Subframe Logics and Confinal Subframe Logics above K 4

T h e o r e m 3.1.13 The subframe logics above K4 are precisely the subframe logics defined in Fine [85]. Therefore any subframe logic above K4 has fmp and is a union-Sf-splitting of K4 and SK4 has a basis.

P r o o f. We denote the set of subframe logics defined by K . Fine by V. We know from Corollary 3.1.12 that a logic 0 is in V if and only if there is a set of finite rooted frames F with 0 = A '4 ({S f 1(^)|^ G F } ) . It follows that SK4 D V and that V is the set of union-S/-splittings of K4. By Fine [85] all logics in V have fmp and therefore all union- S/-splittings of K4 have fmp. By Prop. 2.4.27, all logics in SK4 are union-5/-splittings o f K4. Hence SK4 = V. By Prop. 2.4.21, SK4 has a basis.-!

The similarity between the subframe-formulas introduced in Fine [85] and frame- formulas introduced in Fine [74a] is completely explained: Both of them are splitting- formulas; the frame-formulas are splitting-formulas for Z K 4 and the subframe-formulas are splitting-formulas for S K4.

We denote the operation Sfc<m by Cf. C f is good a operation on G fr(K4) and the C/-logics are exactly the confinal subframe logics defined by M . Za k h a r y a sc h e v. Za-

k h a ryasc he v associates with a finite transitive and rooted frame Q a formula a((/,0, _L) such that for all H e Gfr(K4): 6 ^€ F r {C f(H )) iff H £ o (0 ,0 ,-L ). It follows that K4/clG = K4(-ya(G,9, L ) ) and that ->a(G,9, J.) and ( D ^ Ag —*• ->po)C* are deductively equivalent above K4. Hence the canonical formulas ->a(£,0, ± ) are, up to deductive equivalence, the C'/-splitting-formulas. For a finite transitive frame G with root x we have the following situation:

(:Th(G), K4(aU)a c —► ->px) ) is a splitting-pair in ZK4.

(T h {C f(G )),K4{{pU)&g -,pr )<?/)) }s a splitting pair in CfK4.

(T h (S f(G )),K4^{((d U )^ ß —} ->px) sf ) ) is a splitting-pair in SK4.

It is shown in Za k h ar ya sc h ev [92] that all confinal subframe logics are union-C/- splittings oi K 4 and therefore have the finite model property. A lot o f consequences follow immediately from the propositions we proved in The Use of Splittings. For instance it follows that Sp c j k a{ A ) is always an interval. For let A = K4/clF . Then it follows from Proposition 2.4.30 that Spcjka{A) = [Ar4/m c/(F), A].

45

3.2 Basic Splittings of SAfn 3.2 B a s ic S p litt in g s o f SAfn

Given two subframe logics 0 , A with 0 € SA there are mainly two questions we try to answer in the following chapters:

(1) Is 0 an iterated 5/-splitting of A by finite frames?

(2) Is 0 strictly Sf-complete above A?

By Proposition 2.4.26, we know that a positive answer for (1) implies a positive answer for (2) if 0 is complete. If 0 is Ä-persistent and A is m-transitive even the converse direction holds: For suppose that 0 is Ä-persistent. Then, by the finite embedding property of F r(Q ), there exists a set of finite rooted 0-frames F such that

F r(Q ) = { h e Fr(A)\G % h for Q e F }.

But then 0 V S fm(£ ) for G € F. Define 0 i := A ( { S f m(£)|£ € F } ) . Now it is easy to check that F r (6) = F r(Q i) and it follows that 0 is strictly ^/-complete above A if and only if 0 = © i if and only if 0 = A / ^ F . We conjecture that this is also true for not weakly transitive logics but have found no way to prove this.

D efin ition 3.2.1 Suppose that F = (/, (rj|t < « ) ) and Q — (g, (s.jt < » ) ) are finite n- frames and x £ g. Then F is an x-a rrow su bfram e of Q, F < x G, if f = g, r, C s, for i < n, and x is a root of T . T is a strict x-a rrow su bfram e of Q, in symbols T < x Q, if F <xG and F ± Q .

We explain this definition by an example. Consider the frame G = | x ^ Let x be the reflexive point and y be the irreflexive point in G- Then the frame F = •-—x| is not an x-arrow subframe of Q but is a y-arrow subframe of G- The reason is that x is not a root of F .

