. Looally G-free ~ g e b r a s are defined by the following axioms: 1) ~ (x~ . . . . .
xnd=q~dx~o,...,
x,~o),
2) r
.....
x . i ) = q ~ ( y ~. . . . .
3)
.....
0 ~ S~, i = 1 . . . . .
s;
ni
4) &, V
f(x,
Yn~)--~ V A z~=y~o, ~=~ . . . . .
O~S~j=l
~;
.....
xz,...,Xm)
~ x,
where
The f o r m u l a ~(x~ . . . . , x~) i s in the expressions Ni(x),
(VYl . . . . . Yni)(X@(pi >( (Yl . . . . . Y;ni)), \
f is
an arbitrary
term containing
x.
called ~r [3], if it is formed using the connectives Em p , Dp, m R ( x ~ , . . . ,Xn) , w h e r e p = { i t , .. ., &}~--{l . . ... s}, N~(x)~-~
Np(x)~Nil(X)
d~ . . . a~u
E p - ~ ( U g l . . . . . Ym) X ( ~ \
Np(yi)&
A
where
fi,
{x~ . . . . .
gj,
and hk are
terms
in
x ~ . . . . } = .@J, a n d v a r i a b l e s
m}, 5 ~ { 1 ,
. . . , s}.
The e l e m e n t
a is
the
variables
in
{x~ . . . . .
irreducible
{xt, . . . , x~, yt . . . . . xa~. . . . } do n o t in
the
y~}, {xa~ . . . . .
appear
algebra
~,
in
if
xai . . . .
}N
]~, g~, hG y~, e~,~{l . . . . .
~Nt(a)&...&N~(a).
THEOREM 1 [ 3 ] . For any formula @xt, . . . , x~) i n a n RPC o f s i g n a t u r e ~ , we c a n e f f e c t i v e l y construct a standard f o r m u l a ,%(x~ . . . . , x,~) , e q u i v a l e n t to the original formula with respect to the elementary theory of the class of all locally S-free algebras.
Remark [3N .
1--
In fact,
For any locally S-free algebra
by axioms
2) a n d 4) , f o r
ynplyj~l,t<~]~n~}
x )}[=[{
t---]~l.
any
i~{t,
~
and any set p = { i t , . . . , i~}ve{t . . . . .
..., s} I l z / / ~ l ~
On t h e o t h e r
hand,
s} I N ~ ( ~ ) [ -
(~Ay~, . . . . yni)((p~(y~ . . . . . Yni) =
{x[~
(~y~ . . . . .
Ynr
.....
gnu)=
if Let hk are
Rs(x~ . . . . .
terms
x~, !1~. . . . . Yrn,)~(~ym,+~ . . . . . Ym)( A x~i=/=gi& A Y~h=/=hh& ~ Np.~(y~)~, w h e r e g i a n d \ i=l / in variables i n t h e s e t {x~, . . , x~,y~, . . . , y ~ } , a n d i f P i = @, t h e n N p i ( x ) - ~ . x = x .
LEMMA 1. Let n be a natural number. Then for any formula of the form R s ( x l , . . . , x ~ Y~,'-',Ym') we can effectively find a natural number n(Rs), such that for any locally S-free algebra ~ with n irreducible elements, and for any set at, ..., a ~ N , either I{
Rs (a~. . . . .
az, yi . . . . . Ym')} I ~ n (Rs), o r
1{
Ym'> ] ~ ~ Rs (aa . . . . .
a~, yi . . . . .
Fro')} l = [ ~1.
869
Proof. For any i~{I .... , m'} , let Pi = {1,...,s}. If m' ~< n, then as n(Rs) we may take the number of displacements of the n elements by m'. If n ~< m', then as n(Rs) we may take the number of permutations of m' elements. Then there exists i~[i, ..., m'} such that pi T x {1,...,s}.
Then let
~0~{i ..... s } ~ p ~ .
We show that for any
Ym')l~ ~ Rs (al, ..., az, Yl, ..., Ym')}=/=~, then
]{(yl .... Ym,>13~
3
and a~..... a ~
, if {
Rs(al ..... az, yl .... ym')}[=131.
