具有唯一左单位的二元关系半群P_Γ(Λ×Λ)
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摘要
设Λ是任意的非空集合,Γ是集合Λ上的半格,PΓ(Λ×Λ)是集合Λ上的半格Γ确定的二元关系半群.利用半格的性质,讨论了半群PΓ(Λ×Λ)的最大左单位;通过左单位的构造方法,获得了半群PΓ(Λ×Λ)的左单位是唯一左单位的充分必要条件.
Let Λ be an arbitrary nonempty set and Γ be a semilattice on the set Λ,and then PΓ( Λ×Λ) is a semigroup of binary relations determined by the semilattice Γ on the set Λ. The greatest left unit in the semigroup PΓ( Λ×Λ) is discussed with reference to the properties of semilattice. The necessary and sufficient condition for the greatest left unit of the semigroup PΓ( Λ×Λ) to be the unique left unit is obtained according to the construction of the left unit.
Let Λ be an arbitrary nonempty set and Γ be a semilattice on the set Λ,and then PΓ( Λ×Λ) is a semigroup of binary relations determined by the semilattice Γ on the set Λ. The greatest left unit in the semigroup PΓ( Λ×Λ) is discussed with reference to the properties of semilattice. The necessary and sufficient condition for the greatest left unit of the semigroup PΓ( Λ×Λ) to be the unique left unit is obtained according to the construction of the left unit.
引文
[1] PLEMMONS R J,WEST M T. On the semigroup of binary relations[J]. Pacific Journal of Mathematics,1970,35:743-753.
[2] PLEMMONS R J,SCHEIN B M. Groups of binary relations[J]. Semigroup Forum,1970,1:267-271.
[3] SCHWARZ S. On idempotent binary relations on a finite set[J]. Czechoslovak Mathemtical Journal,1970,20:703-714.
[4] SCHWARZ S. On the semigroup of binary relations on a finite set[J]. Czechoslovak Mathemtical Journal,1970,20:632-679.
[5] KONIECZNY J. The semigroup generated by regular Boolean matrices[J]. Southeast Asian Bulletin of Mathematics,2002,25:627-641.
[6] MCKENZIE R N,SCHEIN B M. Every semigroups is isomorphic to a transitive semigroup of binary relations[J].Transactions of the American Mathematical Society,1997,349:271-285.
[7] CHASE K. New semigroups of binary relations[J]. Semigroup Forum,1979,18:79-82.
[8] CHASE K. Sandwish semigroups of binary relations[J].Discrete Mathematics,1979,28:231-236.
[9]林屏峰.集合Λ上的半格Γ确定的二元关系半群PΓ(Λ×Λ)的基本性质[J].西南民族大学学报(自然科学版),2012,38(4):529-532.LIN P F. Some properties of semigroup PΓ(Λ×Λ)of binary relations determined by the semilatticeΓon the setΛ[J]. Journal of Southwest University for Nationlities(Natural Science),2012,38(4):529-532.
[10] HOWIE J M. Products of idempotents in certain semigroups of transformations[J]. Proceedings of the Edinburgh Mathematical Society,1971,17(2):223-236.
[11] HIGGINS P M,HOWIE J M,RUSKUC N. Generators and factorizations of transformation semigroups[J]. Proceedings of the Royal Society of Edunburgh:Section A,1998,128:1355-1369.
[12]赵平,林屏峰.半群SPK-(n,r)的秩[J].华南师范大学学报(自然科学版),2012,44(4):37-39.ZHAO P,LIN P F. The rank of the semigroup SPK-(n,r)[J]. Jouranl of South China Normal University(Natural Science Edition),2012,44(4):37-39.
[13]林屏峰.集合Λ上的半格Γ确定的二元关系半群PΓ(Λ×Λ)的Green-关系[J].西南民族大学学报(自然科学版),2013,39(1):26-29.LIN P F. Green’s relations of semigroup PΓ(Λ×Λ)of binary relations determined by the semilatticeΓon the setΛ[J]. Journal of Southwest University for Nationlities(Natural Science),2013,39(1):26-29.
[14]林屏峰,曾伟,曾纯一.集合Λ上的半格Γ确定的二元关系半群PΓ(Λ×Λ)的幂等元[J].山东大学学报(理学版),2013,48(8):36-40.LIN P F,ZENG W,ZENG C Y. Idempotents of semigroup PΓ(Λ×Λ)of binary relationa determined by the semilatticeΓon the setΛ[J]. Journal of Shandong University(Natural Science),2013,48(8):36-40.