Troughout the following three chapters F (G) denotes a finite frame (/, (r,|i < « »

( (9 ,(s ,jt < n ))) with root 0. In the monomodal case we write and r (s ) instead of rj (s j).

We write (1C —* G) if G is a p-morphic image of K and (F —►G) if F is a set of frames such that for any IC € F the frame G is a p-morphic image of 1C.

D e fin itio n 3.2.2 Suppose that 7f = (h, (<li|t < n), A ) is a refined n-frame and suppose that b C h is finite such that hb is rooted. Further suppose that (Aj —*• G) via a p-morphism p and F < 0 G- In this case we write ( F < 0 G « - hb). A set {a v|y € g} C A recogn izes ( F <o G <— hb) in H if

( R £) Fory e g : p-1 (y ) C ay.

( RO) For y,z e g : ay Oiaz = h if y ri z.

(Ä O j For y,z e g . ay —> az h if y z.

46 3 SUBFRAME LOGICS

(R n ) For y,z e g : ay C\az = 9 i f y ^ z .

( F <o Q *— hb) is reco gn izab le in 'H if there exists a set which recognizes ( F <o Q *— hb) in H. We write

(G - hb) if F = G , ( F <o G — hb) if G - hb,

(G * hb) if G — F and Q a hb, ( F <o h ) if G — hb.

Note that we allow b to be not internal bnt that we do not allow b to be infinite. Let H = (h, A) G Rfr and let Q be finite with root 0. We have (1) =» (2), where

(1) There is a finite set 6 C h such that (Q <— hb) is recognizable in H.

(2) There is a set {ax\x G </} C A with Vg[ax\x G g) = h and ao ^ 0.

Let Q = (g, (s,|i < n )). Suppose that {ax\x € fir} recognizes ( Q <— hb) = (6 <o 6 ^<- hb) in H. Then r* = s, in (R O ) and Vg[ax\x G g]=- h follows from the definition of V^. Further a o 2 p " 1(O )# 0 .

The main tool for proving that a frame Q 5/-splits a logic A will be the following

O b serva tio n 3.2.3 Suppose that A is an R-persistent subframe logic and Q G Fr/(A) is rooted. Then Q Sf-splits A and A/S*Q is R-persistent if there is a finite set of finite rooted frames Fg C F r(A ) with {Fg —*■ Q) such that

R I There exists anm £ w such that for all h G F r ( A ) : Ifd(to)V<; A po is consistent in h, then there exists AC G Fg with IC C h.

R II YH € Rfr{A ) and AC € Fg : If AC a hb for a set b C h then (G ♦— hb) is recognizable in H.

In this case Fr{A/s-fG) = {h G Fr(A)|AC % h for K G Fg) and A/s' g = ACSf“ ^ ) ) = A(D("*>VC - -po).

P r o o f. Take an m G u> which satisfies (R I) and suppose that (□ ("*) A p A po) } q is consistent in H G Rfr (A). Then there is a subframe H \ Q H with Apo consistent in Ji\. By (R I) there is a AC G Fg with AC C h. Hence, by (R H ) and the implication (1)

=> (2 ) above there is a set {a x |z G g ) C A with Vg[ax\x £ g] = h and ao # 0. But then ( □ ( n)A p A po) i q is consistent in H for all n £ u>. It follows from the splitting theorem that Q 5/-splits A. The other claims follow immediately with Lemma 3.1.10.H

We need some more technical notations:

47

Definition 3.2.4 For a finite T-cycle free frame Q = (g, (stj i < n )) define the depth of x e g in Q by

dpg(x) = 0 Vj/ £ T rg(x) : x = y.

dpg(x) = A: + 1 3y £ T rg(x) : dpg(y) = fc andVy £ T rg (x ) : dpg(y) < k o r x = y.

The depth of Q is defined by dp(G) := max{dpg(x)\x £ p }. The inverse depth of x £ g is d p -(x ) := dp(G) - dpg(x).

3.2 Basic Splittings of SMn

Lem m a 3.2.5 (1) Suppose that TL is refined, that ( F < 0 G <— hb) and that T is cycle free. Then ( F < 0 G hb) is recognizable in TL.