In
r,t
fact, let bl .... ,bm be realizations formula Rs(~l,...,al, Yl,-..,Ym') Let A =
of the subformula Aa~i=/=gi&
(see Remark
I).
of the
with respect to the variables Yl,...,Ym in the model N.
{z[([~zl, ..., z~)(z= (pio(zI..... zh))] . Clearly,
i~ and [ A I = [31
Ayvh~hh&AN~(~0
any element of A realizes Npi(y i) for i =
By Lemma 2 of [3], only a finite number of elements of A
do not realize, with respect to yiT , a formula obtained from A a~i=/=gi& A Y~hs/=hk&
~ Npi(yi) ~i
by replacing the variables Yl ..... ff~-1, Y~+1 .... , g m the statement of the proposition.
by bl, ...,bib_l, bi~+1, ..., b m .
Thus we have
Remark 2. For any formula of the form Rs(x I..... xz, ffl,...,Y,,') , we can effectively find a natural number n(Rs) such that for any locally S-free algebra 3 with a set of irreducible elements of cardinality l~l , and for any a~..... a ~ 3 , either I{(gl..... Y m ' > l ~ ~ Rs(al ..... az, or [{
Yl ..... .qm')}l
of Learns ], we can show that if {
gm') l ~ ~ R s (a~, . . . , LEMMA 2.
a~, g~, . . . ,
and
any
, then ]{
Then for any formula of the form Rs(x~ ..... :ct, y~ ..... Ym,) we can effec-
tively find a natural number n(Rs) T ( ~<)
Rs(a~ ..... a~,y~ ..... Ym,)}=/=~
ym')} [ = l !)l l.
Let T(<~) be a theory of locally S-free algebras with the axiom (Qox)(N~ ......~(x))&
](Ix)[N{~ ......}(x), x = x ] .
ym,)13~
9 In fact, as in the proof
a ......
R s ( a 1. . . .
a~3
, either
such that for any locally S-free algebra Ym')l 3 ~ R s ( a ~ ,
I{
,a~, Yi . . . . . . ym,)}l----x , w h e r e
whether for all models
3
•
...,a~,gi .....
...... /(~)], 13I} 9
~I
of the theory
ym,)}l
We c a n
of the theory T(~ <), we have the equation I{(Y~....
a~,y I..... ym,)}l-:131 for i{
Proof.
that
As in the proof of Lemma
gl..... gm,)}=/=~,
moreover,
LEMMA 3. T(xl
-'"
y m , ) } l = l N { ~ ...... }(~)1.
. . . . ,a~,g~ .... , g m ' ) } [ ~ 0 .
In this case, if there
with the condition Pi ~ {],...,s}, as in Lemma
Rs(a~ ..... a~;y~ ..... gm')}l:I31-
I N{~ ...... ~ (3) 1 = I { < ~ ,
a~, y~ . . . . .
~'>
13 ~
~s(~
.....
if this is JRs (a~.....
I, we can show that if {
then I{
exists ~ { i .... , m'} ym,>l~lb
t~s(a~ . . . . .
I{
learn
Y m ' ) l ~ l ~ R s ( a ~ .....
not so, then for all models of the theory T(~<), it follows from {{
o r I{
effectively
~, then we have l{
If there are no such i, then in this case clearly ~, ~ .....
Y~')}
I.
