Index (C•)α, 9 C•, 1, 7 Nd-graded complex, 9 MinGen(in≺(a)), 13 Path(c, c ), 8 m(w), 12 acyclic matching, 2, 7 infinite, 9 algebra associative, 29 Cohen-Macaulay, 43 differential graded, 18 exterior, 31, 33 Koszul, 17, 25 semigroup, 25 alphabet, 15 Anick resolution, 11, 14, 17, 21, 23 bimodule, 30 binomial, 25 Cartan complex, 26 cellular chain complex, 8, 36 chain complex, 1, 7 cellular, 8, 36 complete intersection, 18, 29, 32 complex Nn-graded, 9 multigraded, 9 critical vertex, 8 CW-complex, 35 P -graded, 35 regular, 37 cycle directed, 2, 7 deterministic finite automaton, 24 DFA, 24 differential, 7 differential graded algebra, 18 directed cycle, 2, 7 edge, 1 path, 2 weighted graph, 1, 7 discrete Morse theory, 1, 2, 7 algebraic, 1, 2, 8 edge weight, 1, 7 exterior algebra, 31, 33 f-vector, 36 free module, 7 fully attached, 14, 15, 23, 24, 30 Gr¨obner basis, 11 infinite, 27 non.commutative, 21 quadratic, 17, 21, 25 reduced, 11, 18, 31 representation, 15, 21 graph directed weighted, 7 Hilbert series, 25, 32 Hochschild complex, 29, 58 Hochschild homology, 29 ideal, 11 Borel fixed, 35, 37 complete intersection, 18 generic initial, 37, 40 initial, 11 monomial, 37 non-commutative lattice, 21 p-Borel fixed, 35, 54 principal Borel fixed, 37 principal Cohen-Macaulay p-Borel fixed, 49 principal p-Borel fixed, 41 two-sided, 21, 30 initial ideal, 11 Koszul algebra, 17, 25 Koszul complex, 26 language, 15, 17 regular, 25 73

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