(2) Suppose that TL £ Rfr((tiuT ), that F £ Fr(oonT ) is T-cycle free and that ( T <o G * - hb)- Then ( F <o G h ) is recognizable in TL.

P ro o f. (1) S u p p o s e t h a t TL= (h ,A ) is re fin e d a n d t h a t p : hb — ►G is a s u rje c tiv e p -m o r p h is m fo r a fin ite bC hsuch t h a t hbis r o o t e d . F o r y £ g P~l (y) is fin ite a n d gis fin ite ; th e re fo re th e re e x is ts a set {c y\y £ g } C A such t h a t cy D p~l (y ) f o r y £ g a n d cj/incV2 = 0 fo r 2/i # V2- F o r yl A y2th e re e x is ts a set d*(j/liJ/2) € Aw it h d }y iflo) 2 P~\y2 ) a n d p ” 1 (2/1 ) C N o w de fin e f o r y £ g:

by := Cj/ n A *> * < n } n Aj m < « } •

It is e a s y to check t h a t {by\y £ g } C A sa tisfie s t h e c o n d it io n s ( R € ) , ( R □ ) a n d ( R H ).

W e n o w d e fin e a set {a y\y £ g } b y in d u c tio n o n dpp(y). S u p p o s e t h a t ayis d e fin e d fo r

dpr(y) < n • T h e n d e fin e f o r dpp(y) = n:

ay := by H z ).

T h i s is w e ll-d e fin e d b e c a u s e fo r y, z £ g yrt z im p lie s dpp(y) < dpr(z) sin c e F is cy c le fre e . W e s h o w th a t {ay\y £ g } re c o g n iz e s ( F < 0 G <— hb). ( R H ) is o b v io u s b e c a u s e

ay ^C byfo r y £ g. ( R □ ) fo llo w s fr o m ( R □ ) fo r {by\y £ / } . ( R O ) a n d ( R € ) a r e p r o v e d b y in d u c tio n o n dp^(y): S u p p o s e t h a t ( R O ) a n d ( R € ) a r e s h o w n f o r a ll z £ g w it h

dpp(z) < n a n d t h a t dpjr(y) = n . I f z\ £ aya n d yrt* z t h e n , b y d e fin it io n , z\ £ O ta z . It fo llo w s t h a t ay—►O ta - = h. N o w p “ 1( y ) C ayfo llo w s f r o m p~l {z ) C a z f o r y r±z, b y in d u c tio n h y p o t h e s is .

( 2 ) T h e c o n s t r u c t io n is a n a lo g o u s : D e fin e { 6 y |y £ g } a s a b o v e . T h e n d e fin e b y in d u c tio n o n dp?(y)a set { a y |y £ g ) C A :

ay by Pi P|{Ot'az|y ri y ^ z}.

Again this is well defined and (R fl) and (R □ ) follow immediately. For ( R O ) and (R £ ) notice that from Ti £ R fr(® nT ) follows ay C O tay for all y £ g and i < n. H

48 3 SUBFRAME LOGICS

It follows immediately that Fg is a finite set o f finite, cycle free rooted frames. By Corollary 3.2.6, (G, Fg) satisfies condition (R II). Define m := dp(G). For (RJ) suppose that h G Ft

3.2 Basic Splittings of SAfn 49

follows with Theorem 1.5.2 from

C%nT = f){T/i(5/(.T))|.T a finite reflexive n-tree } C Th{S f(G ))

and T h (S f(F) ) % T h (S f(G )) for all finite reflexive n-trees T . This follows from the fact that any p-morphic image of a finite T-cycle free frame is T-cycle free. H

E xam ples. T = K/Sf\x\and T.B\ = T/Sf{^=^\ = K/Sf\x\ /SJ | ^ 7 | .

C o ro lla ry 3.2.8 A finite rooted frame splits K n if and only if it Sf-splits K n.

Corollary 3.2.8 does not hold for T because T has only one splitting: (T fiflT] ),£ ).

C o ro lla ry 3.2.9 (1) Suppose that G is a finite n-tree. Then F r{K n/s^G) = {h\G £ h}.

