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D IAGRAMMES

I. S AIN

Weak products for universal algebra and model theory

Diagrammes, tome 8 (1982), exp. no2, p. S1-S15

<http://www.numdam.org/item?id=DIA_1982__8__A2_0>

© Université Paris 7, UER math., 1982, tous droits réservés.

L’accès aux archives de la revue « Diagrammes » implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impres- sion systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

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WEAK PRODUCTS FOR

UNIVERSAL ALGEBRA AND

MODEL THEORY

1« Sain

Abs tract»

Weak products of arbitrary universai algebras are intro- duced. 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 introdu- ced in the literature so far (e# g. <9> et <10> ) are not universal algebraic at ail. For any similarity type t we dénote by M the class of ail algebras of type t . In any class M of similar algebras the weak products as defined e. g. in Gratzer <9> do not exist, except in the trivial case when t consists of a single constant symbol ony (i. e# when t « { <cfo> } )• Further, neither rings with unit élément, nor 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

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notion of weak products introduced in <9> are highly unsactis- factory (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 alge- braitts. 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 is a similarity type such that in the larity class M of ail i

bra has a minimal subalgebra.

similarity class M of ail algebras of type t every alge-

Remark?

There are two ways of achieving this: either t contains at least one constant symbol or else the empty algebra is not ex- cluded from M • Hère we do not care which one is the case.

Let O. € M for each i € I , for some set I.

>P H. dénotes the product of the algebras in the usual sensé,

iei

x

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

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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 a ,

i€I x

- now Pw 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. is defined to be the subalgebra of .ET Cl. with

i€I X 1 § I X

W

universe P A. • i€I x

Clearly, Pw Cl. i s unique.

i€I X

PROPOSITION.

ion 1 is correct in the sensé that

1€I X

(i) Définition 1 is correct in the sensé that P A. is a subuniverse of P fi. •

i€I X

(ii) P Cl. is a subdirect product, if the minimal subalgebra w i€l x

W

M of P fi. is nonempty, e. g. if t contains a constant i€l x

symbol.

The proof is left to the reader.

Let K c M . I. e. K is a *lass of algebras of type t.

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W

P K dénotes the class of ail weak products of possibly infi- nité families of algebras in K s

P*k = { Pw fi s { fi s i ç I } is a subset of K l ,

i€I X

*

Po K dénotes the class of weak powers of éléments of K . w 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 , Pw and Pow as operators on the class M of ail universal algebras of some fixed similarity type 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 the operator H correlates another class H K <= M . Juxtaposition of names of operators dénotes composition. Namely, HSP is the operator correlating with each K c Mt the class HSP K , see p. 109 of <10> , or <9> .

The statement " HH = H •• means that for every type t and every class K c= Mt we claim HH K = H K •

See T.0.2.23 of <3L0>. On the other hand, SH £ HS means that there exist a type t and a class K c M such that

SH K * H S K , cf. 0.2.19 of <10> . See also <17> .

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

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respectively.

PROPOSITION 2.

(o) P * PW ,

(i) HSUp PW a HSP = HSU P , SUp PW s SPUp * SUp P , (ii) HSPW Up * HSP ,

SPW Up ft SPUp ,

(iii) HSPW K is not first order axiomatisable, for some KEM t ,

(iv) HSPW , SPW , HPW , PW , HSPW Up , SPW Up are not closure operators,

though, HSUp Pw and SUp PW are closure operators, (v) HSPow préserves the formulas of the following shapes

V ( A e.)

i <<x i < j < a J

where a is an arbitrary ordinal and {e. : j < a } is a set of équations, and {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 e > V ( A e ) n€N n i < a i < j < <X J where N is an arbitrary set, a is an ordinal and e , e. are équations (of type t ), and

fe • e. : n€N , j <oc } contains a finite set of

L n j J

variable only.

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Proof.

Notations if Q , (h are operators, then Q 5 Q^ means that Q K 5 (X K for every K , see <17> .

Proof of (i).

It is known that HSP = HSUp P , see e. g. <10> 0.4.64. To prove SUp PW s SUp P = SPUp we shall use the following lemma.

Lemma 1.

