Geodesic
On canonical bases of spaces with a~well ordered basis and a~distinguished family of subspaces
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 23, Tome 400 (2012), pp. 215-221

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Let $V$ be a vector space with a well ordered basis and $\mathfrak I$ a family of subspaces of $V$ closed under intersections. An analogue of Groebner basis is defined for subspaces from $\mathfrak I$. It is shown that in Noetherian case such basis always exists and is unique. 
@article{ZNSL_2012_400_a11,
     author = {A. V. Yakovlev},
     title = {On canonical bases of spaces with a~well ordered basis and a~distinguished family of subspaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {215--221},
     publisher = {mathdoc},
     volume = {400},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_400_a11/}
}
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A. V. Yakovlev. On canonical bases of spaces with a~well ordered basis and a~distinguished family of subspaces. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 23, Tome 400 (2012), pp. 215-221. http://geodesic.mathdoc.fr/item/ZNSL_2012_400_a11/