D IAGRAMMES

I. S AIN
Weak products for universal algebra and model theory
Diagrammes, tome 8 (1982), exp. no 2, p. S1-S15
<http://www.numdam.org/item?id=DIA_1982__8__A2_0>

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Diagrammes, Volume S, Paris 1982«

WEAK PRODUCTS
FOR
UNIVERSAL ALGEBRA
AND
MODEL THEORY

1« Sain

Abs tract»
Weak products of arbitrary universai algebras are introduced. The usual notion f-rr groups nd rings is a spécial case.
Some universal algebraic properties are proved and applications
to cylindric algebras aire considered.

Introduction»
The universal algebraic notions of weak products introduced in the literature so far (e# g. <9> et <10> ) are not
universal algebraic at ail. For any similarity type
dénote by

M

class

of similar algebras the weak products as defined

M

the class of ail algebras of type

t we

t . In any

e. g. in

Gratzer <9> do not exist, except in the trivial case

when

t

consists of a single constant symbol

t «

{ <cfo> } )• Further, neither rings with unit élément, nor

ony

(i. e# when

Boolean algebras hâve weak products in the sensé of <9>« At the
saroe time, thèse weak products play an important rôle in récent
literature see e. g. Monk <19> (while they do not exist in the
sensé of <9> )• Thus we conclude that the universal algebraic

S 2
notion of weak products introduced in <9> are highly unsactisfactory (they do not exist in any nontrivial similarity class
of algebras ! ) . Hère we suggest an improved version which
exists in most cases.
Hère we would like to mention one misconception of many algebraitts. Some of them believe that weak products are used in
algebra only to obtain structure theorems. I. e. t they believe
that the only purpose of weak products is structure theorems.
Perhaps this may hold in group theory, but weak products play
an important rôle in Boolean algebra theory and for Boolean
algebras no structure theorem holds with weak products.
Hence the above quoted pejudice of some algebraists is false.

Throughout, t
larity class
similarity

M

is a similarity type such that in the
of ail ialgebras of type

t

every alge-

t

contains at

bra has a minimal subalgebra.

Remark?
There are two ways of achieving this: either

least one constant symbol or else the empty algebra is not excluded from

M

Let O. € M
>P H.

iei

x

• Hère we do not care which one is the case.

for each

i € I , for some set

I.

dénotes the product of the algebras in the usual sensé,

cf. <9> or <10> D.0.3.1.
The following définition généralises R.0.3.60 of <L0> p. 104 .
It is also a généralisation of <9> p. 139 • Note that the
Boolean algebras do not hâve infinité weak products in the
sensé of <9> but they do in the sensé of Définition 1 below.
Weak products of Boolean algebras proved to be rather useful

S 3
in e. g. Monk <19> .

DEFINITION 1.
The weak product

P

Ci.

of the System

< CL s i € I >

of

algebras is defined as follows:
- let

M

dénote the universe of the minimal subalgebra of

P ax ,

i€I

- now

-

P w A. = {f€ P A. s ( 3 g €M)(£ i€I : f(i) t g(i)} finite)}
i€I x
i€I x

w
P Q.
i€I X

is defined to be the subalgebra of .ET Cl.
1§I

with

X

W

universe

P A. •
i€I x
Clearly,
P w Cl. i s unique.
i€I X

PROPOSITION.
(i) Définition 1 is correct in the sensé that
subuniverse of
w
P Cl.
i€l x

(ii)

P A.
1€I X

is a

Pfi.•
i€I X

is a subdirect product, if the minimal subalgebra

W

M

of

P fi. is nonempty, e. g. if
i€l x
symbol.

t

contains a constant

The proof is left to the reader.

Let

K c M

. I. e. K

is a *lass of algebras of type

t.

S 4
W

P K

dénotes the class of ail weak products of possibly infi-

nité families of algebras in K s
P*k = { P w fi s { fi s i ç I } is a subset of K l ,
X
i€I
*

w
Po K dénotes the class of weak powers of éléments of K .
We shall use the notations H K , S K , P K as defined in <9>
et <10> .
Up K

dénotes the class of ultraproducts of éléments of K

see <10> .
We shall consider

H , S , P , Up , P w

on the class

of ail universal algebras of some fixed

M

similarity type

Po w

and

as operators

t . See <17> , <10> p. 89 above T.0.3.17 ,

<4> , <13> p. 387 , or <9> p. 152 §23 .
Namely, to any class K c M
another class

H K <= M

the operator

K c Mt

correlates

. Juxtaposition of names of operators

dénotes composition. Namely, HSP
with each

H

the class

is the operator correlating

HSP K , see p. 109 of <10> ,

or <9> .
The statement

" HH = H •• means that for every type

every class K c= Mt

we claim

SH K

*

t

and

HH K = H K •

See T.0.2.23 of <3L0>. On the other hand, SH £ HS
there exist a type

t

and a class

K c M

means that

such that

H S K , cf. 0.2.19 of <10> .

