On representations of Lindenbaum algebras

Boman Sik o r sk i

(Warszawa)

On representations of Lindenbaum algebras

In [5] I have given a topological characterization of open theories, announced earlier in [4]. The proof of this characterization was based on the Bieger representation theorem for Lindenbanm algebras of pred­

icate calculi, proved in [2] and [3]. Another proof of that theorem was given in [1] in a more general formulation.

In this paper I give a new proof of the topological characterization of open theories. This proof is not based on the Bieger theorem but inci­

dentally I quote also a new proof of the Bieger theorem.

§ 1. Terminology and notation. We shall consider a fixed two-val­

ued first-order predicate calculus ^ containing the following primitive symbols: an infinite set V of free individual variables denoted by the letters x , у (with indices), an infinite set (disjoint from V) of bound in­

dividual variables denoted by the letters £, y, a set of functors, a non­

empty set of predicates, the logical connectives (or), r\ (and), -> (if...

then...), — (not), and the existential and universal quantifiers pj and P |.

The cardinals of the sets of variables, functors and predicates are arbitrary.

The set of all terms and the set of all formulas in Sf will be denoted by T and F respectively. Terms will be denoted by r, and formulas — by a, /?, y, with indices if necessary.

• Any formula of the form л(тх, rm), where л is an m-argument predicate and are terms, is said to be elementary. The set of all elementary formulas will be denoted by E.

The symbols 2Г and will denote formalized theories based on У (except the proof of 4.2 where the language of S'* is different). A theory is said to be open if it can be axiomatized by a set of open formulas, i.e.

formulas without quantifiers.

Symbols w, r\j — will also denote set-theoretical operations and the Boolean join, meet and complement in any Boolean algebra. Symbols pi and P| will also denote the infinite join ard meet in any Boolean algebra A.

Symbol C will denote both the set-theoretical inclusion and their Boolean analogue in any Boolean algebra. Symbols V and /| will denote the unit

R oczn ik i PTM — P race M atem atyczn e V II 7

element and the zero element in any Boolean algebra

Symbol

will always denote the two-element Boolean algebra.

§ 2. The Lindenbaum algebra. The symbol L(.T) will denote the Lindenbaum algebra of a theory i.e. the Boolean algebra obtained from F by identification of formulas a, @ if and only if both a -> /9 and /9 -» a are theorems in ЗГ. For every formula a in F, the symbol \a\y will denote the corresponding element in L(-T). We recall that the join, meet and complement in L{.T) are defined by the equalities

(1) \a\&- ^ Щг — l« w \a\r ^ \PW = \a r\ ft]#-, — \a\&- — | — a}#-*

Moreover

(2) |a|^-C if and only if a -> f is a theorem in (3) a is a theorem in F if and only if \a\j- = V . We recall also that, for every formula a(x),

(4) I U « ( £ ) L r = L U I « ( T ) k , 1 П | « ( | ) | ^ = n , e r | a ( t ) k

where a(£) and a(r) denote respectively the result of substitution of a bound individual variable | or a term r for the free individual variable x in the formula a(x).

The infinite joins and meets (4) are said to be infinite joins and meets corresponding to logical quantifiers, respectively.

A Boolean homomorphism h from the Boolean algebra L{3T) into another Boolean algebra

is said to be a Q-homomorphism provided it preserves all infinite joins and meets corresponding to logical quantifiers*

i.e. provided

(5) M l U « ( £ ) k ) = L U M | a ( T ) | ^ ) , М И Л О Ш И = П « г Л ( | а ( т ) |^ ) .

A theory 2Г' is said to be weaker than if every theorem in У is also a theorem in ZF.

2.1. * I f У is weaker than ZT, then the formula h{\a\r.) = \a\y (a in F) defines a Q-homomorphism from В { У ) into L(У).

The proof follows immediately from (3) and (4).

2.2. I f Q-homomorphisms hx and h2 from F[£T) into a Boolean algebra A assume the same values on all elements \a\#- where a is an elementary formula, then hx = h2.

In fact, the set of all \a\r such that h^la]#-) = h2(\а\&) is a subalgebra

of L(^~), closed with respect to forming the infinite joins and meets

(4) and containing all \a\r where a is elementary. On the other hand*

L(&~) is the only subalgebra satisfying the conditions just mentioned.

Hence it follows that hx(\a\s-) = h2(\a\$-) for аЛ «•

§ 3. The space of all Q-filters. A filter V in L{3~) is said to be a Q- filter provided V is prime (i.e. maximal) and the natural homomorphism from L(&~) onto the two-element Boolean algebra L { ^ ) / v is a Q-homo- morphism.