(2) For a finite reflexive n-tree G Fr($)nT/s*G) — {h € Fr(oonT)|(y £ h}

Some remarks concerning this proof: That a finite rooted cycle free frame .S/-splits K n follows from B l o k s result that these frames split K n. But let us note once more that we cannot conclude that K n/S^G is ß-persistent even if we use the fact that K n/G is ß-persistent. Nevertheless there is a more elegant proof for the irreflexive case of both facts via unravelling. In this case the set Fqcan be defined as the set of p-morphic images K of the total unravelling of G such that G is a p-morphic image o f K\ this set is finite because G is cycle free. Such a simple definition is not possible for the reflexive case, even if we use reflexive unravelling. It is in general not possible to construct a set of T-cycle free frames Fqfor a T-cycle free frame G such that

(I ) F r(T / SfG) = { h e Fr(T)\K % h for K € Fq}.

As an example define G •— I^E G Sf-splits T because G is finite, rooted and T-cycle which is not free. But it is easy to prove that any set Fqwith (I ) must contain

T-cycle free.

T h e o r e m 3.2.10 (1) There are2N° union-Sf-splittings of K and K.AU2^. (2) There are2N° union-Sf-splittings o f T and T.AU3^.

P r o o f. (1) For n e u define Gn = {<7« , rn) by

9n '•=

{(0, m)|m < n} U {(1, m)|m < n} U {0,

xn, yn, vn,

tun}.

r n •—

{((0j

n } i x n )i

((0» w)> J/n)> ((1» n)j vn)j ( ( 1 » ^n)> (0, (0,0)), (0, (1,0))}

u = *+ < « } u {((!,*)> (i> i))li = * + i , i < « } .

All Gn are cycle free and therefore 5/-splits K and KJSiM is a union-5/-splitting o f K for

50 3 SUBFRAME LOGICS

M C {Qn\n 6 w }. We show that A '/ ^ M ^ K/s*N for M N . This follows if Gm is not subreducible onto Qn^for m ^{^} n. Suppose that there exists b C gm^{with p} : (Gm)b — *■ {/„.

Then m > n. We may assume that (Gm)bis rooted. Now it follows from the definition of p-morphism that if p (x) has k successors then x has at least k successors. It follows that 0, (0, m), ( l , m ) € b. Now it is obvious that 6 = gm. Hence p is a p-morphism from Gm^onto Gn• It follows by induction on dpgm(x) that dpgm(x) = dpgn(p (x)) and from this that m = » .

All Gn^are K.AU2-frames and 5/-split K.Alt2 because they 5/-split K.

Again K.Alt2/SiM # K.Alt2/Sf N for M # N .

(2) For define Hn = (hn, s„) by hn := gnand sn := rn U {(x ,x )| x € hn). All Hn are T-cycle free and reflexive and therefore 5/-split T. The same arguments as above show that T / ^ M ^ T/SfN for M ^ N and M , N C {W n|rt € w }. The claim for T.Alt3 follows immediately. H

Let T = (t, ( < t |i < n ))((t, {<,• |t < « ) ) ) always denote a (reflexive) n-tree with root 0.

D efin ition 3.2.11 For a (reflexive) n-tree T define (T ) := { T € Fr\T <0 F ). Then ( ( T ) , <0) is a finite partially ordered set with minimum T. For T € ( T ) the T-closure of F is F - := {fC e (T)\K <0F ) and the strict T -closure of F is F < := F - — { T } . A set F C ( T ) is T-closed if F e F implies F - C F.

Note that the set F - = {K, € (T)|K. <0 F } depends on T . Nevertheless we will often write F - without referring to T , if this causes no confusion.

If we try to show that a specific logic is an iterated 5/-splitting it is useful to have a canonical axiomatization of this logic. Such a canonical axiomatization should show the geometrical meaning of the axioms. One possible approach are the sketch omission logics introduced in Kracht [90b]. Indeed, it can be shown that the logics defined in the following proposition are sketch omission logics.

