The collected mathematical papers of Arthur Cayley.

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A MEMOIR ON ABSTRACT GEOMETRY.

459

14. Any two or more relations may be composed together, and they are then
factors of a single composite relation; viz. the composite relation is a relation satisfied
if, and not satisfied unless, some one of the component relations be satisfied.
15. The foregoing notion of composition is, it will be noticed, altogether different
from that which would at first suggest itself. The definition is defective as not
explaining the composition of a relation any number of times with itself, or elevation
thereof to power; which however must be admitted as part of the notion of composition.
16. A k-fold relation which is not satisfied by any other k-fold relation, and
which is not a power, is a prime relation. A relation which is not prime is composite,
viz. it is a relation composed of prime factors each taken once or any other number of
times; in particular, it may be the power of a single prime factor. Any prime factor
is single or multiple according as it occurs once or a greater number of times.
17. A relation which is either prime, or else composed of prime factors each of
the same manifoldness, is a regular relation; a k-fold relation is ex vi termini regular.
An irregular relation is a composite relation the prime factors whereof are not all of
the same manifoldness.
18. A prime k-fold relation cannot be implied in any prime k-fold relation different
from itself. But a prime k-fold relation may be implied in a prime more-than-k-fold
relation,-or in a composite relation, regular or irregular, each factor whereof is more
than I-fold; and so also a composite relation, regular or irregular, each factor whereof
is at most k-fold, may be implied in a composite relation, regular or irregular, each
factor whereof is more than k-fold. In a somewhat different sense, each factor of a
composite relation implies the composite relation.
19. A composite relation is satisfied if any particular one of the component relations
is satisfied; but in order to exclude this case we may speak of a composite relation as
being satisfied distributively; viz. this will be the case if, in order to the satisfaction of
the composite relation, it is necessary to consider all the factors thereof, or, what is the
same thing, when the reduced relation obtained by the omission of any one factor whatever is not always satisfied. And when the composite relation is satisfied distributively,
the several factors thereof are satisfied alternatively; viz. there is no one which is
throughout unsatisfied.
20. A composite onefold relation is never distributively implied in a prime k-fold
relation-that is, a prime k-fold relation implies only a prime onefold relation, or at
least only implies a composite onefold relation improperly, in the sense that it implies
a certain prime factor of such composite onefold relation. Conversely, every k-fold relation which implies distributively a composite onefold relation is composite.
21. Any two or more relations may be aggregated together into, and they are then
constituents of, a single aggregate relation; viz. the aggregate relation is only satisfied
when all the constituent relations are satisfied. The aggregate relation implies each of
the constituent relations.
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