Download cylindric algebras and algebras of substitutions^) 167

TRANSACTIONS OF THE
AMERICAN MATHEMATICALSOCIETY
Volume 175, January 1973
CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS^)
BY
CHARLES PINTER
ABSTRACT.
Several
new formulations
of the notion of cylindric
algebra
are
presented.
The class
CA of all cylindric
algebras
of degree a is shown to be
definitionally
equivalent
to a class of algebras
in which only substitutions
(together with the Boolean +, •, and —) are taken to be primitive
operations.
Then
CA is shown to be definitionally
equivalent
to an equational
class of algebras
in which only substitutions
and their
taken to be primitive
operations.
1. Introduction.
substitutions
In the theory
ate
Specifically,
defined
in
(A)
Just
as (A) defines
ments,
fined
in terms
of algebraic
certain operations
of degree
and diagonal
a and
k, À < a,
in terms
Thus,
substitutions
in [6] that
(a) in every
and diagonal
and diagonal
it is clearly
in which
elements
possible
are taken
then the opera-
if* ¿A-
of cylindrifications
that both cylindrifications
called
elements.
by the formula
S£x = cK(x - dKX)
of substitutions.
logic
algebras/2)
by S^, is defined
itK=X;
substitutions
it is known
with +, •, and —) are
of cylindrifications
algebra
denoted
S*x = x
(together
of cylindric
terms
if SI is a cylindric
tion of k-for-K substitution,
conjugates
ele-
may be de-
to develop
a theory
to be the only primitive
opera-
tions.
It is proved
S\x
= 1, and (b) in every
K < a.
It is noted
in [6] that
mension-complemented
0, 1, S\)K
conditions.
A simple
cylindric
algebras
There
tions.
\<a
in this
The definition
lindrifications
Received
only
algebras
wnere
tne
and elegant
is more than
d^
is the least
as
sense
has been
by the editors
new treatment
fot each
of di-
structures
operations
subject
by Anne Preller
cylindrifications
to certain
in [l0].
in terms
has the disadvantage
of dimension-complemented
algebras;
of substitu-
of yielding
cy-
alternative
October 4, 1971.
AMS (MOS) subject classifications
(1970). Primary 02J15.
Key words
algebra,
and phrases.
x such that
„S^x
of dimension-complemented
obtained
upon (b), above,
in the case
a possible
^X are unary
element
c x = 2.
algebraic
characterization
one way of defining
based
CAa,
(a) and (b) suggest
cylindric
(A, +,-,-,
CAa,
dimension-complemented
Cylindric
cylindrification,
diagonal
element,
alge-
braic logic.
( ) The work reported
in this paper was done while the author held an NSF Faculty
Fellowship.
The author is indebted
to the referee
for valuable
suggestions,
especially
regarding
an important
simplification
of the material
presented
in §3.
( ) For notation
and terminology,
see Henkin,
167
Monk and Tarski
[6],
Copyright © 1973. American Mathematical Society
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
CHARLES PINTER
168
definitions
are available,
is not needed.
algebras
Using
in terms
In this
these,
any one of these,
we describe
without
which
somewhat
weaker
In a recent
every
take
complementedness
to characterize
an especially
paper,
of dimension
all cylindric
William
Using
these
is definitionally
ideas,
simple
Craig
complementedness.
algebras
Using
in terms of sub-
form in the presence
Í2] has shown
of a condition
that in a cylindric
algebra
) and that cylindrifications
in terms of substitutions
a set of axioms
equivalent
(together
in terms
complementedness.
entirely
we derive
cylindrifications
of cylindric
S y has a conjugate/
which
for defining
characterizations
may be defined
conjugates
two methods
than dimension
substitution
elements
of dimension
it is possible
the assumption
we get two simple
stitutions,
the assumption
of substitutions.
paper,
of substitutions
where
[January
to CAa,
with the Boolean
and diagonal
and their
for an equational
and in which
conjugates.
class
of algebras
substitutions
and their
+ , • , and - ) are the only primitive
opera-
tions.
All the cylindric
to be of degree
sented
a>
elsewhere,
tions
we shall
and related
2.
For the case
considered
examine
2. Substitution
here
mulation
is as follows:
Definition.
and cylindric
algebra
are assumed
hold which
For the cases
algebras.
will be pre-
<x= 0, 1, the ques-
By a "substitution
with "substitution"
By a substitution
algebra
mean a system
21 = (A, + , • , - , 0, 1, S^ )
Boolean algebra
and the S1^ ate unary operations
ditions
results
paper
are trivial.
algebras
mean a Boolean
in this
ct= 2, simpler
in the form of an abstract.
we shall
2.1.
algebras
operations.
of degree
a,
briefly
for-
an SAa, we
x^a where (A, +,.,-,
on A which
algebra"
Our precise
0, l) is a
satisfy
the following
con-
for all x, y £ A and all k, X, p., v < a :
(Sj) SKxix + y) = S$x + S^y,
= -S£x,
(52) S£(-x)
(53) S£x = x,
(s4) SCS%x= SÏSÏx,
(S,) S^Sfx = S^x, if k4K
(S6) S$ S$x = S^S'ix,
Our notion
in the latter,
algebras,
algebras
formations
tions
itK4ri,v
of substitution
some additional
algebras
conditions
as we have just defined
invented
S* may be compared
0~) For the notion
of conjugate,
to that of Preller
are assumed.
