Geodesic
Strong non-noetherity of polynomial reduction
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 73-76

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It is well known result by A. Reeves and B. Sturmfels, that the reduction modulo a marked set of polynomials is Noetherian if and only if the marking is induced from an admissible term order. For finite sets of polynomials with non-admissible order, there is a constructive proof of existence of infinite reduction sequence, although the finite one is might still be possible. On the base of our specialized software for combinatorics of monomial orders, we have found some examples, for which there is not any finite reduction sequence. This is what we call “strong” non-noetherity. Bibl. – 3 titles. 
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N. Vassiliev; D. Pavlov. Strong non-noetherity of polynomial reduction. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 73-76. http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a3/