一类有限半格的自同态半环
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摘要
研究两条有限链直积上自同态半环的性质。利用有限链直积上的两种二元运算,给出了两条有限链直积的子集构成自同态像集的充要条件,证明了自同态半环的乘法半群是正则半群。通过对有限链直积上的自同态进行分解,得到了自同态半环可由其乘法半群的幂等元集生成;推广了有限链上自同态半群的一些结果。
This paper studies the properties of the endomorphism semiring over the direct product of two finite chains.Using two binary operations on the direct product of two finite chains, a sufficient and necessary condition under which the subset of the direct product of two finite chains is the codomain of the endomorphism is given. Next, it proves that the multiplicative reduct of the endomorphism semiring is a regular semigroup. Finally, by decomposing the endomorphism of the direct product of two finite chains, it obtains that the endomorphism semiring can be generated by the idempotents of its multiplicative reduct. Some known results about the endomorphism semigroup over a finite chain are expanded.
This paper studies the properties of the endomorphism semiring over the direct product of two finite chains.Using two binary operations on the direct product of two finite chains, a sufficient and necessary condition under which the subset of the direct product of two finite chains is the codomain of the endomorphism is given. Next, it proves that the multiplicative reduct of the endomorphism semiring is a regular semigroup. Finally, by decomposing the endomorphism of the direct product of two finite chains, it obtains that the endomorphism semiring can be generated by the idempotents of its multiplicative reduct. Some known results about the endomorphism semigroup over a finite chain are expanded.
引文
[1]盛德成.抽象代数[M].北京:科学出版社,2003.
[2] Howie J M.An introduction to semigroup theory[M].London:Academic Press,1976.
[3] Azienstat A J.The defining relations of the endomorphism semigroup of a finite linearly or dered set[J].Sibirsk Mat,1962,3(2):161-169.
[4] Gomes G,Howie J M.On the ranks of certain semigroups of order-preserving transformations[J].Semigroup Forum,1992,45(1):272-282.
[5] Yang X.Products of idempotents of defect 1 in certain semigroups of transformations[J].Communication in Algebra,1999,27(7):3557-3568.
[6] Howie J M.Products of idempotents in certain semigroups of transformations[J].Proceedings of the Edinburgh Mathematical Society,1971,17(3):223-236.
[7] Howie J M.The subsemigroup generated by the idempotents of a full transformation semigroup[J].Journal of the London Mathematical Society,1966,41(1):707-716.
[8] Fernandes V H,Jesus M M,Maltcev V,et al.Endomorphisms of the semigroup of order preserving mappings[J].Semigroup Forum,2010,81(2):277-285.
[9] Fernandes V H.The monoid of all injective order preserving partial transformations on a finite chain[J].Semigroup Forum,2001,62(2):178-204.
[10] Jezek J,Kepka T,Maroti M.The endomorphism semring of a semilattice[J].Semigroup Forum,2009,78(1):21-26.
[11] Jezek J,Kepka T.The semiring of 1-preserving endomorphisms of a semilattice[J].Czechoslovak Mathematical Journal,2009,59(4):999-1003.
[12] Monico C.On finite congruence-simple semiring[J].Journal of Algebra,2004,271(2):846-854.
[13] Trendafilov I,Vladeva D.Idempotent elements of the endomorphism semiring of a finite chain[J].C R Acad Bulgare Sci,2013,66(5):621-628.
[14] Zumbragel J.Classification of finite congruence simple semirings with zero[J].Journal of Algebra and Its Applications,2008,7(3):363-377.
[15] Bashir R E,Hurt J,Jancari K A,et al.Simple commutative semirings[J].Journal of Algebra,2001,236(1):277-306.
[2] Howie J M.An introduction to semigroup theory[M].London:Academic Press,1976.
[3] Azienstat A J.The defining relations of the endomorphism semigroup of a finite linearly or dered set[J].Sibirsk Mat,1962,3(2):161-169.
[4] Gomes G,Howie J M.On the ranks of certain semigroups of order-preserving transformations[J].Semigroup Forum,1992,45(1):272-282.
[5] Yang X.Products of idempotents of defect 1 in certain semigroups of transformations[J].Communication in Algebra,1999,27(7):3557-3568.
[6] Howie J M.Products of idempotents in certain semigroups of transformations[J].Proceedings of the Edinburgh Mathematical Society,1971,17(3):223-236.
[7] Howie J M.The subsemigroup generated by the idempotents of a full transformation semigroup[J].Journal of the London Mathematical Society,1966,41(1):707-716.
[8] Fernandes V H,Jesus M M,Maltcev V,et al.Endomorphisms of the semigroup of order preserving mappings[J].Semigroup Forum,2010,81(2):277-285.
[9] Fernandes V H.The monoid of all injective order preserving partial transformations on a finite chain[J].Semigroup Forum,2001,62(2):178-204.
[10] Jezek J,Kepka T,Maroti M.The endomorphism semring of a semilattice[J].Semigroup Forum,2009,78(1):21-26.
[11] Jezek J,Kepka T.The semiring of 1-preserving endomorphisms of a semilattice[J].Czechoslovak Mathematical Journal,2009,59(4):999-1003.
[12] Monico C.On finite congruence-simple semiring[J].Journal of Algebra,2004,271(2):846-854.
[13] Trendafilov I,Vladeva D.Idempotent elements of the endomorphism semiring of a finite chain[J].C R Acad Bulgare Sci,2013,66(5):621-628.
[14] Zumbragel J.Classification of finite congruence simple semirings with zero[J].Journal of Algebra and Its Applications,2008,7(3):363-377.
[15] Bashir R E,Hurt J,Jancari K A,et al.Simple commutative semirings[J].Journal of Algebra,2001,236(1):277-306.