Regularity of transformation semigroups defined by a partition
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Let $X$ be a nonempty set,and let $\mathscr{F}=\{Y_{i}:i\in I\}$ be a family of nonempty subsets of $X$ with the properties that $X=\bigcup_{i\in I}{}Y_{i}$, and $Y_{i}\cap Y_{j}=\emptyset$ for all $i,j\in I$ with $i\neq j$. Let $\emptyset\neq J\subseteq I$, and let $T_{\mathscr{F}}^{(J)}(X)=\left\{\alpha\in T(X):\forall i\in I\exists j\in J, Y_{i}\alpha\subseteq Y_{j}\right\}$. Then $T_{\mathscr{F}}^{(J)}(X)$ is a subsemigroup of the semigroup $T\left(X,Y^{(J)}\right)$ of functions on $X$ having ranges contained in $Y^{(J)}$, where $Y^{(J)}:=\bigcup_{i\in J}{}Y_{i}$. For each $\alpha\in T_{\mathscr{F}}^{(J)}(X)$, let $\chi^{(\alpha)}:I\rightarrow J$ be defined by $i\chi^{(\alpha)}=j\Leftrightarrow Y_{i}\alpha\subseteq Y_{j}$. Next, we define two congruence relations $\chi$ and $\widetilde{\chi}$ on $T_{\mathscr{F}}^{(J)}(X)$ as follows: $(\alpha,\beta)\in\chi\Leftrightarrow \chi^{(\alpha)}=\chi^{(\beta)}$ and $(\alpha,\beta)\in\widetilde{\chi}\Leftrightarrow \chi^{(\alpha)}|_{J}=\chi^{(\beta)}|_{J}$. We begin this paper by studying the regularity of the quotient semigroups $T_{\mathscr{F}}^{(J)}(X)/\chi$ and $T_{\mathscr{F}}^{(J)}(X)/\widetilde{\chi}$, and the semigroup $T_{\mathscr{F}}^{(J)}(X)$. For each $\alpha\in T_{\mathscr{F}}(X):=T_{\mathscr{F}}^{(I)}(X)$, we see that the equivalence class $[\alpha]$ of $\alpha$ under $\chi$ is a subsemigroup of $T_{\mathscr{F}}(X)$ if and only if $\chi^{(\alpha)}$ is an idempotent element in the full transformation semigroup $T(I)$. Let $I_{\mathscr{F}}(X)$, $S_{\mathscr{F}}(X)$ and $B_{\mathscr{F}}(X)$ be the sets of functions $\alpha$ in $T_{\mathscr{F}}(X)$ such that $\chi^{(\alpha)}$ is injective, surjective and bijective respectively. We end this paper by investigating the regularity of the subsemigroups $[\alpha]$, $I_{\mathscr{F}}(X)$, $S_{\mathscr{F}}(X)$ and $B_{\mathscr{F}}(X)$ of $T_{\mathscr{F}}(X)$.
Abstract
Let $X$ be a nonempty set,and let $\mathscr{F}=\{Y_{i}:i\in I\}$ be a family of nonempty subsets of $X$ with the properties that $X=\bigcup_{i\in I}{}Y_{i}$, and $Y_{i}\cap Y_{j}=\emptyset$ for all $i,j\in I$ with $i\neq j$. Let $\emptyset\neq J\subseteq I$, and let $T_{\mathscr{F}}^{(J)}(X)=\left\{\alpha\in T(X):\forall i\in I\exists j\in J, Y_{i}\alpha\subseteq Y_{j}\right\}$. Then $T_{\mathscr{F}}^{(J)}(X)$ is a subsemigroup of the semigroup $T\left(X,Y^{(J)}\right)$ of functions on $X$ having ranges contained in $Y^{(J)}$, where $Y^{(J)}:=\bigcup_{i\in J}{}Y_{i}$. For each $\alpha\in T_{\mathscr{F}}^{(J)}(X)$, let $\chi^{(\alpha)}:I\rightarrow J$ be defined by $i\chi^{(\alpha)}=j\Leftrightarrow Y_{i}\alpha\subseteq Y_{j}$. Next, we define two congruence relations $\chi$ and $\widetilde{\chi}$ on $T_{\mathscr{F}}^{(J)}(X)$ as follows: $(\alpha,\beta)\in\chi\Leftrightarrow \chi^{(\alpha)}=\chi^{(\beta)}$ and $(\alpha,\beta)\in\widetilde{\chi}\Leftrightarrow \chi^{(\alpha)}|_{J}=\chi^{(\beta)}|_{J}$. We begin this paper by studying the regularity of the quotient semigroups $T_{\mathscr{F}}^{(J)}(X)/\chi$ and $T_{\mathscr{F}}^{(J)}(X)/\widetilde{\chi}$, and