Finiteness and Noetherianity of Gröbner–Shirshov bases for commutative dialgebras beyond finite generating sets and Noetherian coefficients

The Gröbner–Shirshov basis theory for commutative dialgebras developed by Chen and Zhang proves that every ideal of the free commutative dialgebra $Di[X]$ has a unique reduced Gröbner–Shirshov basis for an arbitrary generating set $X$ , but all of its finiteness and algorithmic conclusions — Noetherianity of $Di[X]$ , existence of finite Gröbner–Shirshov bases for every ideal, termination of the Buchberger–Shirshov completion algorithms, and decidability of ideal equality — are proven only when $X$ is finite and the ground ring is a field, while the Noetherianity transfer to the polynomial diring $Di_R[X]$ over an associative coefficient ring $R$ is proven only when $R$ is left Noetherian and $X$ is finite. The problem asks to determine the decidability picture in the uncovered regime of an infinite generating set $X$ , and the Noetherianity, finiteness, and termination picture for coefficient rings $R$ that are not left Noetherian (including the converse direction of the Noetherianity transfer).

Background

A dialgebra (in the sense of Loday's digebras, motivated by the theory of Leibniz algebras, which are a non-antisymmetric analogue of Lie algebras) is a vector space with two associative bilinear products $\dashv$ and $\vdash$ satisfying three further compatibility identities (Definition 1 of Chen–Zhang): $x \dashv (y \vdash z) = x \dashv (y \dashv z)$ , $(x \dashv y) \vdash z = (x \vdash y) \vdash z$ , and $x \vdash (y \dashv z) = (x \vdash y) \dashv z$ ; a commutative dialgebra is a dialgebra in which both products are separately commutative, i.e. $x \dashv y = y \dashv x$ and $x \vdash y = y \vdash x$ . The commutative analogue of a semigroup here is a dimonoid (also called a disemigroup): a set with two multiplications satisfying the dialgebra axioms without linear structure. The free commutative dialgebra $Di[X]$ on a set $X$ over a field $k$ has as a basis the free commutative dimonoid of Zhuchok, whose elements are bar-monomials $\lfloor a \rfloor$ — words $a$ in the pairwise commuting letters of $X$ with a marked centre (the bar) — with $\dashv$ and $\vdash$ multiplying the underlying words while keeping the centre of the left, respectively right, factor. More generally, for an associative unital ring $R$ , the polynomial diring $Di_R[X]$ (a diring being the linear-algebraic analogue with two ring structures) is the free left $R$ -module on this basis with coordinatewise addition and the two products induced from the dimonoid.

A Gröbner–Shirshov basis (GS basis) of an ideal $I$ in such an algebra is the non-commutative-algebra analogue of a Gröbner basis: a generating set $S$ such that every element of the algebra has a unique irreducible normal form modulo one-step reductions by $S$ , which happens exactly when all 'compositions' (analogues of S-polynomials) of pairs from $S$ reduce to zero. The key structural tool is the Composition–Diamond lemma, which for dialgebras was established for free (non-commutative) dialgebras by Bokut, Chen, and Liu (2010) and in a sharpened form by Zhang and Chen (2017). For commutative dialgebras, Chen and Zhang (2019) proved the Composition–Diamond lemma and, using a monomial-centre ordering (a well order on the bar-monomials compatible with both products), showed: every ideal $I$ of $Di[X]$ has a unique reduced GS basis, for an arbitrary set $X$ with no finiteness assumption (Theorem 19); if $X$ is finite, then $Di[X]$ is Noetherian (i.e., its ideals satisfy the ascending chain condition) and every ideal has a finite GS basis (Corollary 24, Theorem 25, Corollary 33); Buchberger–Shirshov completion (Algorithm 29, computing a GS basis from a finite input, and Algorithm 31, reducing it) terminates for finite $X$ (Lemmas 30, 32); consequently the word problem for finitely presented commutative dialgebras and, for finite $X$ , the ideal-equality problem are decidable (Theorems 34, 35); and, over an associative unital ring $R$ that is left Noetherian, with $X$ finite, the polynomial diring $Di_R[X]$ is left Noetherian (Theorem 23).