In addition,
related
[3], with the difference
in the sense
with
is similar
them, are closely
by P. R. Halmos
to replacements
andzi 4 A.
of Halmos
the operations
due to Jonsson
to the transformation
that we restrict
[5, p. 289].
Sir) of [3] where
and Tarski
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
in LIO], but,
substitution
[7],
Thus,
trans-
our opera-
r is restricted
see
our Definition
to
2.5.
1973]
CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS
replacements.
Then
that
the operations
tially
the same
scends
three
our axioms
S(t)
as
Halmos'
the conditions
immediate
(S,)—(S,).
In every
substitution
vation,
we shall
lindric
(and in every
write
observation
Indeed,
S
Now it follows
in every
the range
cylindric
Combining
Axiom
S2, whichtran-
algebra,
we adopt
algebra
c
= rg k
of S\,
if operations
of À. Indeed,
In view
of this
obser-
k / X.
of SA to the range
algebra,
5A
it
S\cK=
c
of cK in cy-
and cKSx = 5A for
Consequently,
for every
from the elementary
properties
cKx is the least
y £ range
(**)
cylindric
rg K for the range
relates
range
in [3]
is essen-
to replacements—namely
tot all X, p /= k.
k / X (see [6, 1.5.8 and 1.5.9]).
(*)
of Halmos'
of substitution
of S* , for k / X, is independent
S* = range
henceforth
algebras.
Instead
(S,)
for our purposes.
algebra
that range
An analogous
SI.
to the condition
and our axiom
of this axiom restricted
by (A)), the range
from (S,)
endomorphisms,
in our notion
is all we need
follows
k, À < a,
Axiom
assumed
consequences
This
are defined
(S.) and (S2) correspond
be Boolean
169
k < a.
of cylindrification
cK such
(*) and (**), we get the following
that
that y > x.
important
property
of cylindric
alge-
bras:
(2.2)
ex
(2.2) clearly
stitutions
is the least
shows
in an algebra
y £ rg k such that y > x.
that cylindrifications
which
does
can be expressed
not need
to be dimension
in terms
of sub-
complemented.
The
proposition
(2.3)
¿
shows
that diagonal
we consider
elements
substitution
(77I) for each
(77-2) for every
and introduce
(B) cKx
is the least y such that S£y = 1
likewise
algebras
can be so expressed.
having
This
x £ A and K < a, {y £ rg k: y > xj has a least
k, X < a, {x: SAx = lj
has a least
that
element,
and
element,
for them operations
cK and dKX by means
is the least
of \y £ rg k: y > xj, and
element
suggests
the properties
of the definitions
(C) dKX is the least element of [*: ^xx = llThe notion
algebras
has shown
similar
Before
to introduce
to introduce
clearly
similar
to (B) holds
quantifiers
cylindrifications
in the work of P.-F.
in every
in transformation
in substitution
Jurie.
In [8], Jurie
polyadic
algebra,
algebras
satisfying
and that
a con-
to (77I).
going
2.4. Lemma.
subject
(77I) appears
that a condition
it may be used
dition
that (B) may be used
satisfying
to condition
on, we consider
some consequences
Let 21 = (A, + , . , -,
(nl).
If c
of (77I) and (rr2).
0, 1, SA)K x<a be a substitution
is defined
by (B), then
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
the following
algebra
statements
170
CHARLES PINTER
[January
hold for all x £ A and all k, X, p < a:
(ii)
^xck
(iii)
= CK'
^V^
cK0=0,
(iv)
Proof.
(vii)
x<cKx,
Furthermore,
cK5^x < S*cKx
ifn4X,p.
from (77I) and (B) that if y = cKx, then
that if y e rg k, then y = cKy.
c
cKx < cKy,
S^x<cKx,
(viii)
It is clear
(a) range
* - y implies
y e rg k, and also
Thus,
= rg k.
it follows
from (SA and (B), respectively,
that
(b) S'{SKX= SKXand cKcK=cK.
Now (i) and (ii) follow
immediately
By (S.) and (S,),
algebra
S\
of 21. In fact,
Halmos
quantifier
of 2I( and therefore
[4, §4]).
