DEFINITION of NRC, NRCT and NRCZT NETSLet it be clear that I don't like nets and I consider that the nrcztnets I'm going to define below are not really useable patterns for a human player, at least in their most general form. But are the most general chains of ALSs (and other AAAAAHHHHAAAHHAAAHHHH LSs) really useable?
Anyway, as this may be a matter of taste and I already spoke of them several times, saying that it is easy to define them, the least I can do is give such a definition. Although this is intuitively obvious, we need a few technicalities if we want to formalise it unambiguously.
Remarks:

as was the case for nrcztchains, nrcztnets do not rely on subsets and on links between subsets;
 given a candidate C, nrcztchains were the most general first order chain patterns that could justify the elimination of C; nrcztnets are the most general first order net patterns that can justify the elimination of C; ("first order" means not relying on subsets);
 the following definitions can be restricted to nrct or nrc nets; they can also be restricted to any of their 2D counterparts.
If you don't know the basics of graph theory, it may be appropriate to start by reading the following elementary introduction:
http://en.wikipedia.org/wiki/Graph_%28mathematics%29An nrcztchain was defined as a sequence (i.e. a completely or linearly ordered set) of 2Dcells (each in either of the rc, rn, cn or bn spaces). It can also be seen as a sequence of (alternatively left and right linking) candidates, but the easiest way to generalise to nets is using the cell view.
Now, as an informal introduction to the following definitions, let me just say that they provide for the possibility for having several right linking candidates in some of the cells and they explain how they can be dealt with. This is very far in spirit from allowing subsets in chains (such as in Grouped NLs).
Definition: an oriented 2Dcell is a 2Dcell whose candidates are labelled according to the rules:
 it has one and only one leftlinking candidate (or llc)
 it has one or more rightlinking candidates (or rlc)
 it has zero or more additional candidates
(given a 2Dcell in a real puzzle, its candidates can be labelled in different ways as llc, rlc or additional, but a choice has to be made for a net to be completely specified; notice that, at this stage, additional candidates are NOT labelled as t or z)
Definition: a 3Dnet is a partially ordered set (called a poset by mathematicians) of oriented 2D cells from any of the 2Dspaces, i.e. either rc, rn, cn or bn spaces (notice that, as I speak of a set and not a bag, these cells are different); this structure must satisfy the following conditions:
 if an oriented 2Dcell has more than one rightlinking candidate, it is called branching,
 if an oriented 2Dcell has more than one immediate predecessor in the poset, it is called merging,
(notice that a cell can be both merging and branching)
 if an oriented 2Dcell has no predecessor in the poset it is called a source,
 if an oriented 2Dcell has no successor in the poset it is called a sink ,
 if an oriented 2Dcell is neither branching nor merging nor a source nor a sink, it is called regular,
 for every oriented 2Dcell C1 (but those in the sinks), each of its rightlinking candidates has a unique distinguished nrclink to the left linking candidate of an immediate successor of C1 in the poset;
(notice that any candidate may have several nrclinks (e.g; along a row and a block) to another given oriented 2Dcell , but one has to be chosen for a net to be completely specified);
(this is the main connectivity condition);
 if different rightlinking candidates in possibly different oriented 2Dcells have their distinguished nrclinks pointing to the (unique) leftlinking candidate of the same oriented 2Dcell C2, then C2 is a merging cell;
 notice that it is not necessary to add a condition on leftlinking candidates being nrclinked to rightlinking ones: this is already part of the definition of 2Dcells;
With a 3Dnet, one can associate a directed acyclic graph (DAG):
 edges: the edges of the graph are all the couples (oriented 2Dcell C, leftlinking candidate of C) and (oriented 2Dcell C, rightlinking candidate of C)
(instead of taking merely all the left and all the right linking candidates, these technicalities provide for the possibility for a candidate to appear several times, but only in different paths);
 vertices:
 for every oriented 2Dcell with leftlinking candidate C1 and for each of its rightlinking candidates C2, the graph has a vertex with head C1 and tail C2;
 for every oriented 2Dcell C, for each of its rightlinking candidates C1 and for the distinguished nrclink of C1 to an oriented 2Dcell C2, the graph has a vertex with head C1 and tail the leftlinking candidate of C2;
 the graph has no vertex other than those defined in the above three clauses.
With these definitions, only leftlinking candidates are merging points and branching occurs only with the existence of several rightlinking candidates.
Notice that the additional candidates are not considred as belonging to this graph.
Definitions: a target of a 3Dnet is any candidate that is nrclinked to the leftlinking candidates of all its sources and to all the rightlinking candidates of all its sinks.Remarks:
 as was the case for the other 2D or 3D chains I have introduced, the target does not belong to the net;
 a 3Dchain is a 3Dnet with a single source, a single sink and no branching or merging; said otherwise, its branching factor is 1;
 2D chains implied a restriction on their first cell: it had to be bilocal or bivalue (this is not an arbitrary restriction, it is a consequence of the 2D constraints); nrcztchains somehow relax this restriction, as their first cell may contain additional zcandidates, although they are rarely needed in practice; with nrcztnets, multiple rightlinking candidates (branching) allow to use any cell as a first cell, and having a set of sources allows having several first cells.
Definition: path, maximal path, initial path. Given a 3Dnet,
 a path between two left or right linking candidates C1 and Cn is any path in the associated graph;
 a maximal path is a path in the associated graph, between the leftlinking candidate of a source and the rightlinking candidate of a sink
 given a candidate C, an initial path to C is any path with head the leftlinking candidate of a source and with tail C.
A path thus defined is a 3Dchain.
Given a path, a rightlinking candidate in a cell that has more than one is called a branched candidate.
Given 2 candidates, in a 3Dchain there is a single path between them. But in a 3Dnet, there may be 0, 1 or more paths beween them. This is the main structural difference between a chain and a net.
Given a candidate, there may be 1 or several initial paths to it.
As was the case for for 2D or 3D chains, only some types of 3Dnets are interesting (those that lead to a contradiction on the target). These conditions are generalisations of the condition on nrc(z)(t) chains: the nrcbivalue (bivalue/bilocation) condition modulo the previous rightlinking candidates and target.We just have to be careful when we formalise them.
Definition: an nrcztnet built on a candidate Z is a 3Dnet admitting Z as a target (in the general sense defined above) and which satisfies the nrczt condition:
 for any cell in the net, for any rightlinking candidate C in this cell, for ANY initial path to C, this 3Dchain would be an nrcztchain or an nrcztlasso if, at any branching point in this chain, we forgot the other rightlinking candidates.Notice that, as there may be several initial paths to C, each additional candidate may have different justifications along each of these paths (it may appear as a tcandidate in some paths and as a zcandidate in others). This is why additional candidates were not a priori labelled as t or z.
Theorem (nrcztnet theorem): given an nrcztnet based on a target Z, Z can be eliminated.
Notice that nrcztnets are
nonanticipative patterns.
Enough for one day.
In a next post, I'll prove that nrcztnets (together with ECP and Singles) subsume all the rules defined in my book and programmed in SudoRules:
 Subsets (Naked, Hidden and SuperHidden),
 elementary interactions row/block and column/block
 all the (h)xy(z)(t) and nrc(z)(t) chains
But, remember the first sentence of this post: this is not to mean that nrcztnets should replace all the other patterns.
Edited 02/04/08: improved the definition of a 3Dnet.
Edited 02/05/08: added the possibility of lassos in nrcztnets.