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freely generated inductive set∗
CWoo†
2013-03-22 2:42:23
In the parent entry, we see that an inductive set is a set that is closed
under the successor operator. If A is a non-empty inductive set, then N can be
embedded in A.
More generally, fix a non-empty set U and a set F of finitary operations
on U . A set A ⊆ U is said to be inductive (with respect to F ) if A is closed
under each f ∈ F . This means, for example, if f is a binary operation on U
and if x, y ∈ A, then f (x, y) ∈ A. A is said to be inductive over X if X ⊆ A.
The intersection of inductive sets is clearly inductive. Given a set X ⊆ U , the
intersection of all inductive sets over X is said to be the inductive closure of X.
The inductive closure of X is written hXi. We also say that X generates hXi.
Another way of defining hXi is as follows: start with
X0 := X.
Next, we “inductively” define each Xi+1 from Xi , so that
[
Xi+1 := Xi ∪ {f (Xin ) | f ∈ F, f is n-ary}.
Finally, we set
X :=
∞
[
Xi .
i=0
It is not hard to see that X = hXi.
Proof. By definition, X ⊆ X. Suppose f ∈ F is n-ary, and a1 , . . . , an ∈ X, then
each ai ∈ Xm(i) . Take the maximum m of the integers m(i), then ai ∈ Xm for
each i. Therefore f (a1 , . . . , an ) ∈ Xm+1 ⊆ X. This shows that X is inductive
over X, so hXi ⊆ X, since hXi is minimal. On the other hand, suppose a ∈ X.
We prove by induction that a ∈ hXi. If a ∈ X, this is clear. Suppose now
that Xi ⊆ hXi, and a ∈ Xi+1 . If a ∈ Xi , then we are done. Suppose now
a ∈ Xi+1 − Xi . Then there is some n-ary operation f ∈ F , such that a =
∗ hFreelyGeneratedInductiveSeti created: h2013-03-2i by: hCWooi version: h41666i Privacy setting: h1i hDefinitioni h03B99i h03E20i
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f (a1 , . . . , an ), where each aj ∈ Xi . So aj ∈ hXi by hypothesis. Since hXi is
inductive, f (a1 , . . . , an ) ∈ hXi, and hence a ∈ hXi as well. This shows that
Xi+1 ⊆ A, and consequently X ⊆ hXi.
The inductive set A is said to be freely generated by X (with respect to F ),
if the following conditions are satisfied:
1. A = hXi,
2. for each n-ary f ∈ F , the restriction of f to An is one-to-one;
3. for each n-ary f ∈ F , f (An ) ∩ X = ∅;
4. if f, g ∈ F are n, m-ary, then f (An ) ∩ g(Am ) = ∅.
For example, the set V of well-formed formulas (wffs) in the classical proposition logic is inductive over the set of V propositional variables with respect
to the logical connectives (say, ¬ and ∨) provided. In fact, by unique readability of wffs, V is freely generated over V . We may readily interpret the above
“freeness” conditions as follows:
1. V is generated by V ,
2. for distinct wffs p, q, the wffs ¬p and ¬q are distinct; for distinct pairs
(p, q) and (r, s) of wffs, p ∨ q and r ∨ s are distinct also
3. for no wffs p, q are ¬p and p ∨ q propositional variables
4. for wffs p, q, the wffs ¬p and p ∨ q are never the same
A characterization of free generation is the following:
Proposition 1. The following are equivalent:
1. A is freely generated by X (with respect to F )
2. if V 6= ∅ is a set, and G is a set of finitary operations on V such that there
is a function φ : F → G taking every n-ary f ∈ F to an n-ary φ(f ) ∈ G,
then every function h : X → B has a unique extension h : A → B such
that
h(f (a1 , . . . , an )) = φ(f )(h(a1 ), . . . , h(an )),
where f is an n-ary operation in F , and ai ∈ A.
References
[1] H. Enderton: A Mathematical Introduction to Logic, Academic Press, San
Diego (1972).
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