See also <17> .

Recall from e. g. <13>, <12> or <15> Thm. 3 that
and

SPUp

HSP

are the closure operators of generating the smallest

variety and generating the smallest quasivariety respectively.
I. e. : HSP K

and

SPUp K

are the smallest classes containing

K and axiomatisable by équations and equational implications

S 5
respectively.

PROPOSITION 2.
(o) P * P W ,
(i) HSUp P W a HSP = HSU

P ,

SUp P W s SPUp * SUp P ,
(ii) HSPW Up * HSP ,
SP W Up ft SPUp ,
(iii) HSPW K

is not first order axiomatisable, for some

KEMt ,
(iv) HSPW , SP W , HP W , P W , HSPW Up , SPW Up are not
closure operators,
though, HSUp P w
(v) HSPow

SUp P W

and

are closure operators,

préserves the formulas of the following shapes

V ( A
i <<x
where

a

e.)

i < j <a

is an arbitrary ordinal and

is a set of équations, and

J

{e. : j < a }

{e. : j < a }

contains

a finite set of variables only (i. e. let

p

be a

formula of the above shape, then K |s 3

implies

HSPow K |r 3 ),
(vi) SPoW

préserves ail the formulas of the shapes
A
n€N

where
e

>

n

V
(
A
e )
i< a
i < j < <X J

is an arbitrary set, a

, e. are équations (of type

fe
L

N

e

n

• e. : n€N , j <oc }
j

variable only.

J

is an ordinal and
t ), and

contains a finite set of

S 6
Proof.
Notations if
Q K 5 (X K

Q , (h

are operators, then

Q 5 Q^ means that

for every K , see <17> .

Proof of (i).
It is known that

HSP = HSUp P , see e. g. <10> 0.4.64. To

W

prove

SUp P s SUp P = SPUp

we shall use the following

lemma.
Lemma 1.
Let

f
P

and

r
P

dénote the operators of taking ail finite

products and ail reduced products respectively. Let
f
r
an operator such that P c Q c S P 1 . The
Then s

Q be

SUp Q a SPUp .

Proof of lemma 1.
Some notations %
- let K c M
Univ K =

, then

{ ( A e.
i€I x

>

V p. )s
j€J J

K 1^

( Ae.

> VP.)
J

and
£ei , Pj s i€I , j€J }
is a finite set of
équations of type
Qeq K =
_ if

L

£ ( h e.
i I

> p ) : ( A e±

is a set of formulas then

of ail models of

Md £

t

} ,

> p) € Univ K } ,
dénotes the class

£ •

Now, SUp Q K a Md Univ Q K , by <10> T.0.3.83
or Thm. 3 (v) of <15> .

and C.0.3.70

S 7
It is not hard to prove thats
p f K 1= ( A e .

>

V

1

j€J

(se)

p.) and J is finite
2

jjnply

( 3 j€J) K l « ( A e . —•

> p..) ,

(see e. g. Lemma 5 of <15> ).
Now (x) , Pf ç Q c SPr , and the fact that

SPr

préserves

quasiequations (i. e. éléments of Qeq 0 ) imply that:
Md Univ Q K = Md Qeq K .
It is known that

Md Qeq K = SPUp K , see e. g. <12> ,

<15> Thm. 3 (vi).
QED of lemma 1.
Since

P f c p c SP r

and

Pf c P w c sPr , lemma 1 implies

SUp P W = SUp P s SPUp .
By this (i) is proved.

Proof of (o), (ii) and (iv).
To prove (o), (ii) and (iv) it is enough to prove s
HSPW Up £ P
We shall fix a class K

HSPW Up p P W P W .

of algebras for whichs

HSPW Up K £ P K
Let the type

and

and

HSPW Up K £ PW P W K .

t bes

t = { (0,0),(1,0),(^,1),(^,1) s i € O» } .
Now 5

K=£n€M

: A= {0,1} and for every i€ CO
n » = f . 0 = 0 and Q }s ^ 1 s 1 } .

S 8
Lemma 2.
w
" a " of an arbitrary algebra Cl € HSP Up K ,

For every élément
either

£f.a s i€ u> }

is finite or

£gia s i€ co} is finite.

Proof of lemma 2.
w
It is enough to prove lemma 2 for every Cl € P K , since the
operator

HS

Let Cl =

P W Cl.
i€I

Let

"prserves" the above property and U p K = K .
and

{ 0. » l € l ) c K .

a € A , a = < a. s i€I > .

Now, either

{ iÇI Î a. ^ 0 }

is finite or

{ i€I s a i £ 1 }

is finite.
Now, K l = { f.O = 0 , g.l = 1 •• i€u>}

complètes the proof

of the lemma.
QED of lemma 2.
Now we def ine a System < fî. î if w + w
Let

>

of algebras of

i,j € to .