Let Xn) ~ (~/1
/
where ti, tj, gi, fi, hk are terms in variables in the set {x~,...,Xn, Y~,...,Ym}. Then T(x~,...,x n) is equivalent, with respect to the elementary theory of the class of locally S-free algebras, to the formula V T i ( x ~ , . . . , x n ) , where Ti(xl,...,x n) is of the form R(x~,..., X n ) 9
Proof. We first show that ~ t~=t~ may be effectively transformed into the disjunction of conjunctions of the form x i = v i, Yi = wi, where v i and w i are terms. If ti = tj and t i, tj are terms of nonzero length
(i.e. contain functional
(t~..... t~), t~=cp~(t~ ..... t~), ~ k < ~ s ;
axioms
870
the
then for some ~
otherwise T is an identically false formula
ti~T~
(axiom 3). By
]) and 2), ~(t~ ..... t~)=(p~(t~ ..... t~)<=> V
~(~/~(~,=~)), then
symbols),
formula
where ~i, ~j are terms. we h a v e
obtained
may b e
A (t~=ti0) 9 Thus /~t~=t~ is equivalent to O~S h /=1 Since A & ( B V C ) is equivalent to ( A & B ) V ( A & C ) , transformed
into
the
form
V A(~=~)
9
We n o w
reveal each conjunction
in our formula in an analogous way, and once again obtain the form
V ~ (zi~T~), where r~ and ~j are terms. We continue this process further. After a finite number of steps, we obtain the required formula, since at each step the lengths of the terms decrease. If now at some conjunction of our formula we have two equations Yi = wi and Yi = w~, then replacing the second by w~ = wi, we clearly obtain an equivalent formula (here we assume that the length of the terms w!, w'.' is not equal to 0). Having made these substitutions, we once again apply the process described above. In the formula we then obtain we make the same substitutions, and once again apply the process, etc. We finally obtain a disjunction of conjunctions, in which, if we have Yi = wi, and the l~ngth of the term w~ is not equal to 0, then there are no other equations of the form Yi = w i (the term w i having nonzero length) in the conjunction. We denote this disjunction by K, V . . . V K~.
Thus T(xz,... ,Xn) is equivalent
This formula is equivalent
~=~
(~Y~. . . . .
to
to
Y,,,) A x6~ = f~ & A x,,~ #= g~ 3,: A
Y~@ ~,~ &
A N ~ (~) & K~
~=~
9
We now show how each term of the disjunction may be reduced to disjunctions of formulas of the form R(xl,...,Xn). Each formula Ki can be represented in the form K i & Ki, where K i consists of,, equations of the form Yi = wi (the term w i does not coincide with xj, j = I .... , n), and K i consists of equations of the form x i = v i. If in K i we have the equation y~ = Yt, then we can eliminate the quantifier (Nyt) , replacing all the Yt'S in the formula by YT" Make all such eliminations.
We obtain a formula of the form
'~ N~i(Y~)), x~i--/=gi&A y~h=/=h~&~
(~gl.....Yrn)(/~Yi=W~&/~x~i=f~&A \
where in the first conjunction
the terms w i have nonzero length.
Suppose now that in the formula we have the equation Yk = Wk. Then, replacing all the yk's by Wk, we can eliminate the quantifier (aye) , since w k does not contain Yk (otherwise by axiom 4 the formula would be identically false). If we have a term Npk(y k) in the formula, then after the substitution
this takes the form Npk(Wk), which allows us either to remove
or to say that the whole formula is identically false. we obtain a formula of the form
it
Having made all such eliminations,
(~Y~..... Y,,,) ( /~ x~ =f~ & /~ xai=/==g~& /~, y,h=/=hl,& ~=~NPi (Yi))" Now for each x6i appearing either in some term fj or in the second or third conjunctions, we substitute the corresponding x6 i = fi"
fi at all the appearances
of x6i , and eliminate the equation
For the formula thus obtained we repeat this transformation,
and so on.
As a re-
sult we obtain a formula of the form
where the variables x~j do not appear in the terms fi, gi, hk (for all i, j, k) and{x=1 .... }N {x~I.... }-----~.
By axioms
I), 2), and 3), each inequality fj ~ gj is either identically
true,
or can be replaced by an expression A (V(Ti#Ts)) , where T i and Tj are terms of lesser length than the maximal of the lengths of the terms fj and gj.
Using this and the
ruleA&(BVC)~
(A•B) V(A&C), we reduce theT conjunction A ]J=/=gJ to a disjunction of conjunctions of exYT T TT pressions of the form x I ~ Ti, Yi ~ Ti (Ti and ~ are terms). The whole formula has now clearly been reduced to a disjunction of formulas of the form 9
871
(~Y~. . . . . ym) A x~i = 1i & A x~ =/= g1 & A y~ 4= h~ & where x~i
does not appear
in the terms
f~, g~, h~,
.=
{x~I.... }~{x~, . . . } = ~ .
The lemma
is proved.
