Author: Torben Mogensen (---.diku.dk)
Date: 09-24-2003 06:17
While I enjoy your articles about finite structures, I cannot help but correct a few minor imprecisions in the text.
First, a connected, directed, acyclic graph is not necssarily a tree.
For example, the diamond-shaped graph
1
/ \
2 3
\ /
4
where the edges are all directed downwards (for lack of a good arrow symbol in ASCII) is not a tree. The simplest description of a tree is a rooted directed graph where a node has at most one edge pointing to it.
Which brings us to rooted graphs: A graph is rooted if there exist a vertex (the root) from which all other vertices are reachable. This is a very useful notion (especially for directed graphs).
Secondly, a ring is not just a group on top of a group. The multiplicative operator need not have an inverse and the additive operator must be commutative, so it is rather a semigroup on top of a commutative (Abelian) group. Furthermore, the two operators must obey the distributive laws: x*(y+z) = (x*y)+(x*z) and (y+z)*x = (y*x)+(z*x).
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