zbMATH — the first resource for mathematics

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 \(K\langle X\rangle\).

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)