COROLLARY. Any formula of the form R(xz,...,Xn) can be e f f e c t i v e l y reduced to a disjunction of formulas of the same form but in which each equation of the form x i = hi, where h i is a term of n o n z e r o length, occurs no more than once for given xi, I ~< i ~< n. In fact, if we have h~ = h2 and use Lemma 3.
two equations
x i = h~ and x i = h2,
then we replace
the second
by
L E M M A 4. Let k be a natural number. For each formula of the form R(x0, xx,...,Xn) we can e f f e c t i v e l y find a natural number n(R), such that for any locally S-free algebra with k irreducible elements, and for any set a~, a~.... , a ~ , either [R(~, a~..... a~)] = ]91[
or
IR(~, a~, ..., a,,)l ~< n(R). Proof.
We noted
above
that
the formula
R(x0,
x~,...,xn)
can be written
in the form
([~Y~. . . . . gm) ( /~ x~i = ~ & /~ x~=/==gi & A Y'~==~h~' & ~ N'i (Y~)) where fi, gi, and variables
xo~{x=l,... } .
Let
terms in variables i n {xo, x~, ..., do n o t o c c u r i n h i , g i o r h k .
a n d hk a r e in (xal,...}
By t h e
we only have one e q u a t i o n Ym}, then clearly as n(R) sets 7{(~, a~.... , a~) and
corollary
x,, yt, ..., Y,~},Ix=l . . . .
t o Lemma 3 , we may a s s u m e
that
}~{x~l . . . .
in R(x0,
}=~
xl,...,x
n)
of the form x0 = f0. If f0 does not depend on variables in {Yl,..., we may take I. Let f0 depend on the variables Yl,...,Ym'The
are either s i m u l t a n e o u s l y finite, and the c a r d i n a l i t y of the first is less than the c a r d i n a l ity of the second, or both infinite with the same c a r d i n a l i t y (by a x i o m 2). Consider the formula (~m'+l
for which
the
statement
Moreover,
consider
.....
of Lemma
the formula
Ym) ( A %i =/=gi & A g'~h=/=hk & ~i=lNPi (Yi)) ' 4 holds,
/\ x=~=fi ai=PO
by Lemma , which
1.
Let
the c o r r e s p o n d i n g
is obtained
from the first
(x0, x l , . . . , x n) by e l i m i n a t i n g the equation x0 = f0. If this formula ables y i l , . . . , Y i r , then by a x i o m 2), no more than some natural number
number
be n s.
conjunction
R
depends on the varin r of sets of values
of these v a r i a b l e s satisfy the given formula in any model ~ , and for any set a~, ..., a ~ 9 ~ (n r can be found effectively). If {y~ ..... Y~n'}C{Y11 ..... Y~rl ' then as n(R) we may take hr. If {Yl ..... Ym'}~{Yil ..... Yir} = IYJl, .'',YJt}:/= ~ ....
with free v a r i a b l e s
yi~, y~,+~ .....
Yjl,''',YJt"
, then consider
the formula
ym)(\ A a~i--/=g~&Ay~h=/=h~&i~-=-1 Npi(y~)~ ]
By Lemma
I, the statement
of Lemma 4 is true for this for-
!
mula.
Let Let
the c o r r e s p o n d i n g
x 0 ~ { x ~ I.... }.
number
We may
be n s.
suppose
!
Then as n(R)
we may take nr'n s.
that x0 does not appear
in the n o t a t i o n
of the terms
fi, since then by axioms 2) and 4) the c o r r e s p o n d i n g n(R) could be found trivially. In this case, if /1(~, a~..... a ~ ) # ~ , then ]/{(~, a~, ..., a~)[=l~[. In fact, let a0, c~1,...,an, bl .... ,b m be the r e a l i z a t i o n
A x~ == I : & A x ~ i ~ g~ & A
872
,
y~#=hh&~ Npi(yO.