These results parallel classical commutative algebra, where the Hilbert basis theorem and its converse give that a polynomial ring $R[x_1,\dots,x_n]$ is Noetherian if and only if $R$ is; where a polynomial ring in infinitely many indeterminates fails to be Noetherian by the trivial ascending chain of ideals generated by growing finite subsets of the indeterminates; and where Gröbner bases over rings rather than fields are known to be substantially more delicate. None of these classical facts has been transferred to the dialgebra setting in the literature: the paper of Chen and Zhang proves only the one-directional transfer under exactly the hypotheses 'X finite' and 'R left Noetherian', develops its GS-basis theory over a field, and poses no explicit open problems.

Problem Statement

For the free commutative dialgebra (polynomial dialgebra) $Di[X]$ over a field and the polynomial diring $Di_R[X]$ over an associative unital ring $R$ , with Gröbner–Shirshov bases, the monomial-centre orderings, and the Buchberger–Shirshov completion algorithms understood as in Chen–Zhang (2019), determine the finiteness and termination picture outside the regime settled by that paper (existence and uniqueness of a possibly infinite reduced Gröbner–Shirshov basis for arbitrary $X$ being already known, it is excluded here):

Infinite generating sets. For an infinite set $X$ : is the ideal-equality problem of Theorem 35 decidable without the finiteness of $X$ ?
Coefficient rings. For finite $X$ and an associative unital ring $R$ that is not left Noetherian: Over $Di_R[X]$ — a setting in which no Gröbner–Shirshov basis theory has been developed at all — determine, for each of left Noetherianity of $Di_R[X]$ , existence of finite Gröbner–Shirshov-type bases — defined as the natural analogue, over the coefficient ring, of the field-case Gröbner–Shirshov basis, that is, a generating set $G$ such that every element of $Di_R[X]$ has a unique irreducible normal form modulo one-step reductions by $G$ (a one-step reduction by an element $g$ being available exactly when the leading coefficient of $g$ is a unit of $R$ , so that each reduction $f \mapsto f - (\alpha/\mathrm{lc}(g))\,m\,g$ is defined over $R$ , where $\alpha$ is a coefficient of $f$ and $m$ a bar-monomial), and nothing stronger than this normal-form property is required — for finitely generated left ideals, and termination of Buchberger–Shirshov completion of finite inputs, whether it holds for every such $R$ (with a proof), fails for every such $R$ (a single counterexample witness then settles the sub-question, since failure is uniform), or neither — i.e. it depends on $R$ , in which case the dependence must be characterized (for which rings it holds and for which it fails).

As a boundary that is already settled by the classical analogue recalled in the Background, Noetherianity in the infinite- $X$ case is not among the genuinely open problems: the ascending chain of ideals generated by growing finite subsets of $X$ is strictly ascending, so $Di[X]$ is not Noetherian for infinite $X$ . The genuinely open content is the decidability of ideal equality without finiteness of $X$ and the ring-coefficient questions of part 2. Also already settled, and excluded, are the finite- $X$ results of that paper upon which the finite parts of the construction rest — Noetherianity of $Di[Y]$ for finite $Y$ and the existence of finite Gröbner–Shirshov bases there, with termination of the completion and decidability of ideal equality for finite $X$ — together with the existence and uniqueness of the reduced Gröbner–Shirshov basis for arbitrary $X$ . The common objective of the remaining parts is the extension (or delimitation) of the Chen–Zhang finiteness, termination, and decidability theorems to both regimes their hypotheses exclude.

References

Yuqun Chen, Guangliang Zhang, “Gröbner–Shirshov bases for commutative dialgebras”, Communications in Algebra 47(4), 1671–1689 (2019), DOI: 10.1080/00927872.2018.1513017, https://doi.org/10.1080/00927872.2018.1513017 (preprint arXiv:1907.06680, https://arxiv.org/abs/1907.06680)
L. A. Bokut, Yuqun Chen, Cihua Liu, “Gröbner–Shirshov bases for dialgebras”, International Journal of Algebra and Computation 20(3), 391–415 (2010), DOI: 10.1142/S0218196710005753, https://doi.org/10.1142/S0218196710005753 (preprint arXiv:0804.0638, https://arxiv.org/abs/0804.0638)
Guangliang Zhang, Yuqun Chen, “A New Composition-Diamond Lemma for Dialgebras”, Algebra Colloquium 24(2), 323–350 (2017), DOI: 10.1142/S1005386717000207, https://doi.org/10.1142/S1005386717000207 (preprint arXiv:1702.00119, https://arxiv.org/abs/1702.00119)