论文标题
部分Abelian扩展的交换反向半群
The commutative inverse semigroup of partial abelian extensions
论文作者
论文摘要
本文是对群体部分群体理论的新贡献。 First, given a unital partial action $α_G$ of a finite group $G$ on an algebra $S$ such that $S$ is an $α_G$-partial Galois extension of $S^{α_G}$ and a normal subgroup $H$ of $G$, we prove that $α_G$ induces a unital partial action $α_{G/H}$ of $G/H$在$ s $的$ s^{α_h} $的子级别上，$ s^{α_h} $是$α_{g/h h} $ - $ s^{α_g} $的部分galois扩展。其次，假设$ g $是Abelian，我们构建了交换性反向半群$ t_ {par}（g，r）$，其元素是$α_g$ - $ $α_g$ - 优势的abelian Abelian Extensions的等价类别。我们还证明，存在$ t_ {par}（g，r）/ρ$和$ t（g，a）$之间的组同构，其中$ρ$是$ t_ {par}（par}（g，r）$和$ t（g，r）$和$ t（g，a）$ a $ g $ g $ a $ a $ a $ a abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian abelian。结果表明，$ t_ {par}（g，r）$的研究减少到$ g $是循环的情况下。还研究了$ t_ {par}（g，r）$的一组dempotents。
This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action $α_G$ of a finite group $G$ on an algebra $S$ such that $S$ is an $α_G$-partial Galois extension of $S^{α_G}$ and a normal subgroup $H$ of $G$, we prove that $α_G$ induces a unital partial action $α_{G/H}$ of $G/H$ on the subalgebra of invariants $S^{α_H}$ of $S$ such that $S^{α_H}$ is an $α_{G/H}$-partial Galois extension of $S^{α_G}$. Second, assuming that $G$ is abelian, we construct a commutative inverse semigroup $T_{par}(G,R)$, whose elements are equivalence classes of $α_G$-partial abelian extensions of a commutative algebra $R$. We also prove that there exists a group isomorphism between $T_{par}(G,R)/ρ$ and $T(G,A)$, where $ρ$ is a congruence on $T_{par}(G,R)$ and $T(G,A)$ is the classical Harrison group of the $G$-isomorphism classes of the abelian extensions of a commutative ring $A$. It is shown that the study of $T_{par}(G,R)$ reduces to the case where $G$ is cyclic. The set of idempotents of $T_{par}(G,R)$ is also investigated.
