Definition of Finite Random Categories
To give a category it means to give:
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DEFINITION OF FINITE RANDOM CATEGORIES To give a category
~
it rneans to give:
a) A set Ob(t) whose elements are called the objects of
e; b) For every pair of objects X,Y E Ob(e) a set
Home(X,Y) {or simply Hom(X,Y) ) whose elements are called morphisms (or arrows) from X to Y i.e. with the source X and ending Y. In denoting the source and the ending of an arbitrary morphism u
by S(u.) and respectively E(u,), given that uEHom~(X, Y) one
gets S(u}=X and
E(u)= Y. The sets Home(X, Y) are mutuallydisjoint,
i.e. every morphism has a single source and a single ending; c) For every three objects
X,Y,ZEOb(e)
an appli-
cation
Hom-e(X,Y)xHome(Y,Z) __. Home(X,Z) called the composi tion of the morphisms which associates wi th every pair of morphisms from
u.EHom-e,(X, Y), \)"EHomeCY,Z) one morphism
Home(X,Z) being denoted by
V"0 u.
Given that category
e
or
V"U..
is associative the cornpo-
sition of the morphisms is associative,
no matter what the morphisrns
S. Guiasu, Mathematical Structure of Finite Random Cybernetic Systems © Springer-Verlag Wien 1972
9
Identical Morphism. Dual Category
u.EHome(X,Y) ,
U"EHome(Y,Z) ,
Let us suppose Category
'lltEHom~(Z,U).
e to
be a category with
identical morphisms if for every object XE Ob(t:) there exists a morphism
1xEHome(X,X) called the iclentical morphism of the ob-
ix0 u. = u.
ject X , or the identity of X , so that
='l>'
'\t
X. We shall denote an ar'hitrary morphism
by
v 01x
u. with the ending X and every morphism
for every morphism with the source
and
u.EHomeCX,Y)
u.:X___.Y or frequently by
X~Y. We sball also use .M.(e) to denote tbe set of all
e'
morphisms of the category
cM,(f)
i.e.
= UHome(X,Y)
where the union is taken over all objects
X,Y from Ob(t). Obvious-
ly, in a category with identical morphisms there is one-to-one correspondence between the objects of the category dentical morphisms X ,. If be denoted by
e
and the i-
., 1x •
is a category, let the dual category of
e0 ,defined
as follows such that
a) Ob(e 0) = Ob(t) ; b)
e
Homy:O(X,Y) = Home(Y,X)
whatever be the objects X , Y ;
e
10
Definition of FR - Categories c) the composi tion of
to the composi tion of u. A category
and ~
I
\1
in
and u. in
tr
e.
e0 is
is called a subcategory of
equal
e
if
a) Ob(e') c Ob(e) ;
for every pair of objects
X, Y from Ob(e 1) ;
c ) The composition of the morphisms in
~I
~
is in-
dicated, is induced, by the composition of the morphisms in A subcategory
e
I
of
e
e.
is called a full subcate-
gory if
Hom ~AX, Y) = Hom oe(X, Y) whichever be the pair of objects X, Y from Ob(e1 ) A
•
subcategory
e
of 'e is called a rich category
Ob(e 1)
=
Ob(e) .
I
if
Obviously, a subcategory
e
1
of the category
e
which is at the same time both rich and full will coincide with
e. We shall say that a category is an FR-category of ~ -~ (finite random category of ~-type) if: a) The objects are finite sets;
11
Randorn Morphisrn
b) The rnorphisrns are stochastic rnatrices, i.e. no matter what the objects X
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