Now (vi) follows
least
Thus,
by Halmos
complete
[4, Proof
from (iii)—(v) by [6, 1.2.7],
y £ ig k such
rg k is a Boolean
Boolean
of Theorem
sub-
subalgebra
5], c
is a
(iii)—(v) hold.
of (iv), (Sj) and (ii).
element
of 21, hence
by (77I), rg k is a relatively
of 21 (see
sequence
from (a) and (b).
is an endomorphism
It remains
that y > Sx.
and (vii) is an immediate
to prove
(viii):
Now S*cKx
by (ttI),
con-
c^S^x
is the
> S*x by (iv) and (S,);
furthermore, by (ii) and (Sg), if v 4 A, then
SlCKX-
thus,
cKS*x<S*cKx.
The next
SaSvcKx=Sv^cKxetgK;
a
definition
introduces
a concept
which
plays
an important
role in the
next section.
2.5.
algebra
Definition
and /, g £
(Jónsson
and Tarski
[7, Definition
A, we say that g is the conjugate
1.11]).
If A isa
Boolean
of / if, for any x, y £ A,
fix) . y = 0 « X - g(y) = 0.
2.6.
ject
Lemma.
to conditions
spectively,
ii)
(iii)
Let \ A, +,.,-,
(77I) and in2).
If c
and fl^
\<a
be a substitution
are defined
algebra
sub-
by (B) aW (C), re-
then the following hold for all x £ A and all k, X, p. < a;
S^x . dKX= x ■ dKX,
(ii)
S*x = cKix • dKX)
S^dKfl= S\dKX
(iv)
Sx admits a conjugate.^)
Proof, (i) Clearly
which
0, 1, S\'K
is the same as
(4) Theorem
2.6(iv)
if k4 u,
ifx4X,
S^iS^x + -x) = 1, hence by (tt2) and (C), dKX < S^x + -x,
x • dKX < S\x.
is a special
case
Analogously,
S xx ■ dKX < x, so we get (i).
of a more general
Lemma 27].
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
result
proved
in Craig
[2,
1973]
CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS
(ii) By 2.4(i) and 2.6(i), cKS$x . dKX= Sxx . d^ = x . rf^.
ck{x •dKX) = ck(ckSAx
• ¿KX)> so by 2.4(i)
Sxx . cKdKX. But by (C) and 2.4(vii),
cpSdu\'
. Therefore
dp.K) = Sidp.\-
Now by (S,) and (S4), S^S^x
if À /= p, then by 2.6(ii),
=
S^x = 1 =» S*x = 1; thus,
StxldKfl= cfx(dKfJL. d^y) =
Thus-
= 1
by (C), (Sj)and(S2).
It follows by (C) that dKX< S^d
, and analogously, d
(S4), SxdKX < SXS&K^
S^dKX
S\dKfl<
(iv) If k = À, we define
is the conjugate
. cKdKX=
cK¿KA = 1, so cK(x • dKX) = Sxx • 1 = Sxx.
hence Sh\x = 1 =» S^x = 1; symmetrically,
by (C), dK/JL=d
Thus,
and (v), cK(x . dKX) = cKSxx
(iii) If X = p, there is nothing to prove.
SkSkx = S?x'
171
of Sx.
If k/
T{x ■ y = 0 _
(applying S£)
= 5idKX.
Tx by Txx = x; in this
(a) TKxx=cKx.dKX,
and we proceed to show that
< S¡¡_dKX
. Thus, by (Sj)—
X, we define
case,
it is obvious
that
Tx
Tx by
Tx is the conjugate
cKx ■ dKX • y = 0
of Sx:
by (a),
=» cKx • 1 • S$y = 0 by (SA-(S2), 2.4(ii) and (C),
=>x.S*y=0
by 2.4(iv).
Conversely,
x • S£y = 0 =*cK(x • SAy)= 0
by 2.4(iii),
=*cKx • S£y = 0
^cKx-dK\-sZy
In our first
initionally
theorem,
equivalent
(t73) Sxc
Note that
by 2.6(i),
=>TAx-y=0
by (a).
we will show
additional
= c Sx
that the theory
of substitution
□
of cylindric
algebras
which
algebras
satisfy
is def(ttI),
condition:
provided p / k, X, where c
only half of this
(v),
=°
=> cKx • ¿KX • y = 0
to the theory
(772) and the following
by 2.4(i)and
condition
is really
is defined by (B).
needed,
as the other
half is de-
rived in 2.4(viii).
2.7. Theorem,
(i) // \A, +,■,-,
0, 1, c , c^KA'KA<a is a cylindric
and Sx is defined by (A), then \A, +, • , - , 0, 1, Sx )
gebra satisfying
(ii)
A<a is a substitution
al-
(tt1)—(tt3), together with (B) and (C).
Conversely,
which conditions
algebra
if \A, +,-,-,
0, 1, S A)
(77I)—(?73)hold, and if c
.
a is a substitution
algebra
in
and d . are defined by (B) and (C) re-
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
172
CHARLES PINTER
spectively,
then
[j anuary
i A, + , . , - , O, 1, c K, dK>) K x<a is a cylindric
algebra
where
(A) holds.