D efin ition 3.2.12 For a finite n-tree T = ( g, (<,• |t < n )) and T <0 G = (g, (s«|t < » ) ) define

:= A <p* -* “’Pvl1 # y) A f\{P v — OiP*\y < i *) A A<Pv — “ "O.p^lx foy)

P ro p o s itio n 3.2.13 For H € Rfr and T < 0G: A po is consistent in H if and only if there is an F € G- with F Q h. A := A'n( a (<<p(rI)V c< -*■ -.po) j<? an R-persistent subframe logic with F r (A ) = {h € Fr\F % h for F 6 G

-}-P r o o f. Let G = (g,(s,|t < n )). Suppose that h,ß,x\= □<<<p(r ) ) V i;< Apo. It is easy to show that there exists a frame F with T < 0F and b C h such that p : F — *• defined

51

even above ?i-transitive logics. (There is a mistake concerning this point in Kracht[90b].) First, it is readily shown that

O r , ( ( D W ^ - ->p0|T < 0 T < 0 </})

= A-.Trn( V ( a (n)Vjr|T < 0 -P o ).

As an immediate consequence we get that the following two conditions are equivalent:

(1) A\7Vn(DWVfls - “>po)

= K .T r„( V (□^n) V^-|T < 0 -P o )

(2) K .Tr,, ( □ ( ’*) V c< —> -ipo) is strictly 5/-complete above K .T rn.

This equivalence shows the subtle difference between axiomatizations by sketch omission formulas of Kracht [90b] and axiomatizations by 5/-split ting-formulas.

3.2 Basic Splittings of SAfn

E xam p les (I) For m £ lj define Tm := (m + 1 ,5 ), where t S j iff j = t + 1.

Tm = <r*— •__ m^x-«x

Now define := (m + 1 ,< ) and N m := (7|‘ ) - . It follows from Theorem 3.2.7 that N m defines a union__5/-splitting of K and that K/ ^ N m = A '(D (m)'v'isim —►->p0) is R- persistent. This union-5/-splitting will play a fundamental role in our investigation of monomodal 5/ splittings.

52 3 SUBFRAME LOGICS

Let us note two technical corollaries which will be useful for determining 5/-splitting logics.

C o r o lla r y 3.2.14 Suppose that Q G ( T ) . Define M := {h\T £ h for T G G<}- Then for h G M :

G C h if □ (dP(T ))V ß A po is consistent in h.

Hence, if A is an R-persistent subframe logic with F r (A ) D M , then (R I) is satisfied by m = dp(T) and Fg = {G}.

C o r o lla r y 3.2.15 Suppose that Ti G Rfr, that T <o T and that {ay\y € t} recognizes { T < 0 G — hb) in Ti. Further suppose that h G M , M defined as in Corollary 3.2.1^4- Then for any x € a0 there exist b\ C h such that x is a root of hbl and an isomorphism p : G — ►hbi w*th p(0) = x so that {a y|j/ G t j recognizes ( F <o G — hbt) in Ti.

For a frame (g, (<Jj|t < n )) define the tra n s itiv e closure of (<J,|t < n ) by:

(x,y) G (<l,|i < n)* iff there is a path o f length > 0 from x to y.

D e fin itio n 3.2.16 For an n-tree T the set ( T )r C (T ) is defined by : T G ( T )r i f f T < 0 F and r* C ( < j \j < n)* U {(x ,0 )| x G t} for i < n.

It follows immediately that ( T )r is T-closed. For T <o G let (?* denote a fixed maximal cycle free frame with T <o G^ <o G- Then we extend the definition o f d ep th to not cycle free frames G- For y € g define dpg(y) := dpci(y ).

T h e o r e m 3.2.17 Suppose that G € ( T )r and that A is a R-persistent subframe logic until F r{A ) C {A|/C £ h for K. € G<}- Then G Sf-splits A and A/S^G is R-persistent.

P r o o f. Define Fg := {G}- We check (R I) and (R H ). (R I) follows from Corollary 3.2.14.

For (R II) suppose that Ti € R fr(A ) and G — hb for a set 6 C h. By Lemma 3.2.5, there exists a set {&y|y € t } which recognizes ( T <o G — hb) in Ti. By induction on dpg(y) we define a set {a v|j/ € t} C A. Suppose that ay is defined for dpg(y) < m and that dpg(z) = m. Then

o.z :— bz 0 Pl'fOjfly |xs,*y, y ^ z, y ^ 0, i < n ) n W|0, i ^ r}.

az is well-defined because G G ( T )r. We prove that { a y|j/ G t} recognizes (G — hb). First it is easy to show that

( I ) for any &i C k with G a: hbl : { a y|y G t } recognizes ( T < 0 G ^ h ) in Ti if {by\y G <}

recognizes ( T <o G — hbt) in Ti.

