Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 2,3\rangle \) | \( X_2 \prec X_1 \) | \( \{ X_1^3-X_2^2 \} \) |
\( X_1 \prec X_2 \) | \( \{ X_2^2 - X_1^3 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 2,5\rangle \) | \( X_2 \prec X_1 \) | \( \{ X_1^2-X_2^5 \} \) |
\( X_1 \prec X_2 \) | \( \{ X_2^5- X_1^2 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 3, 4, 5\rangle \) | \( X_3 \prec X_2 \prec X_1 \) | \( \{ X_2^5-X_3^4, X_1X_3-X_2^2, X_1X_2^3-X_3^3, X_1^3-X_2X_3, X_1^2X_2-X_3^2 \} \) |
\( X_2 \prec X_3 \prec X_1 \) | \( \{ X_3^4-X_2^5, X_1X_2^3-X_3^3, X_1X_3-X_2^2,X_1^2X_2-X_3^2,X_1^3-X_3X_2 \}\) | |
\( X_3 \prec X_1 \prec X_2 \) | \( \{ X_1^5-X_3^3, X_2X_3-X_1^3,X_2X_1^2-X_3^2,X_2^2-X_1X_3 \}\) | |
\( X_1 \prec X_3 \prec X_2 \) | \( \{ X_3^3-X_1^5, X_2X_3-X_1^3, X_2X_1^2-X_3^2, X_2^2-X_1X_3 \}\) | |
\( X_2 \prec X_1 \prec X_3 \) | \( \{ X_1^4-X_2^3,X_3X_2-X_1^3, X_3X_1-X_2^2,X_3^2-X_1^2X_2 \}\) | |
\( X_1 \prec X_2 \prec X_3 \) | \( \{ X_2^3-X_1^4, X_3X_1-X_2^2,X_3X_2-X_1^3,X_3^2-X_2X_1^2 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 2,7\rangle \) | \( X_2 \prec X_1 \) | \( \{ X_1^2-X_2^7 \} \) |
\( X_1 \prec X_2 \) | \( \{ X_2^7- X_1^2 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 3,4\rangle \) | \( X_2 \prec X_1 \) | \( \{ X_1^4-X_2^3 \} \) |
\( X_1 \prec X_2 \) | \( \{ X_2^3- X_1^4 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 3, 5, 7\rangle \) | \( X_3 \prec X_2 \prec X_1 \) | \( \{ X_2^7-X_1^5, X_1X_3-X_2^2, X_1X_2^5-X_3^4, X_1^2X_2^3-X_3^3, X_1^3X_2-X_3^2,X_1^4-X_2X_3 \} \) |
\( X_2 \prec X_3 \prec X_1 \) | \( \{X_3^5-X_2^7, X_1X_2^5-X_3^4, X_1X_3-X_2^2, X_1^2X_2^3-X_3^3, X_1^3X_2-X_3^2, X_1^4-X_3X_2 \}\) | |
\( X_3 \prec X_1 \prec X_2 \) | \( \{ X_1^7-X_3^3, X_2X_3-X_1^4, X_2X_1^3-X_3^2, X_2^2-X_1X_3 \}\) | |
\( X_1 \prec X_3 \prec X_2 \) | \( \{ X_3^3-X_1^7, X_2X_1^3-X_3^2,X_2X_3-X_1^4,X_2^2-X_3X_1 \}\) | |
\( X_2 \prec X_1 \prec X_3 \) | \( \{ X_1^5-X_2^3,X_3X_2-X_1^4,X_3X_1-X_2^2,X_3^2-X_1^3X_2 \}\) | |
