8

W. J. BLOK AND D. PIGOZZI

Let a be an arbitrary substitution. ThS need not be closed under c , i.e.,

T(T) may fail to be a theory for some T £ ThS. For this reason we define

(TS(T) = Cnsr(T) for each T £ ThS.

On the other hand, we shall see in the next lemma that the inverse image of

a theory under substitution is always again a theory; this is a consequence of

the structurally of \-s. For any T C Fm let r~l(T) = {tp £ Fm : ap € T}.

Lemm a 1.2 Let S = (£,h$ ) 6e an arbitrary deductive system.

(i) T/iS is closed under inverse substitution. (I.e., a~l(T) £ T/iS /or

e-uen/ T £ ThS and every substitution a.)

(ii) as(CnsT) = Cnsa(T) for every T C F m and substitution a.

(hi) CTS zs a join-continuous mapping of ThS into itself (I.e., we have

MVfe/T^Vfe/M^)

for any system of theories Ti and any substitution a.)

Proof. By (10) we have j{Cnso-l(T)) C Cnsr{o--l(T)) C C n ^ T =

T for every theory T. Thus C n s e r - ^ T ) C

T~1{T),

and hence (i) holds,

(ii) follows easily from (10). To establish (iii) we calculate: ^ ( V f e / ^ i )

=

rs{Cns\JiaTij = CnsCT{{jieITi) = Cns\Jia aft) = Cns \JieI rs(Ti) =

Vfe/^^)-!

Observe that condition 2.2 (i) is equivalent to structurally (see Wqjcicki

[48].)

1.2 M a t r i x S e m a n t i c s

By an C-algebra we mean a structure A = (A,u )„$£ where A is a non-empty

set, called the universe of A, and

uA

is an operation on A of rank k for each

connective w of rank k. An C-matrix is a pair A = (A,F) where A is an

£-algebra and F is an arbitrary subset of A; the elements of F are called

designated elements of A. Let M be any class of matrices. Let |=|yj be the

relation that holds between a (possibly infinite) set T of formulas and a single

formula £ , in symbols r | =M y, if every interpretation of p in a member .A of M

holds in A (i.e., is one of the designated elements) provided each tp £ T holds

in A under the same interpretation. For any £(po • • • Pn-i) £ Fm and all

a

0

,... , On-i £ A we write ^ A (ao,..., On-i) for the element of A represented by

(f when the variables p

0 5

• • •

5

Pn-i are interpreted respectively as a o , . . . , On-i-