Subalgebra bases and recognizable properties.(English)Zbl 1017.16042

The paper under review presents some properties of subalgebras with finite standard basis, or SAGBI basis (Subalgebra Analogue to Gröbner Bases for Ideals). The paper considers $$K$$-algebras of the form $$K\langle X\rangle/I$$, where $$I$$ is a monomial ideal and $$X$$ a finite set of indeterminates. Also, the left length-lex order is fixed on $$X$$.
The paper introduces the concepts of SAGBI basis, essential product, H-representation, s-element, reduction and of polynomial relation for a finite set of elements of $$A$$. Also the notions of p-relation for an inessential product and of s-relation for an s-element are given. The definitions of some of these notions can be found elsewhere.
The following results are stated and proved.
Theorem 1. Let $$A$$ be an algebra of the form $$K\langle X\rangle/I$$, where $$I$$ is a monomial ideal and $$X$$ a finite set of indeterminates. Let $$B\subset A$$ be a subalgebra and $$G$$ an essencial set of generators of $$B$$. Then the following are equivalent: (i) $$G$$ is a standard basis. (ii) Any element $$b\in B$$ is reducible to zero. (iii) Any element $$b\in B$$ has an H-representation. (iv) Any s-element has a representation (via $$G$$) with a parameter less than its initial parameter. (v) $$(A,\leq, R_G)$$ is a linear scheme of simplification with the canonization property.
The reader has to look in the references for the definition of canonization property.
Theorem 2. Any polynomial relation among generators $$g_1,\dots,g_N$$ is a linear combination of p-, s-relations.
Theorem 3. Let $$G=\{g_1,g_2,\dots,g_N\}$$ be a SAGBI-basis of the subalgebra $$B$$ of the monomial algebra $$A= K\langle X\rangle/I$$. Then it is a recognizable property that $$G$$ generates a free subalgebra $$B$$.
Theorem 3. Let $$G=\{g_1,g_2,\dots,g_N\}$$ be a SAGBI-basis of the subalgebra $$B$$ of the monomial algebra $$A=K\langle X\rangle/I$$. Then it is a recognizable property that $$A$$ is finite dimensional.

MSC:

 16Z05 Computational aspects of associative rings (general theory) 68W30 Symbolic computation and algebraic computation 16G20 Representations of quivers and partially ordered sets
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