\( X_1 \prec X_2 \prec X_3 \) | \( \{X_2^3-X_1^5, X_3X_1-X_2^2,X_3X_2-X_1^4, X_3^2-X_2X_1^3 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 5, 6, 7\rangle \) | \( X_1 \prec X_3 \prec X_2 \prec X_4 \) | \( \{ X_3^2-X_1^3, X_2^2-X_3X_1, X_4X_1-X_2X_3, X_4X_3-X_2X_1^2, X_4X_2-X_3^2, X_4^2-X_3X_1^2 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 2,9\rangle \) | \( X_2 \prec X_1 \) | \( \{ X_1^2-X_2^9 \} \) |
\( X_1 \prec X_2 \) | \( \{ X_2^9- X_1^2 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 3,5\rangle \) | \( X_2 \prec X_1 \) | \( \{ X_1^5-X_2^3 \} \) |
\( X_1 \prec X_2 \) | \( \{ X_2^3- X_1^5 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 3, 7, 8\rangle \) | \( X_3 \prec X_2 \prec X_1 \) | \( \{ X_2^8 X_3^7,X_1X_3^4-X_2^5,X_1X_2^3-X_3^3,X_1^2X_3-X_2^2,X_1^3X_2-X_3^2,X_1^5-X_2X_3\} \) |
\( X_2 \prec X_3 \prec X_1 \) | \( \{ X_3^7-X_2^8,X_1X_2^3-X_3^3,X_1X_3^4-X_2^5,X_1^2X_3-X_2^2,X_1^3X_2-X_3^2,X_1^5-X_3X_2 \} \) | |
\( X_3 \prec X_1 \prec X_2 \) | \( \{ X_1^8-X_3^3,X_2X_3-X_1^5,X_2X_1^3-X_3^2,X_2^2-X_1^2X_3 \}\) | |
\( X_1 \prec X_3 \prec X_2 \) | \( \{ X_3^3-X_1^7,X_2X_1^3-X_3^2,X_2X_3-X_1^4,X_2^2-X_3X_1 \}\) | |
\( X_2 \prec X_1 \prec X_3 \) | \( \{ X_1^5-X_2^3,X_3X_2-X_1^4,X_3X_1-X_2^2,X_3^2-X_1^3X_2 \}\) | |
\( X_1 \prec X_2 \prec X_3 \) | \( \{ X_2^3-X_1^5,X_3X_1-X_2^2,X_3X_2-X_1^4,X_3^2-X_2X_1^3 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 5, 6\rangle \) | \( X_3 \prec X_2 \prec X_1 \) | \( \{X_2^6-X_3^5,X_1X_3-X_2^2,X_1X_2^4-X_3^4,X_1^2X_2^2-X_3^3,X_1^3-X_3^2 \} \) |
\( X_2 \prec X_3 \prec X_1 \) | \( \{ X_3^5-X_2^6,X_1X_2^4-X_3^4,X_1X_3-X_2^2,X_1^2X_2^2-X_3^3,X_1^3-X_3^2 \}\) | |
\( X_3 \prec X_1 \prec X_2 \) | \( \{ X_1^3-X_3^2, X_2^2-X_1X_3 \}\) | |
\( X_1 \prec X_3 \prec X_2 \) | \( \{ X_3^2-X_1^3,X_2^2-X_3X_1 \}\) | |
\( X_2 \prec X_1 \prec X_3 \) | \( \{ X_2^4-X_1^5,X_3X_1-X_2^2,X_3X_2^2-X_1^4,X_3^2-X_1^3 \}\) | |
\( X_1 \prec X_2 \prec X_3 \) | \( \{ X_1^5-X_2^4,X_3X_2^2-X_1^4,X_3X_1-X_2^2,X_3^2-X_1^3 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 5, 7\rangle \) | \( X_3 \prec X_2 \prec X_1 \) | \( \{ X_2^7-X_3^5,X_1X_3^3-X_2^5,X_1X_2^2-X_3^2, X_1^2X_3-X_2^3,X_1^3-X_2X_3 \} \) |
\( X_2 \prec X_3 \prec X_1 \) | \( \{X_1^7-X_3^5,X_1X_3^3-X_2^5,X_1X_2^2-X_3^2,X_1^2 X_3-X_2^3,X_1^3-X_2X_3 \}\) | |