In the algebra Cl.
f 0

we def ine the opérations

if

f. and

j< i
Identity

l- 1 otherwise
In Q

K.

, .

we def ine
(
" 1

g.(0) =

if

f. and

g. as:

j< i

i

f^ - Identity
L 0

otherwise

n' * P
n. and Oî = P W O, x P W n
x
i€<0+i»
i€<»>
i€«**
Now n 1 € P K , QJ € PV «w
P K and q ç f l '
Let

r

.

g, as s

S 9
For the élément
neither

1

{ f-a

a* = < 0 , 0 , ... , 1 , 1 , ... > = A£
s i€u>}

nor

£ g^» Î iÇCo)

is finite

and fl1 ).

(both in q

Thus, by lemma 2 , neither Cl9 nor fî£ is in HSP Up K ,
proving

HSPW Up £ P

HSPW Up p

and

P W PW .

By this (o), (ii), and (iv) are proved.

Proof of (iii).
Recall from <L0>

that

Lf^

dénotes the class of ail

locally finite dimensional cylindric algebras.
HSPW Lf . = Lf' . but

Lf, .

is not axiomatisable.

Proof of ( v i ) .
Let

3 = ( A e
>
V (
A
e ) ) be a
n
n€N
i < et i < j < a J
formula of the required shape and l e t £x-, . . . ,x m ) b e t h e
set of variables occuring in 3 •
Let Cil25 3 • We hâve to prove
PW a = fif
i€I x
where

Cl.

is O

\=

3

for every

,
i€I •

Suppose that

PWn.
I* ( A e )[*, ... , a] .
x

i€I

n€N

For every projection function
a r (i).
We then hâve

£11=-(

A e
n€N

pj.

we dénote

PJi(ar^

) fa.(i), ... ,a (i)J

b

y

.

Then, since Cl & 3 , we hâve

n i- V (

A

zGa z i < j < a
Thus for every

i€I

there exist s

ejfa^i), ... ,a (i)]
J

u

z.Ga

such that:

S 10

Cl \* (

A.
z. < j < a

e.)
j V a ) , ... »am(i)] .
J

i —

I. e. such that:
Q \* £ e.. s z i < 3 < a î ^ d ) , ... #am(i)3 •
W

Since Cl. is Q for every i€I and a1, ... ,a € P A. ,
there is a finit J ΠI such thats
£ < a 1 (i), ... ,am(i) > : 161) c £ < 8^(1),...,am(i) > s i<=J}(
Let r be the greatest élément of £ z^ s i€I } (it exists
since J is finite).
Now s

n i - £ e j S r < j < ( x 3 [ a ^ i ) , . . . #am<i)l ,
for every

i€J , and therefore also for every

i€I .

This implies s
PW Clt

^

••• > a J >

£ e j s r < j < a }£*i>

since subalgebras and direct products préserve équations and
w
P cSP .
Therefore s
w

p n, >= ( A en —>

i€I

X

n€N n

Since su, ... ,a

\/
z<a

(

A

z < j< a J

e^rV'-'O
L

^

mJ

was arbitrary, (vi) is proved.

(v) is a conséquence of (vi) and the fact that H préserves
positive formulas even if they are infinitary.

Remark.
Properties of the operator HSP
and <L8> .

were investigated in <7>

S 11
Recall that if K

contains finite algebras only then

P K

contains no countable algebras.

PROPOSITION 3.
Let

t

contain a constant symbol.

Let

a* be an infinité cardinal such thats

(3 n e K ) i < )A 1 <a%.
w

Then

i

P K

contains an algebra of cardinality a •

Proof,

Let Q € K be such that
Cl9 =

Let
Now,

PW Cl é
i€a
f
JA | as a 1

1 < |A| < a 1 •

because

jA|>

2

and o^a1 = a1

(since a* is infinité).

3ED.

EXAMPLES.
1. Weak direct products of Boolean algebras hâve been studied
recently by J. D. Monk <19>, <14> , see also p. 20 above
question 50 in <6>.
2. In discussions of various spécial classes of algebras,
in particular in the théories of groups and rings, weak
products actually play a more important rôle than ordinary
direct products, cf. e. g. <10> p. 105.
w
3. P
is specially important for cylindric algebras because
P W Lf = Lf
moreover HSPW Lf = Lf
and the class Lf

a

a

a

a

is the class of ail first order théories when considered
as algebras. See Thm. 5.3. of <2> and V.5, VI.5 of <1>.

a

S 12
PROPOSITION 4.