3.1. I f Q-filters Fx, v 2 contain the same elements \a\$- with a in F, then Vx = V2.

Let hx and h2 be the natural homomorphisms from L( F) onto the two-element Boolean algebra A0 = L{$~)jvx = L(Sf)lv2. Since hx(\a\^-) =

= h2{\a-W) for all elementary formulas a, we have hx = h2 by 2.2. Hence

Pi = *г'( Ю = K'i V ) = ^p «.

The set of all Q-filters in L (У) will be denoted b y ^ ( ^ ) . For every formula a, let ||a||^- be the set

\ \ a \ \ r = { V €&(*-): \ a\ r eV} .

By definition, for every formula a in F and for every F (1) Fe||a||^- if and only if \a\y-eF.

3.2. The mapping H$- defined by the formula

(2) " Hr(\ aW) = |H|^

is a Q-homomorphism from L(^~) into the Boolean algebra of all subsets of &{$-).

For the proof, see e.g. Sikorski [6], theorem 24.6.

Observe that 3.2 states that 11$- transforms infinite joins and meets corresponding to logical quantifiers onto set-theoretical unions and inter­

sections respectively.

Assume the following notation:

L(T) = { |И .г : cuF},

L0(T) = {||a||^-: aeF is an open formula}.

By definition, H is a Q-homomorphism from В(У) onto

The set of all Q-filters in L{-T) will always be considered as a topological space, the class L0{F) being assumed as the basis determin­

ing the topology in By definition, every set 80еЬ0(^ ) is clopen, i.e. both open and closed. A set 8 C JF(S~) is open (closed) if and only if it is the union (the intersection) of some sets $0eL0(^-).

3.3. The space S£{2T) is totally disconnected. More exactly, if Fx, V2

are distinct Q-filters, then there exists an elementary formula a such that

the clopen set \\a\\$- contains exactly one of the points Fx, V2a3?(3~).

For if p7!, V

simultaneously belong or do not belong to every set

\\a\\x where a is an elementary formula, then

= V2 by 3.1 and (1).

Let

be a theory weaker than

For every Q-filter

the set

V ' = { |c t( g r , : |c c |j r e V )

is a Q-filter in L(S''), i. e. an element in L( S' ). This follows easily from 2.1.

We shall identify any Q-filter V e S ( S ) with the corresponding Q-filter V' e S( S' ) just defined. By this convention,

(3) S ( S ) C S ( S ' ) .

Moreover, for every formula a,

(4)

=

Taking as a any open formula, we infer that the topology in S ’{S') coincides with the topology induced on S'(S') by the topology in S ( S ' ) . In other words, S ( S ) is a topological subspace of the topological space S ( S ' ) .

3.4. Let S ' be a theory weaker than S'.

I f Hy, is an isomorphism, and S ( S ) is a dense subset of S ( S ' ) , then every open formula a which is a theorem in S is also a theorem in S ' .

Conversely, if Л х is an isomorphism and if every open formula which is a theorem in S is also a theorem in S ', then S ( S ) is a dense subset of S ( S ' ) .

To prove the first part, let us assume that an open formula a is a theo­

rem in S , i.e. that \a\x is the unit element of S { S ) . Thus \\a\\x = S ( S ) , i.e., by (4), S ( S ) С ЦаЦ^,. Since \\a\\x> is clopen in S ( S ' ) and S ( S ) is dense in S ( S ' ) , we infer that Лх> (\a\x>) — ||aj|.r> = S ’(S'). Since Л х, is an isomorphism, \a\x> is the unit element of L( S' ), i.e. a is a theorem in S' .

To prove the second part, suppose that S ( S ) is not dense in S ( S ' ) , i.e. that there exists an open formula a such that S ( S ) С \\а\\х,Ф S ( S ' ) . Thus a is not any theorem in S ' . By (4), Лх(\а\х) = \\a\\x = S ( S ) . Since Л х is an isomorphism, \a\x is the unit element of L( S) , i.e. a is a theorem in S .

§ 4. Open theories. Denote by L °(^ )th e subalgebra o i L ( S ) com­

posed of all elements \a\x where a is an open formula. Let S ° ( S ) be the Stone space of the Boolean algebra L Q(S), i.e. the set of all prime (i.e.

maximal) filters (70 in S ° ( S ) . For every open formula a, let \\a\\x be the S0t

I H & - = { V 0 e S ° ( S ) : \ a \ x * F 0} .

By definition, for every open formula a and for every prime filter v 0 in L°(S),

(1⁾

\a\&~e Po if and only if P0e||a||!r.

The mapping which to every element \a\ eL^fT) (a open) assigns the set

||a||y- is the Stone isomorphism of the Boolean algebra L°(У) onto the field L°{F) of all clopen subsets of the Stone space J

For every Q-filter

the intersenction of

with the subal- gebra is a prime filter v 0 in L°(,T), i.e. an element of It follows from 3.1 that the mapping which assigns to every Q-filter v e£e{&") the corresponding filter F0eJ5P° (J") just defined is one-to-one. This allows us to identify v with p0. By this identification,

' & { .r) c s e * (fr).

