一类变换半群的左相容元
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摘要
设TX是非空集合X上全变换半群,E是X上非平凡的等价关系,R是X/E的横断面,则TE(X,R)={f∈TX:x,y∈X,(x,y)∈E(f(x),f(y))∈E且f(R)R}是TX的子半群.本文赋予半群TE(X,R)自然偏序关系,通过构造映射的方法,刻画它的左相容元,给出充要条件.
Let TXbe the full transformation semigroup on a nonempty set X,Ebe a nontrivial equivalence relation on Xand Rbe a cross-section of X/E,thenTE(X,R)={f∈TX :x,y∈X,(x,y)∈E(f(x),f(y))∈Eand f(R)R}is a subsemigroup of TX.By constructing suitable mappings,all the left compatible elements were described in the transformation semigroup TE(X,R)endowed with the natural partial order and a necessary and sufficient condition for these elements was given.
Let TXbe the full transformation semigroup on a nonempty set X,Ebe a nontrivial equivalence relation on Xand Rbe a cross-section of X/E,thenTE(X,R)={f∈TX :x,y∈X,(x,y)∈E(f(x),f(y))∈Eand f(R)R}is a subsemigroup of TX.By constructing suitable mappings,all the left compatible elements were described in the transformation semigroup TE(X,R)endowed with the natural partial order and a necessary and sufficient condition for these elements was given.
引文
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[2]Sun L,Pei H S,Cheng Z X.Naturally ordered transformation semigroups preserving an equivalence[J].Bulletin of the Australian Mathematical Society,2008,78(2):117-128.
[3]Singha B,Sanwong J,Sullivan R P.Partial order on partial Baer-Levi semigroups[J].Bulletin of the Australian Mathematical Society,2010,81(2):197-206.
[4]Pei H S,Deng W N.Naturally ordered semigroups of partial transformations preserving an equivalence relation[J].Communications in Algebra,2013,41(9):3308-3324.
[5]Sun L,Wang L M.Natural partial order in semigroups of transformations with invariant set[J].Bulletin of the Australian Mathematical Society,2013,87(1):94-107.
[6]Araujo J,Konieczny J.Semigroups of transformations preserving an equivalence relation and a cross-section[J].Communications in Algebra,2004,32(5):1917-1935.
[7]Sun L,Deng W N,Pei H S.Naturally ordered transformation semigroups preserving an equivalence and a cross-section[J].Algebra Colloquium,2011,18(3):523-532.
[2]Sun L,Pei H S,Cheng Z X.Naturally ordered transformation semigroups preserving an equivalence[J].Bulletin of the Australian Mathematical Society,2008,78(2):117-128.
[3]Singha B,Sanwong J,Sullivan R P.Partial order on partial Baer-Levi semigroups[J].Bulletin of the Australian Mathematical Society,2010,81(2):197-206.
[4]Pei H S,Deng W N.Naturally ordered semigroups of partial transformations preserving an equivalence relation[J].Communications in Algebra,2013,41(9):3308-3324.
[5]Sun L,Wang L M.Natural partial order in semigroups of transformations with invariant set[J].Bulletin of the Australian Mathematical Society,2013,87(1):94-107.
[6]Araujo J,Konieczny J.Semigroups of transformations preserving an equivalence relation and a cross-section[J].Communications in Algebra,2004,32(5):1917-1935.
[7]Sun L,Deng W N,Pei H S.Naturally ordered transformation semigroups preserving an equivalence and a cross-section[J].Algebra Colloquium,2011,18(3):523-532.