[15]林屏峰.集合Λ上的半格Γ确定的二元关系半群PΓ(Λ×Λ)的不可分解元[J].山东大学学报(理学版),2016,51(2):12-15.LIN P F. Non-solvable elements of semigroup PΓ(Λ×Λ)of binary relationa determined by the semilatticeΓon the setΛ[J]. Journal of Shandong University(Natural Science),2016,51(2):12-15.
[16]林屏峰.二元关系半群PΓ(Λ×Λ)的一类左单位的构造[J].西南民族大学学报(自然科学版),2016,42(1):96-98.LIN P F. Construction of a class of left unit of semigroup PΓ(Λ×Λ)[J]. Journal of Southwest University for Nationlities(Natural Science),2016,42(1):96-98.
[17] BIRKHOFF G. Lattice theory[M]. 3rd ed. Providence,Rhode Islan:American Mathematical Society,1976.
[2] PLEMMONS R J,SCHEIN B M. Groups of binary relations[J]. Semigroup Forum,1970,1:267-271.
[3] SCHWARZ S. On idempotent binary relations on a finite set[J]. Czechoslovak Mathemtical Journal,1970,20:703-714.
[4] SCHWARZ S. On the semigroup of binary relations on a finite set[J]. Czechoslovak Mathemtical Journal,1970,20:632-679.
[5] KONIECZNY J. The semigroup generated by regular Boolean matrices[J]. Southeast Asian Bulletin of Mathematics,2002,25:627-641.
[6] MCKENZIE R N,SCHEIN B M. Every semigroups is isomorphic to a transitive semigroup of binary relations[J].Transactions of the American Mathematical Society,1997,349:271-285.
[7] CHASE K. New semigroups of binary relations[J]. Semigroup Forum,1979,18:79-82.
[8] CHASE K. Sandwish semigroups of binary relations[J].Discrete Mathematics,1979,28:231-236.
[9]林屏峰.集合Λ上的半格Γ确定的二元关系半群PΓ(Λ×Λ)的基本性质[J].西南民族大学学报(自然科学版),2012,38(4):529-532.LIN P F. Some properties of semigroup PΓ(Λ×Λ)of binary relations determined by the semilatticeΓon the setΛ[J]. Journal of Southwest University for Nationlities(Natural Science),2012,38(4):529-532.
[10] HOWIE J M. Products of idempotents in certain semigroups of transformations[J]. Proceedings of the Edinburgh Mathematical Society,1971,17(2):223-236.
[11] HIGGINS P M,HOWIE J M,RUSKUC N. Generators and factorizations of transformation semigroups[J]. Proceedings of the Royal Society of Edunburgh:Section A,1998,128:1355-1369.
[12]赵平,林屏峰.半群SPK-(n,r)的秩[J].华南师范大学学报(自然科学版),2012,44(4):37-39.ZHAO P,LIN P F. The rank of the semigroup SPK-(n,r)[J]. Jouranl of South China Normal University(Natural Science Edition),2012,44(4):37-39.
[13]林屏峰.集合Λ上的半格Γ确定的二元关系半群PΓ(Λ×Λ)的Green-关系[J].西南民族大学学报(自然科学版),2013,39(1):26-29.LIN P F. Green’s relations of semigroup PΓ(Λ×Λ)of binary relations determined by the semilatticeΓon the setΛ[J]. Journal of Southwest University for Nationlities(Natural Science),2013,39(1):26-29.
[14]林屏峰,曾伟,曾纯一.集合Λ上的半格Γ确定的二元关系半群PΓ(Λ×Λ)的幂等元[J].山东大学学报(理学版),2013,48(8):36-40.LIN P F,ZENG W,ZENG C Y. Idempotents of semigroup PΓ(Λ×Λ)of binary relationa determined by the semilatticeΓon the setΛ[J]. Journal of Shandong University(Natural Science),2013,48(8):36-40.
[15]林屏峰.集合Λ上的半格Γ确定的二元关系半群PΓ(Λ×Λ)的不可分解元[J].山东大学学报(理学版),2016,51(2):12-15.LIN P F. Non-solvable elements of semigroup PΓ(Λ×Λ)of binary relationa determined by the semilatticeΓon the setΛ[J]. Journal of Shandong University(Natural Science),2016,51(2):12-15.
[16]林屏峰.二元关系半群PΓ(Λ×Λ)的一类左单位的构造[J].西南民族大学学报(自然科学版),2016,42(1):96-98.LIN P F. Construction of a class of left unit of semigroup PΓ(Λ×Λ)[J]. Journal of Southwest University for Nationlities(Natural Science),2016,42(1):96-98.
[17] BIRKHOFF G. Lattice theory[M]. 3rd ed. Providence,Rhode Islan:American Mathematical Society,1976.