斜多项式环、Groebner基与自由结合代数的字母位嵌入
Skew polynomial rings, Groebner bases and the letterplace embedding of the free associative algebra
摘要：
本文从有限或可数集$X$生成的自由结合代数$K< X >$引入一个代数$ι:K< X >\to S$，它嵌入到由交换多项式环$P = K[X\times N^*]$和由适当的自同态$σ:P\to P$生成的幺半群$Σ= < σ>$定义的斜幺半群环$S = P * Σ$中.若$P = K[X]$是可数交换变量集上的多项式环，本文还给出了分次代数$S = \bigoplus_i S_i$的分次双边理想的一般Gröbner基理论，其中$S_i = P σ^i$和$σ:P \to P$是满足与$P$的单项式的序和整除相容条件的抽象自同态.此外，使用一个合适的评分的代数$P$兼容的作用$Σ$，我们得到了一个双射对应，保持Gröbner基地之间的分次$Σ$不变理想的$P$和一类分次双边理想的$S$。通过嵌入$ι$这导致统一，在分次的情况下，交换和非交换多项式环的Gröbner基理论。最后，由于环的普通差分多项式$P = K[X\times N]$适合所提出的理论，人们得到，对于一个适当的分级，Gröbner生成的分次普通差分理想的基也可以计算在运营商环$S$和在有限的步骤，直到一些固定的程度。
英文摘要：
In this paper we introduce an algebra embedding $ι:K< X >\to S$ from the free associative algebra $K< X >$ generated by a finite or countable set $X$ into the skew monoid ring $S = P * Σ$ defined by the commutative polynomial ring $P = K[X\times N^*]$ and by the monoid $Σ= < σ>$ generated by a suitable endomorphism $σ:P\to P$. If $P = K[X]$ is any ring of polynomials in a countable set of commuting variables, we present also a general Gröbner bases theory for graded two-sided ideals of the graded algebra $S = \bigoplus_i S_i$ with $S_i = P σ^i$ and $σ:P \to P$ an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of $P$. Moreover, using a suitable grading for the algebra $P$ compatible with the action of $Σ$, we obtain a bijective correspondence, preserving Gröbner bases, between graded $Σ$-invariant ideals of $P$ and a class of graded two-sided ideals of $S$. By means of the embedding $ι$ this results in the unification, in the graded case, of the Gröbner bases theories for commutative and non-commutative polynomial rings. Finally, since the ring of ordinary difference polynomials $P = K[X\times N]$ fits the proposed theory one obtains that, with respect to a suitable grading, the Gröbner bases of finitely generated graded ordinary difference ideals can be computed also in the operators ring $S$ and in a finite number of steps up to some fixed degree.
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