Proof, (i) Let (A, +,•,-,
0, 1, cK, dKX)K x<a be a cylindric
algebra,
and de-
fine S$ by (A). Then (Sj )-(S6) hold by [6, 1.5.3 and 1.5.10], and (tt2) holds by
[6, 1.5.7]. Next, range cK = rg k by [6, 1.5.8(i) and 1.5.9(ii)L hence by [6.
1.2.4 and 1.2.9], we have (ttI ) and (B). Finally, (jt3) holds by [6, 1.5.8(ii)], and
(C) holds by L6, 1.5.7].
(ii) Let (A, +,-,-,
0, 1, S*)
x<abea
(773) hold, and let cK and dKX be defined
that conditions
substitution
is nothing
ckc\ckx'
to prove;
hence,
1.1.l],
assume
are satisfied.
We will show
Well, (C0) holds
We prove (C4) as follows:
k 4 X. By 2.4(iv),
li k =
x < c x, so cc.x
N°w if P 4=X-1A, then
CKC\cKX
This
in which (77I)—
by (B) and (C) respectively.
(Cn)—(Cy) of [6, Definition
by 2.1, and (Cj)— (CA are given by 2.4(iii)—(v).
À there
algebra
shows
that
cKcxx
< cxc
(CA follows immediately
To prove
(CA,
let
= CKC\Sp\CKX
by 2.4(H),
= ck5mcXckx
by (?73)'
= sK^Xckx
by2.4(i),
= cXS>ßcKX
bV <»3),
= cxcKx
by 2.4(ii).
x; symmetrically,
c c x < c cxx, proving
(CA
from (772) and (C).
c^x
= -ic
— x).
It easily
follows
from (773) and
(S2) that
*A^ = cfö tot p4K,X. Thus, S$c*dKX = cfiSKxdKX=I, so by (n2), dKX<
cfLdKX. On the other hand, it follows from 2.4(iv) that cfidKX < dKX, so dKX=
c^dKX £ ig p, hence,
in particular,
for k 4 p- Consequently,
dKX = S^dKy
Thus, by 2.6(iii),
S^dKß=
dKX
if p 4 k> A, then
cpSdK p-• dp-x) =s xdKu hy2-6^'
= dKX'
which proves (CA.
Lastly,
(C7) follows immediately
(A) follows from 2.6(ii) and (Sj).
Remark.
stitution
then there
family of elements
dric algebra;
A similar
D
It follows from 2.7(Uthat
algebra,
(^^
indeed,
remark
was
the
We will show
of either
ness.
We begin
if \A, +,•,-,
is at most one family
0, 1, S^)K x<a is a subof functions
icK'K<a
and one
^<a such that \A, +, ■, 0, 1, cK, dKXIK x<a is a cylinc
made
and dKX ate completely
for polyadic
for locally finite polyadic algebras
sence
from 2.6(a) and (S2), while
next that
by P.-F.
Jurie
by (B) and (C).
[8], and,
earlier,
by L. LeBlanc L9J.
Theorem
of two conditions
algebras
determined
2.7 can be improved
somewhat
weaker
than
with some definitions:
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
significantly
dimension
in the precomplemented-
<
1973]
CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS
2.8. Definition.
bra.
For each
k < a such
173
Let 21 = \A, + , • , - , 0, 1, 5A)K A<a be a substitution
x £ A, the dimension
set of x, in symbols
that cKx -/ x. 21 is said to be dimension
alge-
Ax, is the set of all
complemented
if, for every
x £ A, Ax 4 a.
2.9. Definition. Let 21 = (A, +,-,-,
0, 1, SA) K x<n be a substitution algebra.
Z(k, X) = \x £ A: Sx x = 1 j. Then 21 is said to be normal if
We write
D
,
,
Z(k, X) = {lj for each k < a.
A cylindric
algebra
If 21 satisfies
a cylindric
each
normal
(7r2) and dKX satisfies
algebra),
k < a.(J)
will be called
then clearly
if it has the same
property.
(C) for all A, X < a (tot example, if 21 is
21 is normal
if and only if 1
*
d x = 1 for
\¿K;\<a
2.10. Definition.
bra satisfying
(nl),
Let 21 = \A, + , . , - , 0, 1, Sx)K x<a be a substitution
and for each
k < a,
let
cK be defined
by (B).
alge-
We say that 21
has the RS property if
(2.11)
for all
cKx=
Z
K
k < a and x £ A.
erty if it satisfies
Equation
2.12.
Six
x¿K;X<a
A cylindric
algebra
A
is likewise
said
to have the RS prop-
(2.11).
(2.11) is due to Rasiowa and Sikorski Lll].
Theorem,
(a) Every
dimension
complemented
substitution
algebra
is
normal.
(b) Every
normal
substitution
algebra
satisfying
(nl) and (tt2) has the RS prop-
erty.
Proof,
such
that
(a) If 21 is not normal,
Sxx = 1 for every
for some p < a.
x = S^x
= S^S
7Ç
f\
(b)
then there
À ?¿ k.