the semigroup $T_{\mathscr{F}}^{(J)}(X)$. For each $\alpha\in T_{\mathscr{F}}(X):=T_{\mathscr{F}}^{(I)}(X)$, we see that the equivalence class $[\alpha]$ of $\alpha$ under $\chi$ is a subsemigroup of $T_{\mathscr{F}}(X)$ if and only if $\chi^{(\alpha)}$ is an idempotent element in the full transformation semigroup $T(I)$. Let $I_{\mathscr{F}}(X)$, $S_{\mathscr{F}}(X)$ and $B_{\mathscr{F}}(X)$ be the sets of functions $\alpha$ in $T_{\mathscr{F}}(X)$ such that $\chi^{(\alpha)}$ is injective, surjective and bijective respectively. We end this paper by investigating the regularity of the subsemigroups $[\alpha]$, $I_{\mathscr{F}}(X)$, $S_{\mathscr{F}}(X)$ and $B_{\mathscr{F}}(X)$ of $T_{\mathscr{F}}(X)$.
참고문헌 (18)
- R. Chinram, "Green’s relations and regularity of generalized semigroups of linear trans-formations", Lobachevskii J. Math, 30(4), pp.253~256 (2009)
- A. H. Clifford, "The Algebraic Theory of Semigroups. Vol. I, Mathe-matical Surveys, No. 7" (1961), American Mathematical Society
- A. H. Clifford, "The Algebraic Theory of Semigroups. Vol. II, Mathematical Surveys, No. 7" (1967), American Mathematical Society
- L. -Z. Deng, "Green’s relations and regularity for semigroups of transformations that preserve double direction equivalence", Semigroup Forum, 80(3), pp.416~425 (2010)
- Preeyanuch Honyam, "SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET", 대한수학회지, 48(2), pp.289~300 (2011), 대한수학회
- P. Honyam, "Semigroups of linear transformations with invariant subspaces", Int. J. Algebra, 6(8), pp.375~386 (2012)
- Y. Kemprasit, "Regular order-preserving transformation semigroups", Bull. Austral. Math. Soc, 62(3), pp.511~524 (2000)
- S. Lei, "Green’s relations on semigroups of transformations preserving two equivalence relations", J. Math. Res. Exposition, 29(3), pp.415~422 (2009)
- W. Mora, "Regular elements of some order-preserving transformation semigroups", Int. J. Algebra, 4(13), pp.631~641 (2010)
- S. Nenthein, "Regular elements of some transforma-tion semigroups", Pure Math. Appl, 16(3), pp.307~314 (2005)
- H. Pei, "Equivalences, α-semigroups and α-congruences", Semigroup Forum, 49(1), pp.49~58 (1994)
- H. Pei, "Regularity and Green’s relations for semigroups of transformations that pre-serve an equivalence", Commun. Algebra, 33(1), pp.109~118 (2005)
- H. Pei, "A note on semigroups of linear transformations with invariant subspaces", Int. J. Algebra, 6(27), pp.1319~1324 (2012)
- H. Pei, "Green’s equivalences on semigroups of transformations preserving order and an equivalence relation", Semigroup Forum, 71(2), pp.241~251 (2005)
- J. Sanwong, "The regular part of a semigroup of transformations with restricted range", Semigroup Forum, 83(1), pp.134~146 (2011)
- J. Sanwong, "Regularity and Green’s relations on a semigroup of transformations with restricted range", Int. J. Math. Math. Sci, 2008, pp.11~ (2008)
- R. P. Sullivan, "Semigroups of linear transformations with restricted range", Bull. Austral. Math. Soc, 77(3), pp.441~453 (2008)
- Ping Zhao, "REGULARITY AND GREEN'S RELATIONS ON SEMIGROUPS OF TRANSFORMATION PRESERVING ORDER AND COMPRESSION", 대한수학회보, 49(5), pp.1015~1025 (2012), 대한수학회