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On the cancellation rule in the homogenization. (English) Zbl 1166.13034
Gröbner-Shirshov bases theory were initiated independently by A.I. Shirshov in 1962. L.A. Bokut 1976 gave an approach to construction of minimal Gröbner-Shirshov basis of any associative algebra (see, for example, [L. A. Bokut and Y. Chen, Southeast Asian Bull. Math. 31, No. 6, 1057–1076 (2007; Zbl 1150.17008)]). This paper presents Gröbner-Shirshov bases for associative algebras by another approach. Let $$K\langle X\rangle$$ be the free associative algebra generated by $$X$$ over a field $$K$$, $$t$$ an additional homogenizing variable. Let $$<$$ be the deg-lex ordering on the free monoid $$\langle X\rangle$$ generated by $$X$$ and extend it to the eliminating ordering on the free monoid $$\langle X,t \rangle$$ which is a monomial ordering on $$\langle X,t \rangle$$. For a subset $$G$$ of $$K\langle X\rangle$$ and $$u=\sum u_i \in G$$ where $$u_i$$ is the homogeneous components with deg$$u_i=i,\;1\leq i\leq k$$, let $$u^*=\sum u_it^{k-i}$$. Then $$u^*$$ is an homogeneous element in $$K\langle X, t\rangle$$. Let $$G^*=\{u^*|u\in G\}\cup \{ tx-xt|x\in X\}$$. Then, $$G^*$$ is a minimal Gröbner-Shirshov basis in $$K\langle X, t\rangle$$ if and only if $$G$$ is a minimal Gröbner-Shirshov basis in $$K\langle X\rangle$$.

MSC:
 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 13N99 Differential algebra 16Z05 Computational aspects of associative rings (general theory)
Keywords:
Gröbner basis; associative algebra