Replacing all the variables xl,...,Xn, Y~,...,Ym by ~ , . . . . ,b~,..., we consider only the conjunction of all the inequalities depending on x0. By Lemma 2 of [3], the eardinality of the set of all x0 satisfying this conjunction, and therefore also the whole formula, coincides with [3i 9 The lemma is proved. LEMMA 5. For any formula of the form R(x0, x~,...,Xn), we can effectively find a natural number n(R), such that for any locally S-free algebra ~ with set of irreducible elements of cardinality I~I , and for any a~, ..., a ~ 3 , either jR(3, a~, ..., a~)[ ~n(~), or !R(~, The proof is analogous erences to Remark 2. LEMMA 6.
to the proof of Lemma 4, replacing
I by ref-
Then for any formula of the form R(x0,...,Xn), we can effectively
find a natural number n(R), such that for any model either [R(~, a~...., a~)] ~n(/~)
a~)i=I~ifor
~
of the theory T(~ <), and any a~,...,
or [R(~, a~, ..., a,~)I = •
can effectively determine whether for all models IR(3, a~, ~
to Lemma
Let T(~ <) be a theory of locally S-free algebras with the axiom (Qox)(N~r ......j(a))&
-](Ix) [N(~......)(x), x = x ] . a~,
references
3
, where ~ ~ { N~.. ~i(3)I~ ]3!}.
We
of the theory T(<~), we have the equation
[R(~, a~..... a,)] >n(R); moreover,
if this is not so, then for all models
of the theory T(~<), it follows from ]R(~, a~, ..., a~,)i>n(R)
that I/~(3, a~..... a~)i = I N ~ ......)(3)I .
Proof. The first part of the lemma is proved analogously to the proof of Lemma 4, using Lemma 2. The second part of the lermna reduces to the analogous problem for formulas of the form
(~ym,+~ ..... y,~)(A xB~r
Ay~r
~A,=~.Np~(~,O) ,
which is effectively decidable, by Lemma 2. Remark_~3. A conjunction of formulas of the form R(x~ ..... Xn) can be transformed into an equivalent disjunction of formulas of the form R(xl,...,Xn). This can be verified directly (using arguments like those at the end of the proof of Lemma 3). ._LEMEA 7_. Let O ( x o , x~, . . . , x~) be a formula of an RPC of signature o . We can effectively find a natural number n(O), such that for any locally S-free algebra 3 with finite number k of irreducible elements, and for any a~...., a z ~ , if [0(~ a~....~ a,~)[>n(O), then I@(3, a .....,
a~)l
=
l~tl.
Proof. By Theorem 1 and Remark 3, we may assume that a disjunction of conjunctions of the form
@(x0, x~, ..., x~)
is of the form of
m i
It is sufficient to prove the lemma for such conjunctions~ conjunction of the form (I)], then n ( O ) = n ( K ~ ) + ...+n(KT). Excluding from the conjunction study of the formula
since if 6 ) ~ I f ~ V . . . V I f ~
[K i is a
terms not containing x0, we reduce our problem to the
O~ (Xo, x~ . . . . .
x~) ~ Npo (Xo) & R (xo . . . . , xt).
(2)
Clearly, if P0 = {1,...,s}, then as n(O~) we may take k. Let P0 ~ (I ..... s}. Let n(R) be such that if I/7(3, a~.... , at)l>n(R), then IR(~, a~, ..., az)l = I~]. Such an n(R) can be effectively found, by Lermna 4. Let the equation x0 = t occur in R(x0, x1,...)x~), where t is a term in variables in the set {xl,...,xl, Yl,...,Ym}. By the corollary to Lemma 3, we may assume that this equation is unique. Suppose that t is of the form ~0(T1 ..... mr) , where m!,...,-~ r are terms, and
~ 0 is a
functional symbol of signature g. Then if f0~p0, then @~(xo, x~ ..., x~) is an identically false formula.~ in view of the definition of Np0(X ). If io~po, then each x0 satisfying R(x0~ al,...,~ l)
satisfies Np0(X0).
Thus if ]0~(3, al.... , az)i>n(R),
then
lOi(3, al..... a~)[=i31.
Suppose that t has zero length. Therefore t coincides either with the symbol of the variable xj, or with the symbol of the variable yj. The first case is trivial; consider the second case. Here O~(x0, x~, ..., x~) is equivalent to the formula @,,(x0, x~, ..., xz) ) which is 873
by NpjUpo(Y~).
obtained from R(x0, xl,... ,x~) by r e p l a c i n g the subformula Npj(yj) n(O1)=n(Oz), and n(O2) can be effectively found, by Lemma 4. Suppose case
that an equation
of the form x0 = t does not
n(@1)=n(/~) . Let ]O1(~, al.... , az)i>n(R) .
