University of Paderborn Department of Mathematics Diploma Thesis ...

46 CHAPTER 3. OPERATIONS ON STABLE IDEALSDefinition 3.1. Let x A := x a 00 · . . . · x arrby{R(x A ) := x a 00 · . . . · x at−1t · x a t+1+1t+1 · . . . · x arrthe right-shift of the monomial x A .∈ M, r < n. Then we call the set R(x A ) defined}| t = 0, . . . , rNote that the set R(x A ) always includes the monomial x a 00 · . . . · x ar−1r · x r+1 . Furthermore,if we have a t = 0, we use a l instead of a t , where l < t is the maximal index, such that a l > 0.Definition 3.2. Let x A := x a 00 · . . . · x arr ∈ M. Then we call the set L(x A ) defined by{}L(x A ) := x a 00 · . . . · xsas+1 · x a s+1−1s+1 · . . . · x arr | s = 0, . . . , r − 1the left-shift of the monomial x A . For consistency, as we will see later, we set L(1) := ∅.Again, we use the convention that if a s+1 = 0, we use a m instead, where m > s + 1, m < n,is the minimal index, such that a m > 0.Remark 3.3. By the Definitions 3.1 and 3.2, we have that for each monomial x AR(x A ) ∩ L(x A ) = ∅and none of the sets R(x A ) and L(x A ) contains the monomial x A itself. Furthermore, notethat the degree of all monomials in the sets R(x A ) and L(x A ) equals the degree of themonomial x A . The left-shift of any monomial of the form x k 0, k ∈ N 0 , is L(x k 0) = ∅. Thisfact will be important, when we consider contractions and expansions in the next section.To get used to the sets R(x A ) and L(x A ) and to see, why they are called right-shift respectivelyleft-shift, we have a look at an example.Example 3.4. Consider the monomial x 2 0x 2 ∈ K[x 0 , x 1 , x 2 , x 3 ]. In the terminology of thedefinition of the set R(x 2 0x 2 ), we obtain for t = 0 the monomial x 0 x 1 x 2 , for t = 1 themonomial x 0 x 2 2 and finally for t = 2 the monomial x 2 0x 3 . Thus, we getR(x 2 0x 2 ) = { x 0 x 1 x 2 , x 0 x 2 2, x 2 0x 3}.Now we determine the elements of the left-shift L(x 2 0x 2 ). For s = 0, we get the monomialx 3 0 and s = 1 provides x 2 0x 1 . Hence, the left-shift of x 2 0x 2 isL(x 2 0x 2 ) = { x 3 0, x 2 0x 1}.In the lexicographic order >, we get for all monomials in R(x 2 0x 2 ):x 2 0x 2 > x 0 x 1 x 2 , x 2 0x 2 > x 0 x 2 2, x 2 0x 2 > x 2 0x 3 ,