convex sets. This nuber is. ( ) Reark 4. P =. Now we are able to state the ain result of the paper. The relative rank of OP(X,Y) odulo O(X,Y) depends of the fact whether both 1 and n belong to Y( or not. ) Theore 5. If 1 / Y or n / Y then rank(op(x,y) : O(X,Y)) =. Proof. It is easy to verify that P := kerη P and η = α P. Thus, A = {α P : P P } is a generating set of OP(X,Y) odulo O(X,Y) by Lea 2, i.e. the relative ( rank ) of OP(X,Y) odulo O(X,Y) is bounded by the cardinality of P, which is by Reark 4. But this nuber cannot be reduced by Lea 3. ( ) Theore 6. If {1,n} Y then rank(op(x,y) : O(X,Y)) = 1+. Proof. LetB OP(X,Y)be a relative generating set ofop(x,y)oduloo(x,y). By Lea 3, we know that P {kerα : α B}. Assue that the equality holds. Note that 1η = a 1 η > a η = nη, i.e. kerη / P and η is not order-preserving. Hence, there are θ 1,...,θ l O(X,Y) B, for a suitable natural nuber l, such that η = θ 1 θ l. Fro rankη =, we obtain kerθ 1 = kerη and rankθ i = for i {1,...,l} and thus, {1,n} Y iplies (1,n) / kerθ i for i {2,...,l}. This iplies θ 2,...,θ l O(X,Y). Since kerθ 1 = kerη / P, we get θ 1 O(X,Y), and consequently, η = θ 1 θ 2 θ l O(X,Y), a contradiction. So, we have verified ( that ) P < B, i.e. the relative( rank of ) OP(X,Y) odulo O(X,Y) is greater than. But it is bounded by 1+ due to Lea 2. This proves the assertion. REFERENCES [1] P. M. Catarino, P. M. Higgins. The onoid of orientation-preserving appings on a chain. Seigroup Foru, 58 (1999), 190 206. [2] V. H. Fernandes, P. Honya, T. M. Quinteiro, B. Singha. On seigroups of endoorphiss of a chain with restricted range. Seigroup Foru, 89 (2014), 77 104. 112
[3] V. H. Fernandes, P. Honya, T. M. Quinteiro, B. Singha. On seigroups of orientation-preserving transforations with restricted range. Co. Algebra, 44 (2016), 253 264. [4] G. M. S. Goes, J. M. Howie. On the rank of certain seigroups of order-preserving transforations. Seigroup Foru, 51 (1992), 275 282. [5] G. M. S. Goes, J. M. Howie. On the ranks of certain finite seigroups of transforations. Math. Proc. Cabridge Philos. Soc., 101 (1987), 395 403. [6] Howie, J. M., Fundaentals of seigroup theory, Oxford, Oxford University Press, 1995. [7] J. M. Howie, R. B. McFadden. Idepotent rank in finite full transforation seigroups. Proc. Royal Soc. Edinburgh, 114A (1990), 161 167. [8] J. M. Howie, N. Ruškuc, P. M. Higgins. On relative ranks of full transforation seigroups. Co. Algebra, 26 (1998), 733 748. [9] D. McAlister. Seigroups generated by a group and an idepotent. Co. in Algebra, 26 (1998), 515 547. [10] N. Ruškuc. On the rank of copletely 0-siple seigroups. Math. Proc. Cabridge Philos. Soc., 116 (1994), 325 338. [11] J. S. V. Syons. Soe results concerning a transforation seigroup. J. Austral. Math. Soc., 19 (1975), 413 425. [12] K. Tinpun, J. Koppitz. Relative rank of the finite full transforation seigroup with restricted range. Acta Matheatica Universitatis Coenianae, LXXXV, No 2 (2016), 347 356. Ilinka Diitrova, Faculty of Matheatics and Natural Science South-West University Neofit Rilski 2700 Blagoevgrad, Bulgaria e-ail: ilinka_diitrova@swu.bg Jörg Koppitz Institute of Matheatics and Inforatics Bulgarian Acadey of Sciences 1113 Sofia, Bulgaria e-ail: koppitz@ath.bas.bg Kittisak Tinpun Institute of Matheatics University of Potsda 14476 Potsda, Gerany e-ail: keaw.030@gail.co 113
ВЪРХУ ОТНОСИТЕЛНИЯ РАНГ НА ПОЛУГРУПАТА ОТ ВСИЧКИ ЗАПАЗВАЩИ ОРИЕНТАЦИЯТА ПРЕОБРАЗОВАНИЯ С ОГРАНИЧЕНО МНОЖЕСТВО ОТ ОБРАЗИ Илинка Димитрова, Йорг Копиц, Китисак Тинпун Нека S е полугрупа и A е подмножество на S. Относителен ранг на полугрупата S по модулaсе нарича най-малкото кардинално число на множество B S, такова че A B поражда S. Означава се с rank(s : A). Нека X е крайна верига, например X = {1 < 2 < < n}. Моноида от всички пълни преобразования на множеството X относно операцията композиция на преобразования се означава с T (X). Едно преобразование α T (X) се нарича запазващо наредбата, ако от x y следва, че xα yα за всяко x,y X. С O(X) се означава полугрупата от всички запазващи наредбата преобразования на X. Преобразованието α T (X) се нарича запазващо ориентацията, ако съществуват подмножества X 1,X 2 X със свойствата X 1 < X 2, X = X 1 X 2 и xα < yα за всяко (x,y) X 2 1 X 2 2 X 2 X 1. Полугрупата от всички запазващи ориентацията преобразования на X се означава с OP(X). Нека Y = {a 1 < a 2 < < a } е непразно подмножество на X. С T (X,Y) се означава подполугрупата {α T (X) Xα Y} на T (X) от всички пълни преобразования наx с множество от образи, съдържащо се вy.обект на разглеждане в настоящата работа е полугрупата OP(X, Y) от всички запазващи ориентацията преобразования на X с множество от образи, съдържащо се в Y. Намерен е относителният ранг на полугрупата OP(X, Y) по модул полугрупата O(X, Y) от всички запазващи наредбата преобразования на X с множество от образи, съдържащо се в Y. 114