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平衡范畴与半群的双序
引用本文:喻秉钧.平衡范畴与半群的双序[J].数学学报,2012(2):321-340.
作者姓名:喻秉钧
作者单位:四川师范大学数学与软件科学学院;四川师范大学文理学院
基金项目:国家自然科学基金资助项目(10871161)
摘    要:研究范畴与半群通过幂等元双序建立的一种自然联系.对每个有幂等元的半群S,其幂等元生成的左、右主理想之集通过双序ω~e,ω~r自然确定两个有子对象、有像且每个包含都右可裂的范畴L(S),R(S),其中态射的性质与S中元素的富足性、正则性有自然对应.利用这个联系,我们定义了"平衡(富足、正规)范畴"概念.对任一平衡(富足、正规)范畴■,我们构造其"锥半群"■,证明■左富足(富足、正则),且每个平衡(富足、正规)范畴■都与某左富足(富足、正则)半群S的左主理想范畴L(S)(作为有子对象的范畴)同构.

关 键 词:平衡(富足  正规)范畴  半群的双序  子对象选择

Balanced Categories and the Biorder in Semigroups
Bing Jun YU.Balanced Categories and the Biorder in Semigroups[J].Acta Mathematica Sinica,2012(2):321-340.
Authors:Bing Jun YU
Institution:Bing Jun YU College of Mathematics and Software Sciences,Sichuan Normal University, Chengdu 610066,P.R.China College of Arts and Sciences,Sichuan Normal University,Chengdu 610101,P.R.China
Abstract:In this paper a natural relationship between categories and semigroups established by the biorder of idempotents in semigroups is investigated.For every semigroup S with idempotents,via the biorderω~e,ω~r,the sets of left and right principal ideals generated by idempotents naturally determine two categories L(S),R(S) with subobjects and images in which every inclusion is right split.The properties of morphisms in these two categories naturally correspond to the abundance or regularity of elements in S.By applying this relationship,the concepts of "balanced,abundant or normal categories" are defined.For each balanced(abundant or normal) category ■,the "cone semigroup"■ of■ is constructed.It is proved that■ is a left abundant (abundant or regular) semigroup,a left abundant(abundant or regular) semigroup (monoid) S is isomorphic with a subsemigroup of■L(S)(with■L(S)),every bal- anced(abundant or normal) category■ is isomorphic with the ideal category L(S) of some left abundant(abundant or regular) semigroup S,as categories with subobjects.
Keywords:balanced(abundant  normal) category  the biorder of a semigroup  choice of subobjects
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