3.2 Basic Splittings of SAfn

53

For (G ~ h ) and { a y|y € t} the conditions (R € ), (R ü ) and (R fl) follow from the corre­

sponding properties of {by\y 6 t}. For (R O ) suppose that z € ay and y$iyi. If y\ ^ 0 then z € 0 ,a yi, by definition of ay. Now suppose that y\ = 0. Then, by definition of ay, y € Oi&o. Take x € 60 with y < , x. By Corollary 3.2.15, there exists bi C h such that has root x and ( T < 0 Q ~ h ^) and this is recognized by {6y|y € *}. By (I), {a y\y € *}

recognizes (T <o G ~ h&J. It follows that x € a0 and therefore that y € O,a0. H For F € (T ) the height of T in ( ( T ) , < 0) is defined in the obvious way so that the height of T is 0. Ft( T ) denotes the set of frames of height i in (T ). For F C (T ) closed define (F )i := (F 0( T ) H F, F i(T ) n F , . . . , F „ (T ) n F ) where n is maximal with F n FW( T ) ^ 0.

Now the following corollary follows by induction.

Corollary 3.2.18 Suppose that F C ( T )r is closed. Then (F ), defines an iterated Sf- splitting of A'n.

A"n/5f ( F ) t is R-persistent and F r (A 'n/ ^ ( F ) , ) = {h € Fr\G g h for G € F }.

We will often abbreviate A / Sf (F ), by A / S^F and say that F defines an iterated ^/-splitting of A.

E xam ple. Define an n-tree T by T := ({O },({0 }| i < n )). Then ( T )r defines an iterated 5/-splitting of K n and <X)nT = K n/ S* ( T )r.

Definition 3.2.19 For a reflexive n-tree T = (*,(<,• |t < n )) define a set ( T )r cj C (T ) by: T € (T)/*c/ iff T € (T ) and rt* C ( < t- |i < n)>* U {(x,0 )|x 6 * } for all i < n.

The proof of the following theorem is anologous to the proof of Theorem 3.2.17.

Theorem 3.2.20 Let T be a reflexive n-tree, G € ( T ) R ef and let A be an R-pcrsistcnt subframe logic above (&nT such that F r (A ) C {h\fC g h for K € GK}> Then G SJ-splits A and A / S*G is R-persistent.

Note that any iterated 5/-splitting of ®nT is an iterated 5/-splitting of K nbecause C*>nT is an iterated 5/-splitting of K n.

E xam p le. We need some conventions for pictures of n-frames. A n-frame ($r, (<l,|i < n » is drawn as ((g> < t)|z < n).

Define 7u := ( 1 1, 1 • •! ) and Gu •= ( I — |, [ » —• [ ).

Go ^ (Tu) and (T ® T ).U \ = T <#T/sfGg- Hence T <#T.Ui and T ® T./d are iterated 5/-splittings of A'2 by finite frames and strictly 5/-complete.

54 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICS

4 The Lattice of Monom odal Subframe Logics

Let us start with a list of the results for the monomodal standard systems. The listed properties will be shown in this chapter with the exception of the result that all monomodal subframe logics above K .A ltn for an n > 0 (and therefore all tabular subframe logics) are

4.1 Basic Monomodal Splittings

55

Therefore there exists ( g , < ) ^ T\ such that any point in g has a successor. But then [T)is a p-morphic image of (<7, < ) € F r (Ai ) , which is a contradiction. For the converse direction we distinguish two cases: Case 1. A D T = K / Sf [x ]. In this case it is clear that [•]5/-splits A and that A / s* [ • ] = £ is A-persistent. Case 2. There exists n £ such that A 2 K / S^N U. Let n be maximal with this property. Define Q := |T]and

Fg := { T £ Frf(A)\F \= A lfi;0T, T is rooted and \F\ < n + 1}.

