. Lemma, Let (X, <) | S be a presentation by operator of A and let F be its reduction family

. Proof, The set Red (F ) is the set of words w such that every sub-word of w belongs to Red (S) From Proposition 2.1.14, a word belongs to Red (S) if and only if it does not belong to lm (R)

<. Assume-that-(-x and . Confluent, Let f ? I(R) We assume that lc (f ) is equal to 1. The kernel of ?F being equal to I(R), (?F ) (f ) is equal to 0. Hence, ?F ) (lm (f )) is equal to (?F ) (lm (f ) ? f )

R. Absence and . Basis, We would like to equip the set RO (G, <) with a lattice structure We cannot use the argument of Section 2.1 because a subspace of KG does not necessarily admit a reduced basis. Indeed, consider G = {g 1 , g 2 , g 3 } ordered such: g 1 < g 3 and g 2 < g 3 . The subspace of KG spanned by g 3 ? g 1 and g 3 ? g 2 does not admit any reduced basis, order to equip RO (G, <) with an order relation, we need the following lemma

. Lemma, Let T 1 and T 2 be two reduction operators relative to (G, <) such that ker (T 1 ) is included in ker, Then, Red (T 2 ) is included in Red

L. Absence and . Structure, The order introduced in 4.1.5 does not induce a lattice structure. Consider G = {g 1 , g 2 , g 3 , g 4 , g 5 } ordered such: g 1 < g 3 , g 1 < g 4 , g 2 < g 3

J. Backelin, S. Cojocaru, and V. Ufnarovski, The computer algebra package Bergman: current state In Commutative algebra, singularities and computer algebra, NATO Sci. Ser. II Math. Phys. Chem, vol.115, pp.75-100, 2002.

R. Berger, Weakly confluent quadratic algebras, Algebras and Representation Theory, vol.1, issue.3, pp.189-213, 1998.
DOI : 10.1023/A:1009918131382

R. Berger, Koszulity for Nonquadratic Algebras, Journal of Algebra, vol.239, issue.2, pp.705-734, 2001.
DOI : 10.1006/jabr.2000.8703
URL : http://doi.org/10.1006/jabr.2000.8703

G. M. Bergman, The diamond lemma for ring theory, Advances in Mathematics, vol.29, issue.2, pp.178-218, 1978.
DOI : 10.1016/0001-8708(78)90010-5

A. Leonid and ?. Bokut, Imbeddings into simple associative algebras. Algebra i Logika, pp.117-142, 1976.

V. Ronald, F. Book, and . Otto, String-rewriting systems. Texts and Monographs in Computer Science, 1993.

B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal, 1965.

B. Buchberger, History and basic features of the critical-pair/completion procedure, Rewriting techniques and applications, pp.3-38, 1985.
DOI : 10.1016/S0747-7171(87)80020-2

C. Chenavier, Confluence Algebras and Acyclicity of the Koszul Complex, Algebras and Representation Theory, vol.152, issue.1, pp.679-711, 2016.
DOI : 10.1007/s10468-016-9595-6
URL : https://hal.archives-ouvertes.fr/hal-01141738

J. Peter, U. Hilton, and . Stammbach, A course in homological algebra, Graduate Texts in Mathematics, vol.4, 1997.

D. Kapur and P. Narendran, A finite thue system with decidable word problem and without equivalent finite canonical system, Theoretical Computer Science, vol.35, issue.2-3, pp.337-344, 1985.
DOI : 10.1016/0304-3975(85)90023-4

E. Donald, P. B. Knuth, and . Bendix, Simple word problems in universal algebras, Computational Problems in Abstract Algebra (Proc. Conf, pp.263-297, 1967.

B. Kriegk, M. Van-den, and . Bergh, Representations of non-commutative quantum groups, Proceedings of the London Mathematical Society, vol.124, issue.1, pp.57-82, 2015.
DOI : 10.1112/plms/pdu042

T. Mora, An introduction to commutative and noncommutative Gr??bner bases, Second International Colloquium on Words, Languages and Combinatorics, pp.131-173, 1992.
DOI : 10.1016/0304-3975(94)90283-6
URL : http://doi.org/10.1016/0304-3975(94)90283-6

H. A. Maxwell and . Newman, On theories with a combinatorial definition of " equivalence, Ann. of Math, vol.43, issue.2, pp.223-243, 1942.

E. Alexander, L. Polishchuk, and . Positselski, Quadratic algebras, 2005.

B. Stewart and . Priddy, Koszul resolutions, Trans. Amer. Math. Soc, vol.152, pp.39-60, 1970.

A. I. Shirshov, Selected works of A. I. Shirshov. Contemporary Mathematicians Translated from the Russian by, 2009.