Then
occur
Therefore
in R(x0,...,xl).
IR(~, al..... al)[==-I~l .
Therefore,
In this we may
assume that x0 does not occur in the terms fiLet R'(x0, a:~,..., at, gl, ..., ym) be the quantifier-free part of R(x0, al, ..., a~) . Let a0, b l , . . . , b m be a r e a l i z a t i o n of this formula in ~ , and let i0={i, ..., s}\po . As in the proof of Lemma 4, it follows from Lemma 2 of [3] that only the finite part of the set B = {z{(~z I..... zr)(z =(P~0 (Zl..... zr))} does not realize R'(x0, a~, ..., az, bl, ..., b~). But I~]. The lemma
since each is proved.
element
of B realizes
Np0(X0)
and
[BI = I~] , then
IOi(N, a~.... , an)] =
L E M M A 8. Let @(x0, xl.... , x~) be a formula of an RPC of signature o. We can e f f e c t i v e l y find a natural number n(@) such that for any locally S-free algebra ~ with set of irreducible elements of c a r d i n a l i t y ]~I , and for any a~.... , a z ~ , if [0(~, a~.... , az)[>n(O) , then ] 0 ( ~ , a~, . . . , at)] =]~]. Proof. As i n Lemma 7, we c o n s i d e r only formulas of the form (2). L e t p0 = { 1 , . . . , s } . Then if in R(x0,...,x~) we h a v e t h e e q u a t i o n xo = t , t h e n t h a s z e r o l e n g t h . O n c e a g a i n we consider the case when t = yj. As i n t h e p r o o f o f Lemma 7, we d e f i n e a f o r m u l a O2(x0, x~, . . . , xt) and look for n(02) , u s i n g Lemma 5 . We h a v e n(O~) = n(@~). I f t h e r e i s n o e q u a t i o n x0 = t in R(x0, xl,...,x~), then n(O~)=n(R) . In fact, i f R'(x0, a~ . . . . , a , b~. . . . , bm) i s d e f i n e d as in t h e p r o o f o f Lemma 7 a n d i f 10~(~, al . . . . . a t ) l > n ( R ) , then only a finite set of irreducible elements does not realize R'(x0, a, . . . . , a~, b~. . . . . bm). L e t P0 a { 1 , . . . , s } . Then the proof repeats the corresponding arguments o f Lemma 7, w i t h references to Lemma 4 replaced by references to Lemma 5. The lemma is proved.
LEMMA
9.
Let
(Ix)[N{~......~(x), x = x ] .
(x)) & 7
find a natural , either
number
[ 0 ( ~ , al . . . . .
effectively
determine
[O(~, a~..... at)[=]~[ of the theory prove
T(~<) be a theory
T(~<),
S-free
Then for any formula
n(O)
such
that
az)] ~
whether
or
[O(~, a~ . . . . .
for all models
it follows
from
part of the lemma we consider again
for all models
~
~
at)[ = z ,
~
with
the a x i o m
of the theory where
•
of the theory
moreover,
if this
IO(~, a~, ..., a,)i>n(O)
(Qox)(N{i .....~)
T(~<),
and any a~, .., at~
...... )(~){, [~[} .
T(~<), we have is not
We c a n
the equation
so, then for all models
that ]@(~, a~, ..., a,)l~- IN{~......)(N)I.