We check conditions (R I) and (R II) for (G,Fg). Let h £ F r (A ) and suppose that a pQ js consistent in h. Then it is easy to show that there exists T £ Fg with T C h. For (R II) suppose that 7i £ Rfr(A) and T ~ hb for a T € Fg and 6 C h. Take a € A with a D b and let a# denote the n+1 -reflexive iteration of a. By Lemma 4.1.2, {a #}

recognizes (G <— hb) in H. H

It follows immediately that 7v(Dn± ) = K / S^N U/ S^ [T ]for n £ u>. Clearly F r (A '(D nX )) contains only cycle free frames. Now A '(ü nX ) is n-transitive and, by a result of Blok [7 8 ], V (/ f(D MX )) is locally finite. It follows that any 0 € S K (B U± ) has the fmp. Hence, by Proposition 2 .4 .2 7 , any 0 £ 5 A '(D nX ) is a union-5/-split ting of A '(D nX).

C o ro lla ry 4.1.4 Any subframe logic 0 with DnX € 0 for ann £ u is strictly Sf-complete and an iterated Sf-splitting of K by finite frames.

Example. Th([ x ]) = K / Sf (x— / Sf [ • ] .

Definition 4.1.5 For a finite tree T define(T )w C (7 } by

(/ ,r ) € ( T )n iff {f i r ) £ (T ) and rC <* U{(z,0)|:r € t} U {(a:,a:)|a: € *}•

T h e o re m 4.1.6 Let Q £ (T )w , n £ and let A be a R-persistent subframe logic with F r { A ) C {/i|AJ g h for fC £ G< U N n }. Then G Sf-splits A and A / S^G is R-persistent.

P ro o f. Define Fg := {G}. Again we check conditions (R I) and (R D ). (R I) follows from Corollary 3.2.14. Now suppose that Ft £ Rfr{A ) and Q ~ h*. Take a set {6y|y £ t } which recognizes ( T < 0 G ^ hb) in Ft. By induction on dpg(z) we define az £ A for z £ t. For a C h let a* denote the n-reflexive iteration of a.

[ (t>z n y ^ y ^ 0} n n { O M ^ o } ) # if z $z z \ bz O V [{O a y\zsy,yt0} n f ){O b 0\zsO} if z z

az is well-defined because Q £ (T )jv. The interesting steps of the proof that { a y|y £ t) recognizes (Q ~ hb) in H are (a ) ay -> Oay = h if y s y and (b ) -> Oao = h if y s 0.

(a ) follows from Lemma 4.1.2 by the definition of oy. (b ) Suppose that y s 0 and z £ ay.

56 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICS

Then there exists x £ bo with z <\ x. By Corollary 3.2.15 there is a set a C h such that ha has root x and Q ~ ha and {6y|j/ £ t} recognizes ( T <o Q — ha). It is easy to check that for c C / i with hc — Q {a y| y £ t} recognizes ( T <o Q — hc) if {&v|t/ £ t] recognizes ( T <o Q — hc). Hence x £ ao and z € Oac». "I

C o r o lla r y 4.1.7 Suppose that F C ( T )m is closed and that n £ lj. Then F defines an iterated Sf -splitting of K/ssN n and A 75/N n/S/F is R-persistent.

F r( ff/5/N n/5/F ) = {h £ Fr\K % h for K £ N n U F }.

E xam p les.

(i) n ( 0 ) = k/s' 0 / « [ ~ g / « [ j = g ( I I ) b; defines an iterated ^/-splitting of K and

K.Bn = A '(D(n)V B - - ~*Po) = A'/5

/Bn-(I II ) For any subframe logic A with A D A / ^ N n for an n £ u the set 1^ defines an iterated 5/-splitting of A and A.Im = A ^ ^ V j - ^ —► ->po) = A / ^ I ^ .

(I V ) Define F := { |x—■*!, |«—-x|, | * - * ♦ !, |x— x|, |x-— | }.

Now T\ = |x—»x| , N i = { 7 i } and F C (7j)/v is closed. It follows that F defines an iterated 5/-splitting of K and K/S*F is A-persistent. By the example above I j defines an iterated 5/-splitting of K/S^F and A / ^ F / ^ I j is A-persistent. It follows immediately that Kh = K/S^F/s^ If. By a result o f Na g le & Th om ason [85] all logics above K5 have the fmp and A5 is 2-transitive. It follows that all subframe logics above K5 are iterated S/-splittings o f K by finite frames.

4.2 S u b fr a m e L o g ic s a b o v e K 4 ( I I )

Define Q\ := . Then T2 <0 Q\- Define F j := Q

&2 := I ?e—x I

Qh := x »4 ^

Gz := G4 ^:= <

C r:=

We assume that the points o f the frames Gi are from left to right 0,1,2, respectively 0,1. Define T R - := F i U {G2 ■ ■ -Gj)- The following proposition follows immediately.

