二元域上线性半群到任意域上线性半群的同态

摘要

群同态是群论研究的主要问题，F2上的半群同态也一直是群论研究者关注的热点问题。为探讨二阶线性半群间的同态问题，本文在引进标准型、延断型、平凡型概念的基础上，通过矩阵计算与群的定义关系，描述了 F2上的线性半群 Mn（F2）到任意域 K上的线性半群Mm（K）的同态形式（n≥ m＞2）。给出了 n＞ m ，乘法半群同态（不必保幺元）为 In，s，r或为φ1的延断（其中：若 X为GL2（F2）的2阶元，φ1（X）＝-1；若 X＝ I2或 X为3阶元，φ1（X）＝1）；当n＝ m时，乘法半群同态（不必保幺元）除 In，s，r外，为标准型或为线性非平凡解同态的延断。这些结果结合文献中已有的关于一般线性群及二阶线性半群的结果，完全描述了二元域上的线性半群到任意域上的线性半群的同态的形式。%Group homomorphism is a major problem in group theory research .The semigroup homomorphism is al-ways the hot point on the F2 in the group theory researchers .In order to study the homomorphism problem of two order linear semigroup ,based on the introduction of the standard type ,delay type and the ordinary type concept ,de-fined the group relationship through the matrix .Describe the homomorphism form between the linear semigroup Mn(F2)andonthearbitrarydomainMm(K)(n≥ m>2).Whenn> m,multiplicativesemigrouphomomorphismis theIn,s,ror φ1extendedfault,(iftheXisthesecondorderoftheGL2(F2),φ1(X)= -1;ifX= I2orXisthethird order ,φ1 (X)=1);If n= m ,the multiplicative semigroup homomorphism is the standard type or linear nontrivial so-lutions homomorphism extended fault except the In,s,r .Combining the results on the general linear group and the two order linear semigroup results in the literature ,these results completely describe the homomorphic form from two element field linear semigroup to arbitrary field linear semigroups .