# ApCoCoA-1:NC.Intersection

## NC.Intersection

Computes the intersection of two finitely generated two-sided ideals over a free associative K-algebra.

### Syntax

```NC.Intersection(Ideal_I:LIST, Ideal_J:LIST):LIST
NC.Intersection(Ideal_I:LIST, Ideal_J:LIST, DegreeBound:INT, LoopBound:INT, Flag:INT):LIST
```

### Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

Before calling the function, please set ring environment coefficient field K and alphabet X through the functions NC.SetFp(Prime) (or NC.UnsetFp()) and NC.SetX(X) respectively. Default coefficient field is Q. For more information, please check the relevant functions.

• @param Ideal_I: a list of polynomials generating a two-sided ideal in K<X>. Each polynomial in K<X> is represented as a LIST of LISTs, which are pairs of form [c, w] where c is in K and w is a word in X*. Unit in X* is empty word represented as an empty string "". 0 polynomial is represented as an empty list. For example, polynomial F:=xy-y+1 in K<x,y> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].

• @param Ideal_J: another list of polynomials in K<X>.

• @return: probably a Groebner basis of the intersection of Ideal_I and Ideal_J.

Since the algorithm used in this function is based on Groebner basis computation, we refer users to NC.BP or NC.GB for information about the following optional parameters:

• @param DegreeBound

• @param LoopBound

• @param Flag

#### Example

```NC.SetFp(); -- set default Fp=F2
NC.SetX(<quotes>xyz</quotes>);
F1 := [[1,<quotes>xy</quotes>], [1,<quotes>z</quotes>]];
F2 := [[1,<quotes>yz</quotes>], [1, <quotes>x</quotes>]];
F3 := [[1,<quotes>zx</quotes>], [1,<quotes>y</quotes>]];
Ideal_I := [F1, F2]; -- ideal generated by {xy+z, yz+x}
Ideal_J := [F2, F3]; -- ideal generated by {yz+x, zx+y}
NC.Intersection(Ideal_I, Ideal_J, 20, 25, 1);
[[[1, <quotes>zyzzz</quotes>], [1, <quotes>zzzyz</quotes>], [1, <quotes>yzz</quotes>], [1, <quotes>zzy</quotes>]], [[1, <quotes>yzyz</quotes>], [1, <quotes>zyzy</quotes>]], [[1, <quotes>zyzyyz</quotes>], [1, <quotes>yzyy</quotes>], [1, <quotes>zzyz</quotes>], [1, <quotes>zy</quotes>]],
[[1, <quotes>yzzyzy</quotes>], [1, <quotes>yzyy</quotes>], [1, <quotes>zzyz</quotes>], [1, <quotes>zy</quotes>]], [[1, <quotes>zzzzyzyy</quotes>], [1, <quotes>zzyyzy</quotes>], [1, <quotes>zzzyzz</quotes>], [1, <quotes>zzyz</quotes>]],
[[1, <quotes>zzyzyyyyz</quotes>], [1, <quotes>zyzyyyy</quotes>], [1, <quotes>yzzzyzy</quotes>], [1, <quotes>zzzyyyz</quotes>], [1, <quotes>yzyyz</quotes>], [1, <quotes>zzyyy</quotes>], [1, <quotes>zzyzz</quotes>], [1, <quotes>zyz</quotes>]],
[[1, <quotes>x</quotes>], [1, <quotes>yz</quotes>]]]
-------------------------------
Note the following information printed by the server shows it is a partial Groebner basis.
===== 25th Loop =====
Number of elements in (partial) Groebner basis G: 19 -- partial Groebner basis before being interreduced
Number of S-elements: 25/113 -- 25 S-elements have been check, and 113 unchecked S-elements
```