A sum cover is a subset S of an Abelian group (A,+) provided every element x in A can be expressed in at least one way as a sum x = s+t where s and t are in S. A sum cover is called strict if the additional condition is added that s is not equal to t.
For example consider the set {0,1,2} is a sum cover of the Abelian group (Z_5,+), the integers modulo 5, because
| n | s+t | |
|---|---|---|
| 0 | = | 0+0 |
| 1 | = | 0+1 |
| 2 | = | 0+2 |
| 3 | = | 1+2 |
| 4 | = | 2+2 |
Note that {0,1,2} is NOT a strict sum cover, because 0 cannot be expressed except as 0+0. A strict sum cover for (Z_5,+) is {0,1,2,3}.
| n | s+t | |
|---|---|---|
| 0 | = | 2+3 |
| 1 | = | 0+1 |
| 2 | = | 0+2 |
| 3 | = | 1+2 |
| 4 | = | 1+3 |
Two sum covers S and T are equivalent if there exists a holomorphy which maps the elements of S to T. A holomorphy is a mapping of the form h(s) = A(s) + B, such that A(s) is an automorphism on the group and B is any element of the group. Holomorphies have the property that the number of distinct sums is constant under their action.
Current research in sum covers includes the following questions.
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Those interested in this topic can contact either Mark Fitch or Robert Jamison.
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