<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
more	O
precisely	O
in	O
automata	B-Application
theory	I-Application
,	O
a	O
rational	B-Application
set	I-Application
of	O
a	O
monoid	O
is	O
an	O
element	O
of	O
the	O
minimal	O
class	O
of	O
subsets	O
of	O
this	O
monoid	O
that	O
contains	O
all	O
finite	O
subsets	O
and	O
is	O
closed	O
under	O
union	O
,	O
product	O
and	O
Kleene	O
star	O
.	O
</s>
<s>
Rational	B-Application
sets	I-Application
are	O
useful	O
in	O
automata	B-Application
theory	I-Application
,	O
formal	O
languages	O
and	O
algebra	O
.	O
</s>
<s>
A	O
rational	B-Application
set	I-Application
generalizes	O
the	O
notion	O
of	O
rational	O
(	O
regular	O
)	O
language	O
(	O
understood	O
as	O
defined	O
by	O
regular	B-Language
expressions	I-Language
)	O
to	O
monoids	O
that	O
are	O
not	O
necessarily	O
free	O
.	O
</s>
<s>
The	O
rational	O
subset	O
of	O
are	O
precisely	O
the	O
regular	B-General_Concept
languages	I-General_Concept
.	O
</s>
<s>
Indeed	O
,	O
the	O
regular	B-General_Concept
languages	I-General_Concept
may	O
be	O
defined	O
by	O
a	O
finite	O
regular	B-Language
expression	I-Language
.	O
</s>
<s>
McKnight	O
's	O
theorem	O
states	O
that	O
if	O
is	O
finitely	O
generated	O
then	O
its	O
recognizable	B-Application
subset	I-Application
are	O
rational	B-Application
sets	I-Application
.	O
</s>
<s>
Rational	B-Application
sets	I-Application
are	O
closed	O
under	O
morphism	O
:	O
given	O
and	O
two	O
monoids	O
and	O
a	O
morphism	O
,	O
if	O
then	O
.	O
</s>
<s>
The	O
intersection	O
of	O
a	O
rational	O
subset	O
and	O
of	O
a	O
recognizable	B-Application
subset	I-Application
is	O
rational	O
.	O
</s>
<s>
A	O
binary	O
relation	O
between	O
monoids	O
M	O
and	O
N	O
is	O
a	O
rational	O
relation	O
if	O
the	O
graph	O
of	O
the	O
relation	O
,	O
regarded	O
as	O
a	O
subset	O
of	O
M×N	O
is	O
a	O
rational	B-Application
set	I-Application
in	O
the	O
product	O
monoid	O
.	O
</s>
<s>
A	O
function	O
from	O
M	O
to	O
N	O
is	O
a	O
rational	O
function	O
if	O
the	O
graph	O
of	O
the	O
function	O
is	O
a	O
rational	B-Application
set	I-Application
.	O
</s>
