hypergraph
A hypergraph or metagraph
is an ordered pair, or couple,
where
is the class of vertices of the hypergraph and
is the class of edges such that
, where
is the powerset
of
(the set of subsets of
)
and is also considered to be a class.
Remark 0.1
A hypergraph is as an extension of the
concepts
of a
graph, colored graph and multi-graph.
A finite hypergraph, with both

and

being sets, is also related to a
metacategory; therefore, it can also be considered as a special case of a
supercategory, and can be thus defined as a mathematical interpretation of
ETAS axioms.
Remark 0.2
A finite hypergraph can also be considered as an example of a simple incidence structure.
Note also that the more general definition of a hypergraph given above avoids well known antimonies of set theory involving `sets' of sets in the general case.
Remark 0.3
Many specific graph definitions (but not all) can be extended to similar specific hypergraph, or multigraph, definitions. For example, let

and

. Associated to any finite hypergraph is the finite
incidence matrix

where
For example, let

, where

and

. Defining

and

in the obvious manner (as they are listed in the sets), we have
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