It follows from (1) that, for every open formula a,

Thus the topology in coincides with the topology induced on the subset by the topology in the Stone space S?Q{ZT). In other words, f£{2T) is a topological subspace of the compact Stone space

4.1. I f is an open theory, then Consequently the space ££{3~) is compact.

First we shall prove that every homomorphism h0 from L° {.T) onto the two-element Boolean algebra A 0 can be extended to a Q-homomor- phism h from L[&~) onto Л0.

In fact, let / be the mapping from the set F0 of all open formulas onto A q, defined as follows

(3) /(a) = h0(\a\$-) (a in F0).

By definition,

(4) f(y) = V for every axiom у in

the set of axioms being composed of open formulas only. Since h0 is a ho­

momorphism, we have, for all open formulas a, (3, / ( « - £ ) —/(«) w/C/5), /к, / ( < W ) = / ( « ) - / ( / ? ) ,

(5⁾

f ( ~ a ) = —/(a),

where w, -o, — denote on the left side the logical connectives and on the right side — the Boolean operations in A 0. It is easy to see that / can be extended to a mapping (denoted by the same letter /) from the set F of all formulas onto A 0, in such a way that equalities (5) holds for all formulas a, ft in F and moreover

/ ( U f «(f)) = l W ( a M ) , / ( r v m ) = r w (a < T )).

(6)

This extension / is uniquely determined by (5) and (6) and the values of / at open formulas (the exact definition of the extension / is by induction with respect to the length of the formula). It follows from (4), (5) and (6) that

f ( y) = V

for every theorem

in

,

the exact proof is by induction on the length of the proof of y. Thus if

\a\j- = i.e. if the formulas a -> (3 and -> a are both theorems in^", then /(a) = /(/?). Consequently, the formula

М1«И = / ( « )

defines a mapping h from L{^~) onto A 0. It follows from (5) and (6) that h is a Q-homomorphism from L{&~) onto A 0. By (3), In is an extension of ha.

To prove the first part of 4.1 let us assume that and that hQ is the natural homomorphism from onto the two-element Boolean algebra A 0 = L° { ^) /

0. A

we have proved, h0 can be extended to a Q- homomorphism In from Ъ{У) onto A 0. The set

V = {\a\r: h(\a\x ) = [/}

is a Q-filter in L {У) (i.e. an element in such that its intersection with L° (,T) is equal to |70. Thus every p0 e J

{2Г) can be identified with а Ге&[Г).

The second part of 4.1 follows from the first and the well-known theorem stating that the Stone space of any Boolean algebra is compact.

We recall that V and T denote the sets of all free individual variables and of all terms in ^*, respectively.

4.2. I f is an open theory and the cardinals V and T are equal, then И$- is an isomorphism.

It suffices to show that if \y\x Ф A > he. if у is and irrefutable for­

mula, then the set ||y||^ is non-empty, i.e. that there exists F e i f ( / ) such that \y\j-eV.

Without any restriction we may assume that

is in the normal prenex form

LJyji • * • Oj-fc ?i> *Jn • • • ? TQfc)?

where x , rp (i = 1 , . . . , Jc) are abbreviations

— (*^1 > • • • > % m ) ) = ( £ i l f • • • i % = ( V i l) • • • »

the symbols Urj* are abbreviations for

^ Hi * * * Нпц J ^ Щ 1 * ’ *

respectively, p ( x , ...) is an open formula, and xu . . . , xm are all free indi­

vidual variables appearing in y.

Add to the formalized language of the theory & some new individual constants and some new {тг + • • • + m^-argument functors

<p{j (i = 1 , . . . , 1c, j = 1 , щ). Let y0 be the following open formula in the extended language

P ( C > 9 l ( ^ l ) > • • • > ф й ( ® 1 i • • • » *®&)) >

where the following abbreviations are used:

C = ((/j , . . . , Gm ) , X i = {%i\ , . . . , ®i mj ) >

ф г (* ® l > • • • > *®<) = ( < f t l 0 ® l ? • • • 1 *®г) f • • • t ф г щ ( ® 1 ) • • • ? * ® i)) >

7г/ ( ® l ? • • • j ® i ) = ф ц (*®ll ? • • • » *®lOTj j • • • j j • • • ? ® i m f )

for г = 1, . .. , к and j = 1, .. . ,%. We assume here that all the variables Xij are distinct from one another, and distinct from ж1? ..., .

