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-
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