is proved a n a l o g o u s l y to Lemma 8, using Lemma 6. the formula of the form (2). If P0 = {1,...,s},
of the theory
that ]O~(~, al, ..., a~)] = IN{I......)(~)] , since
algebras
of the RPC @(x0, x,, ..., x~) , we can e f f e c t i v e l y
for any model
for I'O(~, a~, ...,at)I>n(O);
Proof. The first the second part,
then clearly
of locally
this
T(~<), it follows formula
To
from ]O~(~, a~..... a~)[ >n((~)
is satisfied
only by
irreducible
ele-
ment s. Let P0 z {l,...,s}. If there is no equation of the f o r m x0 = t in R(x0, x~,...,x~), then arguing as in the proof of Lemma 7, we see that if IO~(~, a~..... at)[ >n(O~) , then [O~(~, a~.... , a~)[ = [ ~ I for all models ~ of the theory T(~<). This carries over a n a l o g o u s l y to the case when R(x0, x~,...,x~) has an equation x0 = t, where t is a term of nonzero length. Suppose we have the equation x0 = yj in R(x0, x~,...,x~) (preserving all the n o t a t i o n of Lemma 7). Then the proof of the lemma reduces to the analogous p r o b l e m for the formula O~(xo, x~, ..., x~), and this p r o b l e m is e f f e c t i v e l y decidable, by Lemma 6. The lemma is proved. T H E O R E M 2. The two locally S-free algebras ~ and ~0~ are equivalent in the calculus with the q u a n t i f i e r (<~x), if and only if they either have the same finite number of irreducible elements, or the p r o p o s i t i o n ([x)[N(,......)(x), x = x ] is true in both, or the p r o p o s i t i o n (Qox)(N. ...... )(x)) & n ( l x ) [ N l ~ ...... l(x), x=x] i s t r u e i n b o t h . Proof.
~]~s~D~
is obvious.
Sufficiency. Let Tk(~<) be a theory of locally S-free algebras with an axiom satisfying the existence of p r e c i s e l y k irreducible elements, k = 0, I, .... Let @~ be an arbitrary p r o p o s i t i o n of the enriched language. By Lemma 7 and P r o p o s i t i o n I, f r o m ~ we can effectively construct a p r o p o s i t i o n kb2 in a RPC, equivalent to ~ with respect to Tk(~<). Thus if, ~T~(~), ~ O ~ T ~ ( ~ ) , then ~ ~ 2 and ~ 0 ~ = > ~ . By the c o r o l l a r y to T h e o r e m 4 of 874
[3], we h a v e
~ = ~ 0 ~ 2 .
If T~(~) is a theory of locally S-free algebras with the additional axiom ([x)[N(~......~(x), x = x], then applying Lemma 8 and Proposition I, as above, we reduce the problem of the equivalence of models of the theory Tw(~) to results of [3]. If TN(~)
is a theory of locally S-free algebras with the extra axiom
(Qox)(N{~ ...... ~(x)) & n ( [ x ) [ N ( ~ ...... ~(x), x = x], then the corresponding proof is analogous. We only note that in this case it is necessary to use Lemma 9 and Proposition 2. The theorem is proved. COROLLARY~ Any locally S-free algebra quantifier (~x).
~
has a decidable
theory in the calculus with
Proof. It was shown in Theorem 2 how we may effectively describe a recursive system of axioms for ~. Since in this case we can eliminate generalized quantifiers, the problem of the decidability of theories of the model ~ in a calculus with the quantifier (<~x) reduces to the problem of the decidability of the elementary theory of the model ~, which was positively solved by Theorem 5 of [3]. In conclusion, quantifier (<~x) o
we introduce some more examples of decidable theories
in a calculus with
THEOREM 3. The theory of algebraically closed fields of characteristic in the calculus with quantifier (~x).
0 is decidable
Proof. By Tarski's theorem [7] and the lemma of Sec. 5 of [8], each formula in the language of a RPC in the theory o~ algebraically closed fields is equivalent, with respect to this theory, to a disjunction of formulas of the form h
pA (Xo, Xl . . . . .
X,~) ~ po =/=O & /~ p~ = O,
(3)
i=I
where Pi are polynomials with integral coefficients in the cor!:esponding variables. It was shown in [8] that any elementarily defined subset of any model of the theory of algebraically closed fields of characteristic 0 is either finite or eofinite. Therefore, to apply Proposition I, it is necessary to prove that for any formula of the form PA(X0, xl,...,x n), we can effectively find a natural number n(PA) , such that for any algebraically closed field F of characteristic 0, and for any elements al,...,a,~F , if the set PA(F, al,...,an) is finite, then IPA(F , a1,...,an) 1 <~ n(PA). In fact, since PA(F, al,...,an) is finite, then in (3) we k
must have the second conjunction
A P~=0
. Let m i be the degree of the element x0 in Pi,
i=I
i = I, 2,...,k.
Then clearly, as n(PA) we may take min (ml .... ,mk).
Now, since we may eliminate the quantifier (<~x), and since the elementary theory of algebraically closed fields of characteristic 0 is decidable, we obtain the statement of the theorem. THEOREM 4. decidable.