\( X_3 \prec X_1 \prec X_2 \) | \( \{ X_1^7-X_3^4,X_2X_3-X_1^3,X_2X_1^4-X_3^3,X_2^2X_1-X_3^2,X_2^3-X_1^2X_3 \}\) | |
\( X_1 \prec X_3 \prec X_2 \) | \( \{ X_3^4-X_1^7,X_2X_1^4-X_3^3,X_2X_3-X_1^3,X_2^2X_1-X_3^2,X_2^3-X_3X_1^2 \}\) | |
\( X_2 \prec X_1 \prec X_3 \) | \( \{ X_1^5-X_2^4,X_3X_2-X_1^3,X_3X_1^2-X_2^3,X_3^2-X_1X_2^2 \}\) | |
\( X_1 \prec X_2 \prec X_3 \) | \( \{ X_2^4-X_1^5, X_3X_1^2-X_2^3, X_3X_2-X_1^3,X_3^2-X_2^2X_1 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 6, 7, 9\rangle \) | \( X_1 \prec X_2 \prec X_4 \prec X_3 \) | \( \{ X_2^2-X_1^3, X_4^2-X_2X_1^3, X_3X_1^2-X_4X_2, X_3X_2-X_4X_1, X_3X_4-X_1^4, X_3^2-X_2X_1^2 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 6, 7, 8, 9\rangle \) | \( X_2 \prec X_5 \prec X_1 \prec X_3 \prec X_4 \) | \( \{ X_5^2-X_2^3, X_1^3-X_5X_2, X_3X_2^2-X_1^2X_5, X_3X_5-X_1^2X_2, X_3X_1-X_2^2, X_3^2-X_1X_5,\) |
\(X_4X_2-X_3^2, X_4X_5-X_3X_1^2, X_4X_1-X_3X_2, X_4X_3-X_5X_2, X_4^2-X_3X_5 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 2,11\rangle \) | \( X_2 \prec X_1 \) | \( \{ X_1^2-X_2^{11} \} \) |
\( X_1 \prec X_2 \) | \( \{ X_2^{11}- X_1^2 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 3, 7, 11\rangle \) | \( X_3 \prec X_2 \prec X_1 \) | \( \{ X_2^{11}-X_3^7,X_1X_3-X_2^2,X_1X_2^9-X_3^6,X_1^2X_2^7-X_3^5,X_1^3X_2^5-X_3^4,X_1^4X_2^3-X_3^3,X_1^5X_2-X_3^2,X_1^6-X_2X_3\} \) |
\( X_2 \prec X_3 \prec X_1 \) | \( \{ X_1^{11}-X_3^3,X_2X_3-X_1^6,X_2X_1^5-X_3^2,X_2^2-X_1X_3 \}\) | |
\( X_3 \prec X_1 \prec X_2 \) | \( \{ X_1X_2^9-X_3^6, X_1X_3-X_2^2,X_1^2X_2^7-X_3^5,X_1^3X_2^5-X_3^4,X_1^4X_2^3-X_3^3,X_1^5X_2-X_3^2,X_1^6-X_3X_2 \}\) | |
\( X_1 \prec X_3 \prec X_2 \) | \( \{ X_3^3-X_1^{11},X_2X_1^5-X_3^2,X_2X_3-X_1^6,X_2^2-X_3X_1 \}\) | |
\( X_2 \prec X_1 \prec X_3 \) | \( \{ X_1^7-X_2^3,X_3X_2-X_1^6,X_3X_1-X_2^2,X_3^2-X_1^5X_2 \}\) | |
\( X_1 \prec X_2 \prec X_3 \) | \( \{ X_2^3-X_1^7,X_3X_1-X_2^2,X_3X_2-X_1^6,X_3^2-X_2X_1^5 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 3, 8, 10\rangle \) | \( X_3 \prec X_2 \prec X_1 \) | \( \{ X_2^5-X_3^4, X_1^2X_3-X_2^2,X_1^2X_2^3-X_3^3, X_1^4X_2-X_3^2, X_1^6-X_2X_3 \} \) |
