We now continue with the demo from Subsection
In[14]:= WhatAreNumbers[]
Out[14]= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} In[15]:= ColumnForm[ WhatIsHistory[Out[14]] ] Out[15]= {1, x ** x -> a, {0, 0}, {}} {2, x ** y -> a, {0, 0}, {}} {3, x ** y -> b, {0, 0}, {}} {4, x ** x ** x -> b, {0, 0}, {}} {5, x ** a -> a ** x, {1, 1}, {}} {6, a ** y -> a ** x, {1, 2}, {5}} {7, x ** b -> a ** x, {1, 3}, {6}} {8, a ** x -> b, {1, 4}, {}} {9, a ** a -> b, {1, 4}, {1, 7, 8}} {10, b ** x -> b, {4, 1}, {1, 9}} {11, b -> a, {2, 3}, {}} |
We now describe what the above output means. Notice that each line is a number followed by a
replacement rule followed by a pair of numbers followed by a list of zero or more numbers. The tuples
mean
{relation number, relation, the 2 parents of thr relation, which rules relations were applied to the
S-polynomial}
For example, we say that x**x → a is the first replacement rule, x**y → a is the second replacement rule,etc. The third entry is the two parents of rule There are two cases:
Note that one can use history to do two things. The first thing is to see how relations are derived. The second is to find a smaller subset of the relations mentioned which will generate the same ideal. Note that in the above example, x**a-a**x lies in the ideal generated by x**x-a can be seen by referring to only the above history. Also, note that one can see that a **y - a **x lies in the ideal generated by {x **x - a, x **y - a and x **a - a **x}. This will prove very helpful in finding a subset of a particular set of a generated relations R which generates the same ideal as R. (See Section 30).
Consider the following tree on the nodes {2,3,6,7,8,9}.
Let us suppose that this tree represents the history corresponding to {p2,p3,p6,p7,p8,p9}. That is, p8 is generated as an s-polynomial from p6 and p7, p6 is generated as an S-polynomial from p7 and p9 and p7 is generated as a reduced S-polynomial. The polynomial generated is from p3 and p9 and a reduction step using p2 was used. In the notation of the history output, this would take the form
{
{2,p_2,{0,0},{}}, {3,p_3,{0,0},{}}, {6,p_6,{7,9},{}}, {7,p_7,{3,9},{2}}, {8,p_8,{6,7},{}}, {9,p_9,{0,0},{}}, } |
One sees from the picture that