D IAGRAMMES

J. ROSICKÝ
Théories des bornes
Diagrammes, tome 2 (1979), exp. no 5, p. R1-R3
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Diagrammes, Vol. 2 , 1979-

THEORIES DES BORNES.
Chapitre J: An algebraic description of
ordinals.

J. Rosicky.

The class ON of ail ordinals will be described as a certain free algebra in the category of classes. It gives a way to
consider ordinals in more gênerai situations than in sets.
We will work in a given model of the Godel-Bernays set theory. Dénote by Cl
the category of ail classes in this model
with mappings which are classes as morphisms (of course, Ce
is not a category in our model), Having a class A , P(A) will
dénote the class of ail non-empty subsets of A . Then
P: Ce
f: A

> CZ is a functor where P(f) , for a mapping
> B , is given by the direct images (so, we hâve

P(f)(x) = £f(x) / x € x ] , f or any X £ P(A) ) .
Under a complète semilattice in Cl
we will mean a partially ordered class such that each non empty subset has a supremum. It is well known that complète semilattices are precisely couples (A,h) where A £ Cd and h: P(A)
> A is a
mapping such that

h( \ x } ) = x , for any x 6 A , and

h( U 3£ ) = h( { h(X) / X € 36 }

ï Ç P(A).

) , for any non-empty subset

R 2
A triple

(A,f,h) will be called an o-algebra if

(Afh) is a complète semilattice and f: A

^ A

A € Ce ,

a mapping

such that h( S x,f(x) } ) = f(x) (i. e. x 4 ( ))> f o r ajCïy
x € A. A mapping t: A
> A1 is a homomorphism of o-algebras
(A,f,h) and ( A S f S h 1 ) if t;f = f1 and hf.P(t) = t.h .
We hâve an o-algebra (ON,u,s) where u(x) a x + 1 , for
any x € O N , and s(x) = sup X , for any X C P(ON) .
f

THEOREM.

x

(ON,u,s) is a free o-algebra over one gênerator.

Proof• We will prove that 0 freely générâtes ON as an o-algebra. Let (A,f,h) be an o-algebra and a € A . We hâve to show
that there is a unique homomorphism

t: ON

-> A

of o-alge-

bras such that t(o) = a.
Define t by a transfinite recursion as follows: t(o) = a ,
t(x+l) = f.t(x) , for any x£ON, and t(x) = h.P(t)(x) (hère
x is understood as a set of ail smaller ordinals), for any limit x € ON . If x €_ ON , then t will dénote the restriction
x
of t on x+1 Ç ON . To see that t is a homomorphism of o-algebras it suffices to verify that each t : x+1
> A is a
homomorphism of partial o-algebras (i. e. préserves ail existing opérations). At first, we show that each t préserves ~ .
Indeed, if t does it then t A as wel3 because
'
x
x+1
t(x+l) = f.t(x) > t(x). If x is limit, the induction step is
évident•
It is clear that if t is a homomorphism of partial o-algebras
then t . as well. Assume that x is limit and t a homox+1
y
morphism for any y < x . Then h.P(t)(X) = t.s(X) for any
X £ P(x+1) such that s(X) < x . If s(x) = x , then either x
is the greatest élément of X or X is cofinal with x . Since t préserves ^ $ the same alternative holds for t(x)
and P(t)(x). In the first case, h.P(t)(X) = t(x) is immédiate and in the second one h.P(t)(X) = h.P(t)(x) = t(x) by the

H5
construction of

t . Since

t(x) = t.s(x) , t

is proved to be

a homomorphism.
The uniqueness of

t

is évident.

Remark. The just identified ordinal ob.ject ON , which is completely determined by "the calculus of the functor

P " , may be

compared with the natural number object in toposes (which is
the free unary algebra over one générât or) •

(à suivre J