Because implications can only be transferred between candidates, it is not possible to omit the candidates without placing certain restrictions on the chain. For example, one can restrict the chain to cells which all have a candidate for a single digit. The so called X-Chain then shows implications between the candidates for the selected digit, without explicitly mentioning them. Another method by which candidates could be omitted can be seen in an XY-Chain. This type of chain is limited to cells which only have 2 candidates left. Although it is customary to name the digits involved, a reader, given the candidate grid, is able to verify the chain if only the initial candidate is given.
The implications in a chain can either travel in a single direction (from left to right) or in both directions. A chain that operates in both directions is considered more valuable than a unidirectional chain.
Besides single candidates, it is possible to include more complex structures, such as Almost Locked Sets into a chain. Several alternative structures have been discovered which can be included in chains. Most of these structures use the qualifier almost in their name.
Some people use terminology from Graph theory to name the components of chains. A cell or candidate is called a vertex or node and the link between them is an edge.
There are several notation systems for chains. These all look very similar. Once you've mastered one of the notation systems, it is not so difficult to read and interpret alternative notation systems. Some examples of chain notation systems are:
A chain by itself is not asolving technique. It is merely the proof that accompanies a solving technique. However, several named chain types are recognized by the Sudoku community as valid solving techniques.