In this section we shall identify the states of any extensional
Chu space (*A*,*r*,*X*) with the dual words representing them. Thus instead
of an extensional Chu space being (*A*,*r*,*X*) with
being injective, we have just (*A*,*X*) with *r*(*a*,*x*) being defined
implicitly as *x*(*a*) (application of
to ).

Define a *property* of a Chu space (*A*,*X*) to be any superset of *X*
that is a subset of .
The discrete Chu space
has only
the one property, namely itself, which we identify with the vacuous
property .
In general the properties of
form the set
,
namely all ways of adding new columns to .

Given two sets *A* and *B*, a function
induces a map
of properties sending the property
to the property
.
This is true independently of the choice of
making *A*
into a Chu space (*A*,*X*). When *F* sends every property of
to
a property of ,
i.e. when
implies
,
we call *f* a *homomorphism* of Chu spaces. The following is
easily seen.

**Proof:** If *f* is a homomorphism from (*A*,*X*) to (*B*,*Y*) then it
sends the property *X* to a superset of *Y*. But this is equivalent to
saying that for every column ,
is in *X*, whence *f*
is continuous.

Conversely, suppose *f* is continuous. Let
be a
property of .
We wish to show that
is a
superset of *Y*. But this is just the requirement that every
satisfies
.
Since *f* is continuous we have the
stronger property that
.

This notion of property is quite abstract, and it will therefore be helpful to develop some intuition about it to demonstrate its relationship to more conventional notions of property.

To begin with, its generality notwithstanding, this notion of property
is internal to the Chu space being described, since it is defined in
terms of a fixed carrier *A* and alphabet .
We cannot use it
to express properties that refer to other Chu spaces, such as the
property of being the smallest Chu space meeting some condition.

Some intuition for the scope of this concept of property can be built
up as follows. We call a property *atomic* when it excludes
exactly one state. The intersection of two atomic properties excludes
both their respective states and thus corresponds to conjunction,
yielding a compound (non-atomic) property. By allowing arbitrary
conjunction we can express any property of a Chu space
as a
conjunction of its atomic properties; in particular
itself can be
expressed as the conjunction of all its atomic properties, namely those
states absent from .

When , a natural presentation of an atomic property is as where is a list of the points projected by the property to 1 and lists the remaining points, those projected to 0. If we interpret points as propositions that may be either true (1) or false (0), and take to be the conjunction of its members and the disjunction, then is the logical expression of this atomic property, since it is false in the state excluded by that property and true in all other states.

This language generalizes to larger alphabets by providing a region for
each letter. With three letters one could write something like
,
where every letter appears in
exactly one ,
and describes the state in which each letter in
has value *i*.

Every property of a Chu space can be expressed as a conjunction of atomic properties of , which therefore can serve as the constants of a description language for . But if we insist on atomicity for the constants of our language, it may be unnecessarily rich.

Consider a Chu space having
atomic properties whose excluded
states differ from each other only at one element *a*, which is assigned a
different letter of
in each state. Instead of listing all
of these properties, we could simply take one of them and drop *a*
from it, which would say the same thing. More generally if the space
had
atomic properties which collectively were independent
of *n* points in the same way, we could again simply take one of those
properties and drop any mention of all *n* of those points.

These considerations lead to the notion of an *axiomatization* as
a set of properties, not necessarily atomic, whose conjunction is the
property *X* of being the Chu space (*A*,*X*).

We can take this notion a step further by defining a *language* for
a set *A* to consist of a set of possible properties of *A*. Each such
property is some subset of .
The purpose of such a language
is to specify Chu spaces in a more conventional way than simply by
giving the whole matrix, namely as the conjunction of the available
properties.

For example, take
and regard states as subsets of *A*. We
can specify those Chu spaces representing a partial ordering of the set
*A* by taking the language to consist of one property for each pair
(*a*,*b*) in *A*^{2}. The property associated with (*a*,*b*) would exclude
all states containing *a* but not *b*, and hence express the
relationship .
A partial order can now be defined on *A* by
forming the conjunction of those
properties sufficient to
express that order.

The advantage of this particular language is that each property
can be named by naming just *a* and *b*, which for large *A*
requires many fewer bits of information than needed to identify even
one state let alone the many needed to identify the whole partial
order.

For another example, still with ,
take the language to
consist of one property for each triple (*a*,*b*,*c*), namely the property
whose states are all those satisfying .
With this language
we can equip *A* with the structure of a join-semilattice by listing
one such property for each pair *a*,*b* in *A*, choosing *c* to be
whatever the join of *a* and *b* happens to be in that semilattice.

If we list only a subset of those properties then we will have
specified a partial join-semilattice, one for which the join operation
is defined only for some pairs. Continuous functions from such a
structure will preserve only those joins that exist. In particular we
can specify an arbitrary partial order by listing only those triples
(*a*,*b*,*c*) such that *a* and *b* are comparable in that order, in which
case *c* will always be the larger of *a* and *b*, being their improper
join. Continuous functions from such a structure will then preserve
only the partial order, that is, they will simply be monotone
functions. Even if two incomparable elements happen to have a least
upper bound in that partial order, continuous functions need not
preserve that least upper bound, i.e. they will not recognize least
upper bounds as being ``part of the signature.''