The theory of real-closed fields
in a calculus with the quantifier
(
jProof. By Tarski's theorem [7] and the lemma in Sec. 4 of [8], any formula in the language of an RPC in the theory of real-closed fields is equivalent, with respect to this theory, to a disjunction of formulas of the form pR (xo, x~ . . . . .
x~) ~
A p~ = 0 &
qi > O,
(4)
where Pi and qi are polynomials in the corresponding variables with integral coefficients. It was shown in [8] that if in the real-closed field F pR(F, a~, ..., a~) is an infinite set, where a~..... a ~ F , then IPR(F , al,...,an)[ = JFk. Therefore, to apply Proposition I, it is necessary to prove that for any formula of the form (4) we can effectively find a natural number n(PR ) such that for any real-closed field F and any elements a~.... , a ~ F , if the set PR(F, al .... ,an ) is finite, then ]PR(F, al .... ,an)[ ~ n(PR). We find this number as in the proof of Theorem 3. The statement of the theorem now follows from the decidability of the theory of real-closed fields. THEOREM 5. The theory of the model
(~x),where 875
Proof. It was proved in [9] that the theory of the model
(~y, z)(y ~ z& (rx)[~(x), x < y ] & (Ix)[~(x), x < z]) V (Vy)(~x)(y < x & ~(x)). The theorem is proved. LITERATURE CITED I 9
2. 3. 4. 5. 6.
A Baudisch, "The theory of Abelian groups with quantifier (~x)," Z. Math. Log. Grundle. Math., 23, No. 5, 447-462 (1977). M Weese, "The universality of Boolean algebras with the Hartig quantifier," Lect. Notes Math., 537, 291-296 (1976). A I. Mal'tsev, "~xiomatizable classes of locally free algebras of certain types," Sib. Mat. Zh., 3, No. 5, 729-743 (1962). A Tarski, "Some decision problems for locally free commutative algebras," Not. Am. Math. Soc., No. 13, A-634 (1966). J Doner, "Decidability of locally free algebras with unary operations," Not. Am. Math. Soc., 13, A-634-635 (1966). A G. Pinus, "The expressability of the concept 'an infinite subset' on certain classes of algebraic systems," Novosibirsk (Preprint Inst. Mat. Sib. Otd. Akad. Nauk SSSR)
(1977). 7.
8.
9.
A. Tarski, "Arithmetical classes and types of mathematical systems, mathematical aspects of arithmetical classes and types, arithmetical classes and types of Boolean algebras, arithmetical classes and types of algebraically closed and real-closed fields," Bull. Am. Math. Soc., 55, 63-64 (1949). I. Cowles, "The relative expressive power of some logic extending first-order logic," J. Symbolic Logic, 44, No. 2, 129-146 (1979). M. Krynicki and A. H. Lachlan, "On the semantics of the Henkin quantifier," J. Symbolic Logic, 44, No. 2, 184-200 (1979).
POINTS OF CONTINUITY OF A FUNCTION AND POINTS OF EXISTENCE OF FINITE AND INFINITE DERIVATIVE UDC 517.51
L. I. Kaplan
I.
INTRODUCTION
In this article we solve the problem of the relative disposition of three sets of points -namely, the set of points of continuity, the set of points of existence of finite derivative, and the set of points of existence of infinite derivative -- of areal-valued function of a real variable. This theme goes back to Zagorskii [I]. The present article generalizes the results of [I-3]. Formulation of the Main Result. Let f be an arbitrary function defined on the line R I. Let Q, M, and N denote, respectively, the set of points of discontinuity of f, the set of points at which f has infinite derivative, and the set of the points at which f does not have a finite derivative. Then, as we know [4, 5], the sets Q, M, and N must satisfy the following conditions. I.
Q is an Fo-set and M is an Fo6-set.
II.
N can be expressed in the form
N = N , U N2, N, n N 2 = ~ , N ~ = M , where Nz is a G6-set and N2 is a G6o-set of measure zero. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 24, No. 6, pp. 66-79, NovemberDecember, 1983. Original article submitted August 4, 1981. 876
0037-4466/83/2406-0876507.50
9 1984 Plenum Publishing Corporation