Let P and P dénote the operators of taking ail finite f r products and ail reduced products respectively. Let Q be

c Q c S P1 . The SUp Q a SPUp . an operator such that P c Q c S P . Then s f r

Proof of lemma 1.

Some notations % - let K c M , then

Univ K = { ( A e. > V p. )s K 1^ ( Ae. > V P . )

i€I x j€J J J

and

£ei , Pj s i€I , j€J } is a finite set of

équations of type t } , Qeq K = £ ( h e. > p ) : ( A e± > p) € Univ K } ,

i I

_ if L is a set of formulas then Md £ dénotes the class of ail models of £ •

Now, SUp Q K a Md Univ Q K , by <10> T.0.3.83 and C.0.3.70 or Thm. 3 (v) of <15> .

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It is not hard to prove thats

pf K 1= ( A e . > V p.) and J is finite

1 j€J 2

(se) 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 Pf c p c SPr and Pf c Pw c sPr , lemma 1 implies SUp PW = 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 and HSPW Up p PW PW . We shall fix a class K of algebras for whichs

HSPW Up K £ P K and HSPW Up K £ PW PW K . Let the type 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 } .

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Lemma 2.

For every élément " a " of an arbitrary algebra Cl € HSP Up K , w either £f.a s i€ u> } is finite or £gia s i€ co} is finite.

Proof of lemma 2.

It is enough to prove lemma 2 for every Cl € P K , since the w operator HS "prserves" the above property and U p K = K . Let Cl = PW Cl. and { 0. » l € l ) c K .

i€I

Let a € A , a = < a. s i€I > .

Now, either { iÇI Î a. ^ 0 } is finite or { i€I s ai £ 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 > of algebras of K.

Let i,j € to .

In the algebra Cl. we def ine the opérations f. and g, as s

f 0 if j < i l- 1 otherwise

Identity

In Q , . we def ine f. and g. as:

(" 1 if j < i

g.(0) = i f^ - Identity L 0 otherwise

Let n' * P n. and Oî = P

W

O, x P

W

n . i€<0+i»

x

i€<»> i€«**

V «w

Now n

1

€ P K , QJ € P P K and q ç f l '

r

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For the élément a* = < 0 , 0 , ... , 1 , 1 , ... > = A£

neither { f-a1 s i€u>} nor £ g^» Î iÇCo) is finite (both in q and fl1 ) .

Thus, by lemma 2 , neither Cl9 nor fî£ is in HSP Up K , proving HSPW Up £ P and HSPW Up p PW PW .

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

Proof of (iii).

Recall from <L0> that L f ^ 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 i n 3 • Let Cil

25

3 • We hâve to prove

P

W

a = fi

f

\ = 3 , i€I

x

where Cl. is O for every i€I • Suppose that

P

W

n. I* ( A e )[*, ... , a ] .

i€I x n€N

For every projection function pj. we dénote PJi(ar^ by ar(i).

We then hâve £11=-( A e ) fa.(i), ... ,a (i)J . n€N

Then, since Cl & 3 , we hâve

n i- V (

zGa z

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

i

< j < a

J u

Thus for every i€I there exist s z.Ga such that:

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Cl \* ( A. e.) j V a ) , ... »a

m

(i)] .

z. < j < a

J

i —

I. e. such that:

Q \* £ e.. s z

i

< 3 < a î ^ d ) , ... #a

m

(i)3 •

W

Since Cl. is Q for every i€I and a

1

, ... ,a € P A. , there is a finit J Œ I such thats

£ < a

1

(i), ... ,a

m

(i) > : 161) c £ < 8^(1),...,a

m

(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 ) , . . . #a

m

<i)l ,

for every i€J , and therefore also for every i€I . This implies s

P

W Clt

^ £ e

j

s r < j < a }£*i> ••• >

a

J >

since subalgebras and direct products préserve équations and w

P c S P . Therefore s

p

w

n, >= ( A e

n

—> \/ ( A e ^ r V ' - ' O

i€I

X

n€N

n

z < a z < j < a

J L

^

m J

Since su, ... ,a 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 were investigated in <7>

and <L8> .