P W Lf

= Lf

a

a

Proof.
Let n, € Lf
i
a

for every

i€I .Let

f €

P w A.
i

be arbi-

i6I

trary. By définition 1 there i s a

g € M c

P A. , where
i€I X

M

is the minimal subalgebra of

P fl. , such that
i€I X

g

differ only at finitely many places, i. e.s

f and

£ i € I s f(i) t g(i) } is finite.
By T.2.4.2. of <10>
A f = U £Af(i) s i € I } .
Also (Vi€I) Ag(i) c û g and
Ùs% is finite by
T.2.1.16 of <10> . Since (ViÇI) [ û f ( i ) is finite3
by O. € Lf
i

, we can conclude that also

(X

J\f

is finite.

5ED.
PROBLEM (cf. <10>).
Dénote by

<f\r the class of simple éléments of

HSPW d+r c

Lf

Lf •

, obviously.

Now, is it also true that

HSPW<for = L f ^ ?

The importance and basic properties of the class <Àj" were
discussed in <1>, <2>, <3>, <11> and in <16> .
Continuation of examples.
4. Weak direct sum of vector spaces is a spécial case of weak
w
product P
as defined hère, see <5> p. 42.
5. Direct sums of modules are also a spécial case of weak products.
Direct sums of Abelian groups are also a spécial case, see
e. g. <8>t

S 13
Recall that for groups, rings, semigroups with zéro (annihilât or) P w P w = p w , see also Prop. 5 (i) below.

PROPOSITION 5.
W

(i) Let

V « P V

be a class of algebras in which the

one-element algebra is initial (i. e. every algebra
in V

contains a minimal subalgebra and the minimal

subalgebra has exactly one élément).
Then in V
K c V

we hâve

we hâve

P

W

Pw Pw = Pw
W

, i. e. for every

W

P K = P K .

(ii) For Boolean algebras, P W , SPW , SPW Up , P W Up
are not closure operators.
(iii) For rings < R ; + , . , 0 , 1 >

with unit (ii)

holds.
Proof.
Proof of (i) is left to the reader.
Proof of (ii).
Let

2 ^ s < 2 j fl , U , \ >

dénote the two-element Boolean

algebra, and K » £ j£ } .
Let Cl9 6 SPW K
Then

f

be arbitrary.

f

(V a €A ) [ £ x€Af : x > af } is finite or
£ x€Af s x < a1 } is finite3

But this is not true for éléments of
- let

Cl9 =

PW

Pw Pw K s

2 ,

IÇUI

- then

(3 af € A1 x A f ) £ £x U ' x A '

: x > af } is infinité

and
£x € Af x A1 : x < a* } is infinité^ •
Of course, <
Clearly

is understood in n 1 x n 1 .

K = Up K

and thus

SPW Up K = SPW K .

S 14
Proof of (iii) (for rings with unit).
Let

2 = <

(0,1) i + , . , 0 , 1 >

1

defined by

1+1=0

be the ring with unit

(this is the two-element Boolean

ring).
Def ine <

by:
x < y

iff

x.y = x •

Now the proof given for (ii) works by taking K = £ 2 } .
QED.

PROBLEM.
Find a category-theoretic characterisation of weak products.

REFERENCES.
<3>

Andréka H. , Gergely T. , Németi I. : Purely algebraic
construction of first order logics. Logic Semester 1973
Warsaw Banach Center. Alsos Central Re. Inst. Phis.
Hung. Acad. Sci. KFKI-73-71. 1973 Budapest.

<2>

Andréka H. , Gergely T. , Németi I. s On Universal
Algebraic Construction of Logics. STUDIA LOGICA XXXVI,
1-2, 1977, pp. 9-47.

<3>

Andréka H. , Németi I. s A simple, purely algebraic
proof of the completeness of some first order logics.
ALGEBRA UNIVERSALIS Vol. 5, 1975, pp. 8-15.

<4>

Andréka H. , Németi I. s Formulas and ultraproducts in
catégories. Beitrage zur Algebra u. Geom. 8, 1979,
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<5>

Arbib M. A. , Mânes E. G. s Arrows, Structures and
Functors: The Categorical Imperative. Ac. Press, 1975.

<6>

Van Deuven E. K. , Monk J. D. , Matatyahu R. : Some
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<7>

Eilenberg S. , Schïïtzenberger M. P. % On Pseudovarieties.
Advances in Math. Vol. 19, No 3, 1976, pp. 413-418.

S 15
<8>

Fuchs L. : Infinité Abelian Groups. Ac. Press, 1970.

<9>

Gratzer G. •• Universal Algebra. Second édition, Springer
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<10> Henkin L. , Monk J. D. , Tarski A. : Cylindric Algebras,
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<12> Malcev I. s Algebraic Systems. Akademie Verlag Berlin,
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<3L4> Monk J. D. s On depth of Boolean algebras. Lecture at
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Mathematical Institute of
the Hungarian Academy of Sciences
Budapest, Re^ltanoda u. 13-15
H-1053 Hungary.