L-.\<a s\x
=
A
X¿k;
= 1;' in either
case,
substitution
Kfx<aCK^-¿Kx)
X¿K;X<
= CK (X ' XéK;
thus,
p £ Ax
Lemma.
easily
that
(77I) and (,7r2). Then,
by [6,1. 2. 6],
\<adK\)
by the remark following 2.9.
// (A, + , ■ , -,
the RS property satisfying
satisfying
by2.6(ii),
- cKx
( ) It follows
complemented;
we have a contradiction.
algebra
-^x^x<Jx-dK^
a CAa is normal
x / 1 in A and some k < a
If p = k, then for any X ^ p, x = S^x = Sxx = 1; if p ¿ k, then
x = S^l
fJ.
IS.
Let ** be a normal
2.I3.
is some
But 21 is dimension
0, 1, Sx)
x<a is a substitution
G
algebra
with
(ttI) and (tt2), then (n3) holds.
every
normal
CA
for a<a>,
iff it is dimension-complemented.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
is discrete.
Thus,
for a<(0,
174
CHARLES PINTER
Proof.
We have proved,
Lemma 2.6(iv),
follows
by Jónsson
ditive.
Thus, if K 4 X, p, we have
and Tarski
XucKx
Z
K = S*
My¿K;V<a
= SKSxx+
r
Z
^
Consequently,
cKS
x.
Combining
that each
a conjugate.
It
S1^ is completely
Z
V é K,\\
ad-
S*aSKv,
," v
V <a.
by (S4) and (S6),
^
S"vS*x.
^
S>cKx + S^x
We conclude
1.14]
S^ admits
S*S£x=S*S£*+
M v
,"• A
S«S*x
v¿K,X-,v<a
= V ¿K,X;V<a.
Z
that each
[7, Theorem
SKvx=
Z
v
v¿K;V<a
fi
I anuary
= 2^
^J^x
this with 2.4(viii)
this
section
gives
= c^x,
hence Sfax
us our result.
with the following
<
D
result:
2.14. Theorem, (i) // 21 = (A, + , • , -, 0, 1, cK, dKX)K x<a is a cylindric
gebra
with the RS property
complemented
cylindric
'^> +i • > -,
a normal
substitution
algebra,
Conversely,
resp.
if 21 =(A,
(resp.
(resp.
algebra,
resp.
by (A), then
algebra
a dimension
+,-,-,
a normal
(nl) and (772), and if c
then 21 = (A, +,.,-,
erty
cylindric
5^ is defined
a dimension
21 =
with the RS property
complemented
(resp.
substitution
alge-
(nl) and (tt2), as well as (B) and (C).
with the RS property
isfying
a normal
and
0> 1> $)) K x<a z5 a substitution
bra) satisfying
(ii)
(resp.
algebra)
al-
0, 1, Sx)K x<a is a substitution
SA, resp.
a dimension
and d . are defined
cylindric
algebra,
resp.
SA) sat-
by (B) and (C), respectively,
0, 1, cK, dKX> x<a is a cylindric
a normal
algebra
complemented
algebra with the RS prop-
a dimension
complemented
cylindric
algebra) in which (A) holds.
Thus,
mented,
in the special
or have
(773). Theorem
the theorem
a problem
2.14,
posed
namely
raised
which
Theorem
In fact,
complemented
Theorem
in the theorem
possess
a property
are normal,
dimension
comple-
without
condition
2.7 may be re-stated
algebras,
2.14 provides
of Preller
stronger
[l0],
is germane
a partial
it is required
than dimension
to
solution
that
to
the
complemented-
the property
x, y there
the question
(T) is replaced
by dimension
given an affirmative
answer
by (B), and, correspondingly,
3. Some equivalences
Craig
L2] showed
every
substitution
diagonal
of algebras
for dimension
by Preller;
algebra
(T) For every
Preller
stated
in Pre Her [lOj.
substitution
ness,
case
the RS property,
elements
exists
k such that
S^x = x and S*y = y tot allA<
whether
the theorem
in [10] holds
complementedness
to tfcis question,
that
Preller's
in substitution
that in a polyadic
admits
a conjugate,
may be defined
2.8).
We have
(our Definition
that
(SA be replaced
In a recent
with generalized
of substitutions
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
cK be defined
by (nl).
paper,
diagonal
and that cylindrifications
in terms
condition
with the proviso
algebras.
algebra
in case
a.
William
elements
and generalized
and their
conjugates.
1973]
CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS
In this section
conditions
we shall
which
are shown
characterization
tions
use the ideas
introduced
to be equivalent
of cylindric
algebras
by Craig to formulate
to (tt1)—(tt3),
in terms
175
and thus
of substitutions
several
obtain
another
and Boolean
no-
alone.
