关于半环上格林关系的开同余

秦官伟,任苗苗,邵勇

(西北大学数学系,陕西 西安 710127)

关于半环上格林关系的开同余

秦官伟,任苗苗,邵勇

(西北大学数学系,陕西 西安 710127)

首先给出了由半环的乘法半群上的格林关系所确定的半环开同余的性质和刻画.其次,由开同余出发,得到了六个不同的半环类,并证明了这六个半环类均是半环簇.最后,对半环簇的子簇格上的开算子进行了探讨,得到了一些有趣的结果.

半环;格林关系;开同余;半环簇;开算子

1 引言及预备知识

许多学者研究了半环上的格林关系[3-7].在文献[3]中,作者通过格林关系的开同余代替格林关系本身,研究了半环簇和加法幂等元半环簇格上的开算子.特别地,建立了加法幂等元半环簇的子簇格与此格上关于两开(闭)算子的开(闭)簇的格的直积之间序嵌入.本文运用开同余,研究了半环上乘法格林关系的开同余,半环簇和乘法幂等元半环簇的子簇格上的开算子,给出了若干关于格林关系的半环簇之间的最小上界和最大下界,以及乘法幂等元半环的一个次直积分解.最后,利用开同余,构造了半环簇的子簇格上的一类开算子.

2 主要内容

致谢作者对导师赵宪钟教授的指导表示衷心感谢！

[1]Jezˇek J,Kepka T,Mar´oti M.The endomorphism semiring of a semilattice[J].Semigroup Forum,2009,78:21-26.

[2]Howie J M.Fundamentals of Semigroup Theory[M].Oxford:Oxford Science Publication,l995.

[3]Damljanovi´c N,´Ciri´c M,Bogdanovi´c S.Congruence openings of additive Green′s relations on a semiring[J].Semigroup Forum,2011,82(3):437-454.

[4]Pastijin F,Zhao X Z.Green′s D-relation for the multiplicative reduct of an idempotent semiring[J].Arch. Math.,2000,36:77-93.

[5]Pastijin F,Zhao X Z.Varieties of semirings with commutative addition[J].Algebra Universalis,2005,54:301-321.

[6]Kehayopulu N.Ideal and Green′s relations in ordered semigroup[J].International Journal of Mathematics and Mathematical Sciences,2006(1):1-8.

[7]Hall T E.Congruences and Green′s relations on regular semigroup[J].Journal of Algebra,1973,24:1-24.

[8]Burris S,Sankppanavar H P.A Course in Universal Algebra[M].New York:Springer Verlag,1981.

[9]Sen M K,Guo Y Q,Shum K P.A class of idempotent semirings[J].Semigroup Forum,2000,60:351-367.

On congruence openings of Green′s relations on a semiring

Qin Guanwei,Ren Miaomiao,Shao Yong
(Department of Mathematics,Northwest University,Xi′an 710127,China)

In this paper we fi rstly give properties and characterizations of congruence openings of a semiring that is determined by Green’s relations of multiplicative semigroup of a semiring.Secondly,we obtain six classes of semirings by means of congruence openings,and prove that these classes of semirins are all varieties of semirings.Finally,we investigate open operators on the lattice of all subvarieties of the variety of semirings and obtain some interesting results.

semiring,Green′s relation,congruence opening,variety of semirings,open operator

O153.3

A

1008-5513(2012)05-0668-08

2011-10-19.

陕西省自然科学基金(2011JQ1017);西北大学科学研究基金(NC0925).

秦官伟(1987-),硕士生,研究方向：代数学.

2010 MSC:16D10