\( X_2 \prec X_3 \prec X_1 \) | \( \{ X_3^4-X_2^5, X_1^2X_2^3-X_3^3, X_1^2X_3-X_2^2, X_1^4X_2-X_3^2, X_1^6-X_3X_2 \}\) | |
\( X_3 \prec X_1 \prec X_2 \) | \( \{ X_1^{10}-X_3^3, X_2X_3-X_1^6,X_2X_1^4-X_3^2, X_2^2-X_1^2X_3 \}\) | |
\( X_1 \prec X_3 \prec X_2 \) | \( \{ X_3^3-X_1^{10},X_2X_1^4-X_3^2, X_2X_3-X_1^6, X_2^2-X_3X_1^2 \}\) | |
\( X_2 \prec X_1 \prec X_3 \) | \( \{X_1^8-X_2^3,X_3X_2-X_1^6,X_3X_1^2-X_2^2,X_3^2-X_1^4X_2 \}\) | |
\( X_1 \prec X_2 \prec X_3 \) | \( \{ X_2^3-X_1^8, X_3X_1^2-X_2^2, X_3X_2-X_1^6, X_3^2-X_2X_1^4 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 6, 7\rangle \) | \( X_3 \prec X_2 \prec X_1 \) | \( \{ X_2^7-X_3^6,X_1X_3^2-X_2^3,X_1X_2^4-X_3^4,X_1^2X_2-X_3^2,X_1^3-X_2^2 \} \) |
\( X_2 \prec X_3 \prec X_1 \) | \( \{X_3^6-X_2^7,X_1X_2^4-X_3^4,X_1X_3^2-X_2^3,X_1^2X_2-X_3^2,X_1^3-X_2^2 \}\) | |
\( X_3 \prec X_1 \prec X_2 \) | \( \{ X_1^7-X_3^4, X_2X_3^2-X_1^5, X_2X_1^2-X_3^2,X_2^2-X_1^3 \}\) | |
\( X_1 \prec X_3 \prec X_2 \) | \( \{ X_3^4-X_1^7,X_2X_1^2-X_3^2,X_2X_3^2-X_1^5,X_2^2-X_1^3 \}\) | |
\( X_2 \prec X_1 \prec X_3 \) | \( \{ X_1^3-X_2^2, X_3^2-X_1^2X_2 \}\) | |
\( X_1 \prec X_2 \prec X_3 \) | \( \{ X_2^2-X_1^3,X_3^2-X_2X_1^2 \}\) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 6, 9, 11\rangle \) | \( X_3 \prec X_2 \prec X_1 \prec X_4 \) | \( \{ X_2^3-X_3^2, X_1^3-X_2^2, X_4X_3-X_1^2X_2^2, X_4X_2-X_1^2X_3, X_4X_1-X_2X_3,X_4^2-X_1X_3^2 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 7, 9, 10\rangle \) | \( X_4 \prec X_1 \prec X_2 \prec X_3 \) | \( \{ X_1^5-X_4^2, X_2^2-X_1X_4, X_3X_4-X_2X_1^3, X_3X_1^2-X_2X_4, X_3X_2-X_1^4, X_3^2-X_1^2X_4 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 6, 7, 8\rangle \) | \( X_4\prec X_2 \prec X_3 \prec X_1 \) | \( \{ X_2^4-X_4^3, X_3^2-X_2X_4, X_1X_4-X_3X_2, X_1X_2^3-X_3X_4^2, X_1X_3-X_2^2, X_1^2X_2- X_4^2, X_1^3-X_3X_4 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 6, 7, 9\rangle \) | \( X_4 \prec X_2 \prec X_1 \prec X_3 \) | \( \{ X_2^3-X_4^2, X_1^3-X_2X_4, X_3X_4-X_1^2X_2, X_3X_2^2-X_1^2X_4, X_3X_1-X_2^2, X_3^2-X_1X_4 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 6, 8, 9\rangle \) | \( X_3 \prec X_2 \prec X_1 \prec X_4 \) | \( \{X_1X_3^2-X_4X_2^2, X_1X_2^2-X_4X_3, X_1X_4-X_2X_3, X_1^2X_3-X_2^3, X_1^2X_2-X_3^2, X_1^3-X_4X_2\} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 7, 8, 9, 11\rangle \) | \( X_1 \prec X_4 \prec X_2 \prec X_3 \prec X_5 \) | \( \{X_4^5-X_1^9, X_2X_1^4-X_4^3, X_2X_4^2-X_1^5, X_2^2-X_4X_1, X_3X_1^2-X_4^2, X_3X_4-X_2X_1^2, X_3X_2-X_1^3,\) |
\(X_3^2-X_2X_4, X_5X_1-X_3^2, X_5X_4-X_1^4, X_5X_2-X_3X_1^2, X_5X_3-X_4X_1^2, X_5^2-X_3X_4X_1\} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 6, 7, 8, 9, 10, 11\rangle \) | tbd | tbd |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 6, 8\rangle \) | \( X_1 \prec X_2 \prec X_3 \) | \( \{X_2^5-X_1^6, X_3X_1^2-X_2^3, X_3X_2^2-X_1^4, X_3^2-X_2X_1^2\} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 6,7 \rangle \) | \( X_1 \prec X_3 \prec X_2 \) | \( \{X_3^5-X_1^7, X_2X_1^3-X_3^3, X_2X_3^2-X_1^4, X_2^2-X_3X_1 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 7, 9\rangle \) | \( X_1 \prec X_2 \prec X_3 \) | \( \{X_2^4-X_1^7, X_3X_1^3-X_2^3, X_3X_2-X_1^4, X_3^2-X_2^2X_1 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 6, 9\rangle \) | \( X_1 \prec X_2 \prec X_3 \) | \( \{X_2^2-X_1^3, X_3^2-X_2X_1^3 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 3, 8, 13\rangle \) | \( X_1 \prec X_2 \prec X_3 \) | \( \{X_2^3-X_1^8, X_3X_1-X_2^2, X_3X_2-X_1^7, X_3^2-X_2X_1^6 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 3, 10, 11\rangle \) | \( X_1 \prec X_2 \prec X_3 \) | \( \{X_2^3-X_1^{10}, X_3X_1^3-X_2^2, X_3X_2-X_1^7, X_3^2-X_2X_1^4\} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 7, 8,9\rangle \) | \( X_2 \prec X_3 \prec X_1 \prec X_4 \) | \( \{X_3^7-X_2^8, X_1X_2^5-X_3^5, X_1X_3^2-X_2^3, X_1^2X_2^2-X_3^3, X_1^3-X_3X_2,\) |
\(X_4X_2-X_3^2, X_4X_3-X_1^2X_2, X_4X_1-X_2^2, X_4^2-X_1^2X_3 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 7, 8, 11\rangle \) | \(X_1 \prec X_2 \prec X_4 \prec X_3 \) | \( \{X_2^5-X_1^7, X_4X_1^2-X_2^3, X_4X_2^2-X_1^5, X_4^2-X_2X_1^3, X_3X_1^2-X_4X_2, X_3X_2-X_1^3, X_3X_4-X_2^2X_1, X_3^2-X_4X_1 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 9, 10, 11\rangle \) | \(X_1 \prec X_3\prec X_2 \prec X_4 \) | \( \{X_3^2-X_1^5, X_2^2-X_3X_1^2, X_4X_1^2-X_2X_3, X_4X_3-X_2X_1^3, X_4X_2-X_3^2, X_4^2-X_3X_1^3\} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 4, 6, 11, 13\rangle \) | \(X_1 \prec X_2\prec X_3 \prec X_4 \) | \( \{X_2^2-X_1^3, X_3^2-X_2^3X_1, X_4X_1-X_3X_2, X_4X_2-X_3X_1^2, X_4X_3-X_2^2X_1^3, X_4^2-X_3^2X_1\} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 6, 7, 9,10,11\rangle \) | \(X_1 \prec X_3\prec X_2\prec X_4\prec X_5 \) | \( \{X_3^2-X_1^3, X_2^3-X_3X_1^2, X_4X_1-X_2X_3, X_4X_3-X_2X_1^2, X_4X_2^2-X_3^2X_1,\) |
\(X_4^2-X_2^2X_1, X_5X_1-X_4X_2, X_5X_3-X_4^2, X_5X_2-X_3^2, X_5X_4-X_3X_1^2, X_5^2-X_4X_1^2 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 6, 7, 8, 10,11\rangle \) | \(X_1 \prec X_4\prec X_3\prec X_2\prec X_5 \) | \( \{X_4^3-X_1^5, X_3X_1^2-X_4^2, X_3X_4-X_1^3, X_3^2-X_4X_1, X_2^2-X_3X_1,\) |
\(X_5X_1-X_2X_4, X_5X_4-X_2X_3X_1, X_5X_3-X_2X_1^2, X_5X_2-X_3X_4, X_5^2-X_4X_1^2\} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 6, 7, 8, 9, 11 \rangle \) | \(X_1 \prec X_4\prec X_2\prec X_3\prec X_5 \) | \( \{X_4^2-X_1^3, X_2^3-X_4X_1^2, X_3X_1-X_2^2, X_3X_2-X_4X_1, X_3^2-X_2X_4,\) |
\(X_5X_1-X_3X_4, X_5X_4-X_3X_1^2, X_5X_2-X_4^2, X_5X_3-X_2X_1^2, X_5^2-X_2X_4X_1\} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 6, 7, 8, 9,10 \rangle \) | \(X_1 \prec X_4\prec X_2\prec X_3\prec X_5 \) | \( \{X_4^2-X_1^3, X_2^3-X_4X_1^2, X_3X_1-X_2^2, X_3X_2-X_4X_1, X_3^2-X_2X_4, X_5X_1-X_2X_4,\) |
\(X_5X_4-X_2X_1^2, X_5X_2-X_3X_4, X_5X_3-X_4^2, X_5^2-X_3X_1^2\} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 8, 9, 11, 12\rangle \) | \(X_2 \prec X_5\prec X_1\prec X_3\prec X_4 \) | \( \{X_5^2-X_2^3, X_1^4-X_5X_2, X_3X_2-X_1X_5, X_3X_5-X_1X_2^2, X_3X_1^3-X_2^3, X_3^2-X_1^2X_2,\) |
\(X_4X_2-X_3X_1^2, X_4X_5-X_1^3X_2, X_4X_1-X_2^2, X_4X_3-X_1^4, X_4^2-X_3X_1X_2 \} \) |
Semigroup | Choice of lex-ordering | Gröbner basis |
\( \langle 5, 7, 9, 11, 13\rangle \) | \(X_1 \prec X_3\prec X_2\prec X_4\prec X_5 \) | \( \{X_3^5-X_1^9, X_2X_1^4-X_3^3, X_2X_3^2-X_1^5, X_2^2-X_3X_1, X_4X_1-X_2X_3, X_4X_3-X_1^4, X_4X_2-X_3^2,\) |
\(X_4^2-X_2X_1^3, X_5X_1-X_4X_2, X_5X_3-X_2X_1^3, X_5X_2-X_1^4, X_5X_4-X_2^2X_1^2, X_5^2-X_4X_1^3 \} \) |