LN SATUHAT~ Yz~TIT I~U 
Abstract 
Stephan-Yl ~n Solomon 
Let L be a lan~'uage over m vocmb~Ja~y V ~nd let de~ot~. 
by E (V) the s,t of ml! enuSv~lence relmtlon~ (partitions) 
on V. If ~ ~ E (V) and x ~ V then we sb~ll denote by ~(v) 
the cell of ~ containing the elmment 7. 
Defini%ion I. A p~rtition ~GE(V) is s~id to be smturated 
ev~-VY i (I ~imn) there exlst such elements x~ (J=l,..., 
~-l, i+~ ..... ~) that ~j ~ ~(7.~ ~d 7;...~|., ~i~i.f..~,~ L. 
Our purpose i.,~ to find (in the cese of m finite vocebu- 
I mry) th~ gre,~test smtnrmted p_-rtit~on Z of the Imn~.umme. 
Definition 2. Let~E(V) sod x, y~V. We shall s8y that 
x ~-domimatms y (x3~y) if for every string xI...x~E L 
wh~e xi= x there exist such ~lem~nts x~ . (~=I .... ,i-l, 
+~ ..... n) that ~2 ~'~d~' ' T' • "'~'/-~Y i*~'''x~ ~ L. 
We c~n ~wtroduce now the p~rtitlon ~* , cmlBed tb~ 
asterisk of the pertition ~ : 7~y If both 7 ~-~--~y ~nd 
yPx hold true. 
Theorem I, ~ is ~ smtt)rmt~d pmrtition if mnd only if 
is finer tbmn 
The connection between the ~sterisk ~nd the ~erivative 
of ~ pm~t~t~on is given by: 
In orQer that #'=~,~ it is necessary ~nd 
sufficient thmt 2 be saturated. 
• . o. By u~ing the no%~tlon:~ -J, and 
. we hav.e : 
~Theorem E. Thmrm mxlsts ? nmtur~ number n so that Z--~ 
(where ~ im the improper p~rt~tion of V). 