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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 i Then P K contains an algebra of cardinality a •

Proof,

Let Q € K be such t h a t 1 < | A | < a

1

Let Cl

9

= P

W Cl é

i€a

Now, JA

f

| as a

1

because j

A

| >

2

and o^a

1

= a

1

(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 PW Lf = Lf moreover HSPW Lf = Lf and the class Lf

a 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>.

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PROPOSITION 4. PW Lf = Lf

a a

Proof.

Let n, € Lf for every i€I .Let f € Pw A. be arbi-

i a i 6 I i

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 f and i€I X

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

£ 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) A g ( 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 , we can conclude that also J\f is finite.

i (X

5ED.

PROBLEM (cf. <10>).

Dénote by <f\r the class of simple éléments of Lf • HSPW d+r c 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 product P as defined hère, see <5> p. 42. w

5. Direct sums of modules are also a spécial case of weak pro- ducts.

Direct sums of Abelian groups are also a spécial case, see e. g. <8>t

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Recall that for groups, rings, semigroups with zéro (annihi- lât or) Pw Pw = pw , 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 we hâve Pw Pw = Pw , i. e. for every K c V we hâve PW PW K = PW K .

(ii) For Boolean algebras, PW , SPW , SPW Up , PW 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 be arbitrary.

Then (V af€Af) [ £ x€Af : x > af } is finite or

£ x€Af s x < a1 } is finite3 But this is not true for éléments of Pw Pw K s - let Cl9 = PW 2 ,

IÇUI

- then (3 af € A1 x Af) £ £x U ' x A ' : x > af } is infinité and

£x € Af x A1 : x < a* } is infinité^ • Of course, < is understood in n1 x n1 .

Clearly K = Up K and thus SPW Up K = SPW K .

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Proof of (iii) (for rings with unit).

Let 2 = < (0,1) i + , . , 0 , 1 > be the ring with unit 1 defined by 1 + 1 = 0 (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, pp. 133-151.

<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 questions about Boolean algebras. Preprint 1979 Jan.

Univ. Colorado Boulder, Colorado 80309 USA.

<7> Eilenberg S. , Schïïtzenberger M. P. % On Pseudovarieties.

Advances in Math. Vol. 19, No 3, 1976, pp. 413-418.

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<8> Fuchs L. : Infinité Abelian Groups. Ac. Press, 1970.

<9> Gratzer G. •• Universal Algebra. Second édition, Springer Verlag, 1979.

<10> Henkin L. , Monk J. D. , Tarski A. : Cylindric Algebras, Part I . North Holland, 1971.

<11> Henkin L. , Monk J. D. , Tarski A. , Andréka H. ,

Németi I. : Cylindric Set Algebras. Lect. Notes in Math.

Vol. 883, Springer Verlag, 1981.

<12> Malcev I. s Algebraic Systems. Akademie Verlag Berlin, 1973.

<13> Monk J. D. s Mathematical Logic. Springer Verlag GTM 37, 1978.

<3L4> Monk J. D. s On depth of Boolean algebras. Lecture at the Math. Inst. Hung. Acad. Sci., Dec. 1978.

<15> Németi I. , Sain I. s Cône Injectivity and some Birkhoff type Theorems in Catégories. Universal Algebra (Proc.

Coll. Univ. Alg., Esztergom, 1977). North-Holland, 1981, pp. 535-578.

<16> Andréka H. , Németi I. , Sain I. s Connections between Algebraic Logic and Initial Algebra Semantics of CF Languages. Mathematical Logic in Computer Science (Proc.

Coll. Logic in Programming, Salgotarjàn, 1978). North- Holland, 1981, pp. 25-83 and 561-605.

<17> Pigozzi D. s On some opérations on classes of algebras.

ALGEBRA UNIVERSALIS Vol. 2, 1972, pp. 346-353.

<18> Rosicky J# : Concerning equational catégories. Universal Algebra (Proc. Coll. Univ. Alg., Esztergom 1977). North- Holland, 1981.

<19> Monk J. D. s Cardinal functions on Boolean algebras.

Preprint Univ. Colorado Boulder, 1981.

Mathematical Institute of

the Hungarian Academy of Sciences Budapest, Re^ltanoda u. 13-15 H-1053 Hungary.

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