In this
element,
and the succeeding
then the least
lemma, we record
3.1.
pages,
element
of B will be designated
a few elementary
Lemma.
if B is an ordered
properties
Let A be a Boolean
(i) // / has a conjugate
fg(x) = miniy
Suppose
g(x) = miniy
£ tange
Proof,
for each x £ A,
lj.
x £ A, then g is the conjugate
(a) is an immediate
consequence
For (b ), we have
of Definition
element.
just seen
that
If
of f.
2.5 and the fact that
g(x) < g(x), so by (a), x < fg(x), hence
g(x) £ \y £ A: f(y) > xj; now if f(y) > x, then by (a), g(x) < y, which
For (c), we have
of A.
for each x £ A,
x £ A, \y £ A: f(y) > xj has a least
£ A: f(y) > xj for each
(i)
and let f be an endomorphism
f: y > xj,
£A:/(y)=
that for each
/ is an endomorphism.
In our next
g, then
(d) g(l)=miniy
(ii)
by min B.
of conjugates.
algebra
(a) x < f(y) ~ g(x) < y,
(b) g(x) = min \y £ A : f(y) > xj,
(c)
set and B has a least
x < fg(x),
proves
(b).
so fg(x) £ {y £ tan f: y > xj; now if
y £ ran /, say y = f(z), and y = f(z) > x, then by (a), g(x) < z, so /g(x) < ¡(z) = y, which
proves
(c).
(ii)
Finally,
Clearly
(d) is a special
fg(x) < /(y) =» * < f(y).
a Boolean
case
f(y) > x => g(x) < y.
Thus,
endomorphism,
We introduce
of (b).
By hypothesis,
x < /(y)
<=>g(x) < y; from this
we immediately
two new conditions
(7t4) for all K, X < a,
From Lemma 3.1, with
deduce
and the fact
that g is the conjugate
for substitution
Sx admits
(775) for all K, X < a and every
fg(x) > x, so g(x) < y =>
that / is
of /.
G
algebras:
a conjugate;
x £ A, {y: Sxy > xj has a least
Sx replacing
/, together
element.
with Lemma 2.6(iv),
we imme-
diately get the following result:
3.2.
Theorem.
/t2 every
substitution
algebra,
the following
conditions
are
equivalent:
(i) (77I) and (tt2) hold;
(ii)
(774) holds;
(iii)
(775) holds.
A substitution
gebra
admitting
From 3.1(i)b,
lowing
algebra
satisfying
these
conditions
is called
a substitution
al-
conjugates.
with
two conditions
Sx replacing
are equivalent
(D) Tx is the conjugate
/ and Tx replacing
for every
g, we deduce
SAa admitting
of Sx, tot all k, X < a,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
that the fol-
conjugates:
176
CHARLES PINTER
(E)
[j anuary
Txx = miniy: Sx y > xj, for ail k, X < a and every x £ A.
Similarly,
in every
by 3.1(i)c,
SAa admitting
(F)
with
Sx replacing
conjugates,
/ and Tx replacing
(B) is equivalent
cKx = S^TKxx if À / k, where TX is defined
and by 3.1(i)d,
(C) is equivalent
(G) dKX= Tfl,
equivalent
by (D) (resp. (E)),
to
remarks,
(775)) replacing
The purpose
that
where T$ is defined by (D) (resp. (E)).
By 3.2 and our last
(?74) (resp.
g, it follows
to
Theorems
2.7 and 2.14 can be re-stated
(77I) and (772), (F) replacing
of our next result
is to show that
with
(B), and (G) replacing
(773) may likewise
(C).
be replaced
by
conditions.
We introduce
three
new conditions
for SA's
admitting
conjugates:
(776) If x £ rg K and y £ rg À and x < y, then there is some
such that
2 £ rg k O rg X
x < z < y.
(nl)
&.dKX/ a for all k, X < a, where d x is given by (C).
(778) Si T£= T^SÏ if K / p, v and X / p, where T^ is defined by (D) or (E).
Condition
3.3.
Theorem.
admitting
tively,
(776) is a special
conjugates,
case
of condition
// 21 = \A, +,•,-,
and
then the following
(Ind) in Jurie
0, 1, Sx)K x<a is a substitution
c , d . and Tx are defined
three
L8].
conditions
algebra
by (B), (C) and (D), respec-
are equivalent:
(i) (tt3) holds;
(ii)
(776)and (nl) hold;
(iii)
(778) holds.
Proof. (773) => (776) A (777). If (773)holds, then 21 is a cylindric
2.7.
Thus,
if x £ rg k and y £ rg À and x < y, then
rg À and ex
= c ,c
x = c
c . x £ rg k, so
x < ex
< c
algebra by
y = y; but
c^x £ rgK n rg X. This
proves
c. x £
(776);
(777)holds by [6, I.3.3].
(776) A (777) => (7r3). Assume
(3.4)
If a=
ci
3, this
that (776) and (777) hold.
£ rg v
is the same statement
We have
for all v / k, X.
as (777); thus,
we assume
that
a>
3.
