Given $k$ sets of objects $X_1$, ..., $X_k$, a
$k$-ary relation $R$ on $D(R) = (X_1, \ldots, X_k)$ is a
subset $G(R)$ of the Cartesian product $X_1 \times \cdots
\times X_k$. We refer to $D(R)$ and $G(R)$
as the domain and the graph of the relation,
respectively (alternative notions are that of ground and
figure, respectively). We also refer to $s = (s_1, \ldots,
s_k)$, where each $s_i$ gives the cardinality of $X_i$, as
the size of the relation. Strictly speaking, the relation is the pair $(D(R), G(R))$;
often, relations are identified with their graph. We say that a
$k$-tuple $t$ is contained in the relation $R$ iff
it is an element of $G(R)$.
The characteristic function $f_R$ (sometimes also referred
to as indicator function) of a relation $R$ is the predicate
(boolean-valued) function on the Cartesian product $X_1 \times
\cdots \times X_k$ such that $f_R(t)$ is true
iff the $k$-tuple $t$ is in $G(R)$.
Relations with arity 2, 3, and 4 are typically referred to as
binary, ternary, and quaternary relations,
respectively. An endorelation on $X$ (or binary relation
over $X$) is a binary relation with domain $(X, X)$.
See predicates for the most important
types of endorelations.
Relations with the same domain can naturally be ordered according to
their graphs. I.e., $R_1 \le R_2$ iff $G(R_1)$ is a subset of
$G(R_2)$, or equivalently, if every $k$-tuple $t$
contained in $R_1$ is also contained in $R_2$. This induces a
lattice structure, with meet (greatest lower bound) and join (least
upper bound) the intersection and union of the graphs, respectively,
also known as the intersection and union of the
relations. The least element moves metric on this lattice is the
symmetric difference metric, i.e., the cardinality of the
symmetric difference of the graphs (the number of tuples in exactly
one of the relation graphs).
The complement of a relation $R$ is the relation with
domain $D(R)$ whose graph is the complement of $G(R)$, i.e.,
which contains exactly the tuples not contained in $R$.
For binary relations $R$, it is customary to write $x R y$ iff
$(x, y)$ is contained in $R$. For binary relations $R$
and $S$ with domains $(X, Y)$ and $(Y, Z)$, the
composition of $R$ and $S$ is defined by taking
$x S z$ iff there is an $y$ such that $x R y$ and
$y S z$. The dual (or converse) $R^*$ of the
relation $R$ with domain $(X, Y)$ is the relation with domain
$(Y, X)$ such that $x R^* y$ iff $y R x$.
Package relations implements finite relations as an S3 class
which allows for a variety of representations (even though currently,
only dense array representations of the incidences are employed).
Other than by the generator,
relations can be obtained by coercion via the generic function
as.relation, which has methods for at least logical and numeric
vectors, unordered and ordered factors, arrays including matrices, and
data frames. Unordered factors are coerced to equivalence relations;
ordered factors and numeric vectors are coerced to order relations.
Logical vectors give unary relations (predicates). A (feasible)
$k$-dimensional array is taken as the incidence of a $k$-ary
relation. Finally, a data frame is taken as a relation table. Note
that missing values will be propagated in the coercion.
Basic relation operations are available as group methods: min
and max give the meet and join, and range a
relation ensemble with these two.
Comparison operators implement the natural ordering in the relation
lattice. Where applicable, ! gives the complement, &
and | intersection and union, and * composition,
respectively. Finally, t gives the dual.
There is a plot method for certain
endorelations provided that package Rgraphviz is available.
There is also the notion of fuzzy relations, for which each tuple is
contained in the graph with a certain membership value. (I.e., the
graph is a fuzzy subset of the Cartesian product of the elements of
the domain.) Basic support for fuzzy relations will be added
eventually.