關于半環上格林關系的開同余

秦官偉,任苗苗,邵勇

(西北大學數學系,陜西 西安 710127)

關于半環上格林關系的開同余

秦官偉,任苗苗,邵勇

(西北大學數學系,陜西 西安 710127)

首先給出了由半環的乘法半群上的格林關系所確定的半環開同余的性質和刻畫.其次,由開同余出發,得到了六個不同的半環類,并證明了這六個半環類均是半環簇.最后,對半環簇的子簇格上的開算子進行了探討,得到了一些有趣的結果.

半環;格林關系;開同余;半環簇;開算子

1 引言及預備知識

許多學者研究了半環上的格林關系[3-7].在文獻[3]中,作者通過格林關系的開同余代替格林關系本身,研究了半環簇和加法冪等元半環簇格上的開算子.特別地,建立了加法冪等元半環簇的子簇格與此格上關于兩開(閉)算子的開(閉)簇的格的直積之間序嵌入.本文運用開同余,研究了半環上乘法格林關系的開同余,半環簇和乘法冪等元半環簇的子簇格上的開算子,給出了若干關于格林關系的半環簇之間的最小上界和最大下界,以及乘法冪等元半環的一個次直積分解.最后,利用開同余,構造了半環簇的子簇格上的一類開算子.

2 主要內容

致謝作者對導師趙憲鐘教授的指導表示衷心感謝！

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

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[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.

陜西省自然科學基金(2011JQ1017);西北大學科學研究基金(NC0925).

秦官偉(1987-),碩士生,研究方向：代數學.

2010 MSC:16D10