K, A; for any n / k, v, SxS^dKX = S^SXdKX = 1, so by (?72), dKX<s^KX-
lows by 2.4(i) and (vi) that cvdRX < c^S^d^
2.4(vii),
= S^dKy
Let
v 4-
It fol-
But S^dKX < cvdKX by
hence
(*)
5Xx=cAx
for any 77^ k, 7,.
Now by (777), there
is some
K, X, tot otherwise
dKX = 1, and therefore
assume
for otherwise
that
p/v,
p < a such
that
dKX £ rg p; we may assume
(3.4) certainly
(3.4) holds
immediately.
holds;
Now,
that
similarly,
let
p /
we may
n / k, p, v;
by (S6), SpvndKX = SvnS»dKX= SldKX, so SyKX £ rg p. But by (*), S»dKX =
SyKX,
hence
S^dKX £ rg p. Thus, by (S3) and <S4), SvßdKX= S^d^
dKX, so d . £ rg v. Next, we have
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
= S^kX
=
1973]
CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS
(3.5)
c„c.=c.c„
Note first that
cKcxx
By' (B), z = c..c,x,
K A
hence
CKcxx £ rß ^' so
for all k, X < a.
£ rg X. Indeed,
c c x e rg K, so by (776), there
by 2.4(iv),
is some
ex
< cKcxx;
= ckc\Xm
This
cally, cKc x < c cKx, proving (3.5).
but c^x £ rg À and
z £ rg k Pi rg À such that c.x
c^c.x
< c.x,
K A £ rg° A. Now x —
A
cxckcXx
177
shows
Finally,
so cc^x
X K
that
c c
x < c
< z < c cxx.
< c,c„c,x;
—
A K A
but
c x; symmetri-
if p 4 k, X, then
cuSXx=^\-CKix-dKX)]
by 2.6(H),
= cKcp.{x-dKX]
by3-5-
= cK^cp.x ' dKX^
by 2-4(v)and
= Sxc¿A
by 2.6(H).
3-4>
(778) =» (?73). By (778), (S6) and (F), it p 4 «, X, then SKxCflx = SKxS^T>^x =
S^S^x^S^Slx-c^x.
(773) =»(778).
First,
we note that in every
(3.6)
conjugates,
T*x-cKx.dKk.
Indeed,
(3.6).
it is easily
verified
We have already
that (3.4) holds.
that
proved
ex
that
• d . = minly:
We shall
operations
gebra
(n3) ==>(776) A (777), hence
together
(of degree
algebra
and
algebras
whose
with substitutions
a) with conjugates,
T^ is the conjugate
primitive
and their
briefly
1, Sx, r^)KX<awhere
operations
conjugates.
a SACa,
(A, +,-,-,
ample,
SACamay
(57)
be axiomatized
Indeed,
It is obvious
that
class(
together
) of algebras.
For ex-
with
TKxix + y) = TKxx + TKxy,
(58) x<SÏTÏx,
(s9)
al-
of Theorem 3.2.
it is an equational
by (Sj)—(Sg)
By a substitution
we mean an algebra
of S1^ for all k, X < a.
of SACa is that
are the Boolean
0, 1, Sty KX<a is a substitution
\A, + , • , _ > 0, 1, Sy K x<a W*H satisfy the conditions
The interest
we may assume
G
now consider
+ , -,-,0,
S^y > x\, so by (E) we get
Thus, by (7r3), (3.4) and (3.6), SKX
T£ x = SKx(c^x •dfjLV)=
^XC^-SUau-c^x.d^^T^x.
^.
SAa admitting
and
x>tkxskXx.
suppose
rem 1.14].
21 is an SACa;
then (S7) holds
by Jonsson
Next, - TKxx. T$x = 0, hence - S^T^x
Sxx • S'x - x = 0, so Tx Sx x • - x - 0, giving
(SA.
and Tarski
[7, Theo-
• x = 0, giving (Sg). Finally,
Conversely,
one verifies
imme-
diately that if (S7)-(Sg) hold, then Tx is the conjugate of Sy
From our last remarks,
of substitution
algebras
( ) For the notion
together
with Theorem
with conjugates
of equational
class,
satisfying
see,
2.7, it is clear
that the theory
(778) is definitionally
for example,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
[6, p. 44].
equivalent
178
CHARLES PINTER
to the theory
the strong
of cylindric
sense
tion whether
it is possible
operations
tutions
algebras.
that all added
to define
are the Boolean
equivalent
each
for a class
algebra
additional
of algebras
operations
lindric
algebra
c
this,
is equational
subclass
0, 1, S'y)
class
equivalent
x<a subject
which
and is
any set
to CA , where
to (S,)—(S,)
and
are not equations.