Let be an open theory based on the extended language, such that the formula y0 and all the (open) axioms of T are axioms for .T'. By a well- known theorem, the hypothesis that у is irrefutable in У implies that ЗГ' is consistent, i.e. the Boolean algebra L{.T') and have at least two- elements. Consequently, the Stone space is not empty. By 4.1, the space is not empty, i.e. there exists a Q-filter v'

It is easy to see that the set T' of all terms in 2Г' has the same power as the set V of all individual variables. Thus there exists a one-to-one mapping g from V onto T' such that

(7) g{Xj) = Cj for j = 1, .. ., m.

If a is a formula in and y t , . . . , yn are all free individual variables appearing in a, let a' denote the formula

lg(yi), •••, g( yn)

\ У 1» • * • ? Уп

in .7"', i.e. the result of the substitution indicated. It is easy to see that the mapping which assigns the element \a\jr, to an element \a\#- is a ho­

momorphism from L(&~) into The hypothesis that g maps V onto T' implies that this mapping is a Q-homomorphism. Thus the set

V

= {M<r: |aV ,eF'}

is a Q-filter in L(&~), i. e. Since the formula y0 is an axiom in

<7~', the formula y' (see (7)):

L J r j i * ‘ ' f l Ę f c U r j f c ^ l l ł • • • » % k j k )

is also a theorem in . Thus — / eV \ Hence \yl,rep.

4.3. Suppose V = T. Then in order that 2Г be an open theory it is ne­

cessary and sufficient that S^{^) be compact and Н у be an isomorphism.

The necessity follows from 4.1 and 4.2. We are going to prove the sufficiency.

Let be a theory, based on the same formalized language as вГ, such that the set of all axioms for ZT' is composed of all open formulas which are theorems in 3~. By definition, ST' is open and weaker than By § 3 (3) Z£{2T) is a subset of Since all open formulas which are theorems in F are theorems in S'', the set is dense in S£{^') on account of the second part of 3.4 and the hypothesis that Sy- is an iso­

morphism. Since SPlfF) is compact, S£{ST) is a closed subset of This proves that

Suppose that a is a theorem in i.e. that ЦаЦ^ = s e ^ ) =

Hence it follows, by § 3 (4), that \\a\\n — £?{2T'). Since is open and V — T, the homomorphism H^, is an isomorphism on account of 4.2.

Thus \\a\\$-t = & (^ ') implies that

| / , i.e. that a is theorem in . This proves that T is weaker than ST. Thus is identical with the open theory .

§ 5. The representation theorem for the predicate calculus. We shall now examine the case where is the predicate calculus & mentioned in § 1. By definition, S? is an open theory. By 4.1,

thus ) is a compact space. We shall prove that

5.1. The space is homeomorphic with the Cantor space 2E, i.e.

with the Cartesian product of E two-element Hausdorff spaces.

We recall that E denotes the set of all elementary formulas in S?.

By (1), it suffices to prove that is homeomorphic with 2E.

Let Sf о be a propositional calculus with logical connectives w, r\, — >,

— and with a set F0 of propositional variables, such that V0 — E. Let L(S?0) denote the Lindenbaum algebra of the propositional calculus i.e. the Boolean algebra obtained from the set of all formulas in by identifications of formulas a, (5 if and only if both the implications a — > ft and /1 a are propositional tautologies. As is well known, L(6^0) is a free Boolean algebra with f 0 free generators. Thus the Stone space of L{S^0) is- homeomorphic to the Cantor space 2 F°, i.e. to 2E (see e.g. Sikorski [6] § 14).

On the other hand, let g be a one-to-one mapping from V0 onto E.

If a is a formula in and ax, . . . , an are all propositional variables appearing in a, let a denote the open formula

(1⁾

&{&) = &°{ST)

а

in i.e. the result of the substitution indicated. It is well known that a is a propositional tautology if and only if a' is a predicate tautology.

Hence it follows that the mapping

MN^o) — \&'W

is an isomorphism from L[Sf0) onto L Q(^). Thus the Stone space of L(6^

) is homeomorphic to the Stone space of L°(&*). This proves that is homeomorphic with 2E.

References

E . - >

[1] H. R asiow a and R. S ik o rsk i, On the isomorphism of Lindenbaum algebras with fields of sets, Colloquium Mathematicum 5 (1958), pp. 143-158.

[2] L. R ieger, On free ^-complete Boolean algebras, Fundamenta Mathematicae 38 (1951), pp. 35-52.

[3] — O jedne zaldadni vete matematiclce logilcy, Casopis pro pestovani Matema- tiky 80 (1955), pp. 217-231.

[4] R. S ikorski, A topological characterization of open theories, Bull. Acad. Pol.

Sci. 9 (1961).

[5] — On open theories, Colloq. Math., in print.

[6] — Boolean algebras, Berlin-Gottingen-Heidelberg 1980.