Churned in foam, that outer ocean lashed the clouds; and straight in my white wake, headlong dashed a shallop, three fixed specters leaning o'er its prow: three arrows poising.

He drew a deep breath. ‘Well, I'm back,’ he said.

In the original plan, the SPES survey was structured as a trilogy centered around the second volume, Macaulay's Paradigm and Gröbner Technology: after the first volume, The Kronecker–Duval Philosophy, formulated the task of ‘solving’ not as producing programs which compute the roots but as producing techniques for computing with such roots, the second volume introduced the Gröbnerian technologies needed for effectively and efficiently fullfilling this task. The third and last volume, according to this plan, would have consisted of a part surveying all the recent approaches, mainly based on these techniques, which successfully completed the required task, and a final part covering extensions, applications, anticipations and alternatives to Gröbner bases.

This seventh and last part, Beyond, which grew to a gargantuan size and which is the present volume, covers the extensions of Buchberger's theory and algorithm in three different directions.

• Relaxing commutativity and allowing coefficients from a domain it is possible to export Buchberger theory over an effective ring monoid and to produce a Buchberger algorithm based on the Möller–Pritchard Lifting Theorem.

I cover the preliminary results by Zacharias, Kandri-Rody–Kapur, Pan and Möller on Buchberger theory over a domain; the extension from monoid rings toward weaker algebras (group rings, path algebras, magmas). In particular I cover the recent results by Birgit Reinert which cover function rings via saturation techniques.

I also cover the results of Ore on the construction of quotient fields over a noncommutative ring and on a non-commutative Euclidean algorithm, together with the related results by P. M. Cohn and a Buchberger theory for multivariate Ore extensions.

An intermezzo chapter covers applications of non-commutative Gröbner bases, combinatorial structures over monomial algebras and a (very preliminary) taxonomy of term orderings.