0, 1, S'x)K x<a£ SAa which
\A, + , • , -,
0, 1, c , d x>
.
is a cy-
of SAa,
given
in Birkhoff
if and only if K is closed
and arbitrary
to give an example
direct
under
[l]:
a class
of dimension
then B' = (B,U,n,^,
the formation
products.
of subalge-
For our purposes,
of a CA^ with a subalgebra
a) with base
<d (see
which
it will
is not a CA*.
[6, Definition
1.1.5]).
If we define
CK(X n DkX) tot each X £ B,
0, °>ù>,^>KtX<û,isa
P £ B be the set of all strictly
that is, P = \p £ "&>: k < X => p
< pxj.
CA*.
increasing
sequences
of natural
Let C = ¡0, wu>, P, ~Pj;
(C, U, O, -v, 0, "to, StyK x<ù) is a subalgebra
of B'.
for P ^ 0 and Sx P = 0; but this
numbers,
it is easy to see
that for all k, X < co, S^ 0 = 0, S^Cco) = Mû),Sx P = 0, and S^P
not be discrete,
K of similar
Let B = (ß, U, n, ~, 0, "co, CK, DKA>KiA<Cl)
be a full cylindric
SlX=
Let
of substi-
we will show that
the class of all (A, +,.,-,
images
3.7. Example.
set algebra
only primitive
K is an equational
is definitionally
axioms
we use the criterion
homomorphic
suffice
whose
in
the ques-
and Sxx = c (x • d' x) tot all x £ A and k ¡¿ X. We will prove that
To prove
bras,
which
and d . such that
CA* is not an equational
algebras
that
raises
(the conjugates
in the negative:
(A, +, . , -,
must include
Let CA* designate
admit
such
this question
is a structure
conditions,
K of algebras
and substitutions
as primitive)
equivalent
This
to CAa.
We will now answer
of axioms
they are definitionally
are equational.
a class
operations
are not to be taken
definitionally
In fact,
definitions
[January
= œco. Thus, S =
If S were a CA*^, it would
contradicts
[6, 2.1.9(ii)];
thus
S
is not a CA*ÙJ .
3.8.
we have
Remark.
shown
In Example
that
rem 2.7, condition
3.7, one easily
S is not a CA*^; this
verifies
It might be of some interest to consider algebras
algebras
substitution
and (B) holds.
algebra
with quantifiers,
for example,
(7r5).
it clear
algebras
are easily
The class
details).
derived
\A, +,-,-,
that
But
in Theo-
0, 1, Sx, cK'K x<a to
where \A, +,.,-,
of these
it may be axiomatized
2.4 (i)—(v); (we omit the simple
and cylindric
makes
(?r3) is indispensable.
be called substitution
to be equational;
that S satisfies
observation
0, 1, Sy
algebras
is easily
by (Sj)—(S6)
The relationship
from Theorem
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
between
2.7.
A
together
is a
shown
with
these algebras
1973]
CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS
179
BIBLIOGRAPHY
1. G. Birkhoff,
On the structure
of abstract
algebras,
Proc.
Cambridge
in algebraic
logic,
Studies
Philos.
Soc.
31 (1935), 433-454.
2. W. Craig,
Logic,
Math.
3.
Unification
Assoc.
P. Halmos,
(1956), 363-387.
4.
The basic
in Algebraic
(to appear).
concepts
of algebraic
logic,
Amer. Math. Monthly
63
MR 19, 112.
', Algebraic
(1956), 217-249.
5. -,
and abstraction
of America
logic.
L Monadic
Boolean
algebras,
Compositio
Math.
12
MR 17, 1172.
Algebraic
logic.
II. Homogenous
locally
of infinite degree, Fund. Math. 43 (1956), 255-325.
6. L. Henkin, D. Monk and A. Tarski,
finite
polyadic
Boolean
algebras
MR 19, 112.
Cylindric
algebras,
North-Holland,
algebras
with operators.
Amster-
dam, 1971.
7.
B. Jonsson
(1951), 891-939.
8. P.-F.
and A. Tarski,
Boolean
I, Amer.
J. Math. 73
applications
a l'al-
MR 13, 426.
Jurie,
Notion
de quasi-somme
amalgamée:
Premieres
gèbre boolérienne polyadique, C. R. Acad. Sei. Paris Sér. A-B 264 (1967), A1033—A1036.
MR 37 #3975.
9. L. LeBlanc,
Transformation algebras,
Canad. J. Math. 13 (1961), 602—613.
MR
24 #A1858.
10.
A. Preller,
Substitution
algebras
in their
relation
to cylindric
algebras,
Arch.
Math. Logik. Grundlagenforsch. 13 (1970), 91-96.
11.
H. Rasiowa
and R. Sikorski,
Math. 37 (1950), 193-200.
A proof
of the completeness
theorem
of GödeI, Fund.
MR 12, 661.
DEPARTMENT OF MATHEMATICS, BUCKNELL UNIVERSITY, LEWISBURG, PENNSYLVANIA
17837
DEPARTMENT OF MATHEMATICS,UNIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA
94720
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use