Let $ T(X) $ be the full transformation semigroup on a nonempty set $ X $. For an equivalence relation $ E $ on $ X $ and a nonempty subset $ Y $ of $ X $, let
$ \overline{S}_E(X, Y) = \{ \alpha \in T(X) : \forall x, y \in Y, (x, y) \in E \Rightarrow (x \alpha, y \alpha) \in E, x \alpha, y \alpha \in Y \}. $
Then $ \overline{S}_E(X, Y) $ is a subsemigroup of $ T(X) $ consisting of all full transformations that leave $ Y $ and the equivalence relation $ E $ on $ Y $ invariant. In this paper, we show that $ \overline{S}_E(X, Y) $ is not regular in general and determine all its regular elements. Then we characterize relations $ \mathcal{L} $, $ \mathcal{L}^* $, $ \mathcal{R} $ and $ \mathcal{R}^* $ on $ \overline{S}_E(X, Y) $ and apply these characterizations to obtain the abundance on such semigroup.
Citation: Kitsanachai Sripon, Ekkachai Laysirikul, Yanisa Chaiya. Regularity and abundance on semigroups of transformations preserving an equivalence relation on an invariant set[J]. AIMS Mathematics, 2023, 8(8): 18223-18233. doi: 10.3934/math.2023926
Let $ T(X) $ be the full transformation semigroup on a nonempty set $ X $. For an equivalence relation $ E $ on $ X $ and a nonempty subset $ Y $ of $ X $, let
$ \overline{S}_E(X, Y) = \{ \alpha \in T(X) : \forall x, y \in Y, (x, y) \in E \Rightarrow (x \alpha, y \alpha) \in E, x \alpha, y \alpha \in Y \}. $
Then $ \overline{S}_E(X, Y) $ is a subsemigroup of $ T(X) $ consisting of all full transformations that leave $ Y $ and the equivalence relation $ E $ on $ Y $ invariant. In this paper, we show that $ \overline{S}_E(X, Y) $ is not regular in general and determine all its regular elements. Then we characterize relations $ \mathcal{L} $, $ \mathcal{L}^* $, $ \mathcal{R} $ and $ \mathcal{R}^* $ on $ \overline{S}_E(X, Y) $ and apply these characterizations to obtain the abundance on such semigroup.
| [1] |
R. Chinram, S. Baupradist, Magnifying elements in semigroups of transformations with invariant set, Asian-Eur. J. Math., 12 (2019), 1950056. http://dx.doi.org/10.1142/S1793557119500566 doi: 10.1142/S1793557119500566
|
| [2] |
W. Choomanee, P. Honyam, J. Sanwong, Regularity in semigroups of transformations with invariant sets, International Journal of Pure and Applied Mathematics, 87 (2013), 151–164. http://dx.doi.org/10.12732/ijpam.v87i1.9 doi: 10.12732/ijpam.v87i1.9
|
| [3] | J. Howie, Fundamentals of semigroup theory, Oxford: Clarendon Press, 1995. |
| [4] |
P. Honyam, J. Sanwong, Semigroups of transformations with invariant set, J. Korean Math. Soc., 48 (2011), 289–300. http://dx.doi.org/10.4134/JKMS.2011.48.2.289 doi: 10.4134/JKMS.2011.48.2.289
|
| [5] |
T. Kaewnoi, M. Petapirak, R. Chinram, Magnifying elements in a semigroup of transformations preserving equivalence relation, Korean J. Math., 27 (2019), 269–277. http://dx.doi.org/10.11568/kjm.2019.27.2.269 doi: 10.11568/kjm.2019.27.2.269
|
| [6] | E. Lyapin, Semigroups, Providence: American Mathematical Society, 1963. |
| [7] | S. Nenthein, P. Youngkhong, Y. Kemprasit, Regular elements of some transformation semigroups, Pure Mathematics and Applications, 16 (2005), 307–314. |
| [8] |
H. Pei, Regularity and Green's relations for semigroups of transformations that preserve an equivalence, Commun. Algebra, 33 (2005), 109–118. http://dx.doi.org/10.1081/AGB-200040921 doi: 10.1081/AGB-200040921
|
| [9] |
H. Pei, Equivalences, $\alpha$-semigroups and $\alpha$-congruences, Semigroup Forum, 49 (1994), 49–58. http://dx.doi.org/10.1007/BF02573470 doi: 10.1007/BF02573470
|
| [10] |
H. Pei, A regular $\alpha$-semigroup inducing a certain lattice, Semigroup Forum, 53 (1996), 98–113. http://dx.doi.org/10.1007/BF02574125 doi: 10.1007/BF02574125
|
| [11] |
H. Pei, On the rank of the semigroup $T_E(X)$, Semigroup Forum, 70 (2005), 107–117. http://dx.doi.org/10.1007/s00233-004-0135-z doi: 10.1007/s00233-004-0135-z
|
| [12] |
H. Pei, H. Zhou, Abundant semigroups of transformations preserving an equivalence relation, Algebr. Colloq., 18 (2011), 77–82. http://dx.doi.org/10.1142/S1005386711000034 doi: 10.1142/S1005386711000034
|
| [13] |
L. Sun, J. Sun, A note on naturally ordered semigroups of transformations with invariant set, Bull. Aust. Math. Soc., 91 (2015), 264–267. http://dx.doi.org/10.1017/S0004972714000860 doi: 10.1017/S0004972714000860
|
| [14] |
L. Sun, L. Wang, Natural partial order in semigroups of transformations with invariant set, Bull. Aust. Math. Soc., 87 (2013), 94–107. http://dx.doi.org/10.1017/S0004972712000287 doi: 10.1017/S0004972712000287
|
| [15] |
L. Sun, H. Pei, Z. Cheng, Naturally ordered transformation semigroups preserving an equivalence, Bull. Austral. Math. Soc., 78 (2008), 117–128. http://dx.doi.org/10.1017/S0004972708000543 doi: 10.1017/S0004972708000543
|
| [16] |
L. Sun, Compatibility on naturally ordered transformation semigroups preserving an equivalence, Semigroup Forum, 98 (2019), 75–82. http://dx.doi.org/10.1007/s00233-018-9965-y doi: 10.1007/s00233-018-9965-y
|
| [17] |
J. Symons, Some result concerning a transformation semigroup, J. Aust. Math. Soc., 19 (1975), 413–425. http://dx.doi.org/10.1017/S1446788700034455 doi: 10.1017/S1446788700034455
|
Kitsanachai Sripon, Ekkachai Laysirikul, Yanisa Chaiya. Regularity and abundance on semigroups of transformations preserving an equivalence relation on an invariant set[J]. AIMS Mathematics, 2023, 8(8): 18223-18233. doi: 10.3934/math.2023926
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