[Inquiry] Re: Relations In General

[Jon Awbrey]

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RIG.  Note 4

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3.4.31.3.  Numerical Incidence Properties of Relations

Of particular interest are the local incidence properties of relations
that can be calculated from the cardinalities of their local flags, and
these are naturally enough called "numerical incidence properties" (NIP's).

For example, L is said to be "c-regular at j" if and only if
the cardinality of the local flag L_x at j is c for all x in X_j,
coded in symbols, if and only if |L_x at j| = c for all x in X_j.

In a similar fashion, one can define the NIP's "<c-regular at j",
">c-regular at j", and so on.  For ease of reference, I record a
few of these definitions here:

L is  c-regular at j   iff   |L_x at j| = c for all x in X_j.

L is <c-regular at j   iff   |L_x at j| < c for all x in X_j.

L is >c-regular at j   iff   |L_x at j| > c for all x in X_j.

The definition of a local flag can be broadened from a point x in X_j
to a subset M c X_j, arriving at the definition of a "regional flag".
Suppose that L c X_1 x ... x X_k, and choose a subset M c X_j.  Then
"L_M at j" denotes a subset of L called "the flag of L with M at j", or
"the M at j-flag of L".  The regional flag L_M at j is defined as follows:

L_M at j  =  {<x_1, ..., x_j, ..., x_k> in L  :  x_j in M}.

Returning to 2-adic relations, it is useful to describe some
familiar classes of objects in terms of their local and their
numerical incidence properties.  Let L c S x T be an arbitrary
2-adic relation.  The following properties of L can be defined:

L is "total" at S     iff   L is (>=1)-regular at S.

L is "total" at T     iff   L is (>=1)-regular at T.

L is "tubular" at S   iff   L is (=<1)-regular at S.

L is "tubular" at T   iff   L is (=<1)-regular at T.

If L c S x T is tubular at S, then L is called a "partial function"
or a "prefunction" from S to T, sometimes indicated by giving L an
alternate name, say, "p", and writing L = p : S ~> T.

Just by way of formalizing the definition:

L = p : S ~> T   iff   L is tubular at S.

If L is a prefunction p : S ~> T that happens to be total at S, then L
is called a "function" from S to T, indicated by writing L = f : S -> T.
To say that a relation L c S x T is totally tubular at S is to say that
it is 1-regular at S.  Thus, we may formalize the following definition:

L = f : S -> T   iff   L is 1-regular at S.

In the case of a function f : S -> T, one
has the following additional definitions:

f is "surjective"   iff   f is total at T.

f is "injective"    iff   f is tubular at T.

f is "bijective"    iff   f is 1-regular at T.

Jon Awbrey

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