P ro p o s itio n 4.2.1 A frame h is transitive if and only if Q % h for Q £ T R “ .

T h e o r e m 4.2.2 For n £ u A „ := K/S*T*/s*F i/ stQ2/ss . . . /5/Q7 is well-defined and A „ = K (S f2(G)\g £ T R " U {7 ; * }) = K A f^ T *.

57

58 4 THE LATTICE OF MONOMODAL SUBFRAME LOGICS

Suppose that y € Q-i- Let n g w b e minimal with y6 T r* (x ). Co n Oc2 =

0

and Co = { x } and therefore n > 2. There exists a path x = xq < x\ < . . . < x „ = y with x ,fl Xj for j ^{> t + 1}. It follows by induction that x 1+i <3x, for i > 0. From this follows, by Q2, S3 ^{2 SN} that x,' is reflexive for i > 0. Again it follows by induction that x, < i j for j > 0 because Qa £ 9^{• Hence} y <i x\. Now xj 6 61 n 0 62- Therefore there exists z 662 with xj < z. It follows that z <J i j and z <\ z. Hence xi € cti and y 6 O aj.

The step j = 5 is proved anlogously.

j = 6. Define Fq1 ^{:= { £7}}. (R I) follows from Corollary 3.2.14. (R II) Suppose that Q7 ^~ lib with H e Rfr(0e). Let {60,6i,62} recognize (T2 <0 Ö7 — H) in H. Define a2 := (62n Obi H O60)*, ai := (61 H Oa2 ^{fl O6o)* and} ao := (bo H Octi)*. It is easy to check that {00,01,02} recognizes (Q7 ^~ kb) in Ti.

(I ) is proved. From (I ) and Prop. 4.2.1 follows A n = K4/S* T ’ . The axiomatization of A „ follows immediately. H

4.3 A Chain o f incom plete Subfram e Logics

For the following construction we use ordinal arithmetic and the notion of intervals in the usual way. For two ordinals Q i,Q2 let [01,02] denote the set {7 € ord|aj < 7 < 02}.

Define (h, < ) by h := uu> + 1 and

<

:= {(un + mi,um + m2)|n,mi,m2 €w,mi > m2}

U {(u>a,u;a)|a € u ;+ l,a ^ 0} U {(u>(n+ l),um + m)|n, m € u;}.

Now define 7

i :=

^{(h,<, A), where A := [0}]* denotes the closure of 0 C hwith respect to finite intersection, complement and □. Define

Into

:= u> + 1 = [0,u;] and for n > 0 define

Intn

:= [um + l,u>(n+ 1)]. Then h = |J{/»tn|n € u;} U {ww}. Define a set C by c 6 C iffcCfc and there is an m G u with c C [J{/ntn|n > m} U {wu>} and for m < n either c fl

Intm

is finite and u>(m + 1) £ c or

Intm

— c is finite and u>(m + ¹) 6 c. Define

B := {c|c ^€ C or — c ^€ C}.

Claim 1. A = B.

Proof. B C A. The following equations are easily checked:

1. Into = OD0UD0.

2. 7ntn+1 = —In tn 0 0/nt„.

3. For m € u> : m = □ m+1⁰.

4. For m € u;: {m} = m + 1 D — m.

5. F o r n , m e « : {w(n + 1) + m + 1} = - □ ”*/«<„ fl □ TO+1 In tn.

4.3 A Chain of incomplete Subframe Logics 59

It follows that In tn £ A for n £ u>and that { a } £ A if a = 0 or a is a successor ordinal.

From this follows immediately that B C A. A C ^B: It is clear that B is closed under finite intersection and complement. We show that B is closed under □. For bC

h

^define

mmn(6) € ww if In tnC\b/ 0 by minn(b) := min(br\Intn). Then Obnlnt0

=

[0, mm0(6)+l]

if bn In t0/ 0 and D6 n

Into

^{= {0}} else. We have for

n >

⁰

f [um + 1, m inn(b) +1] if b H In tn 0, un £ b

□6 n In tn = < 0 if un £ 6

{um + 1} if b D In tn = 0 and urn € b It follows immediately that B is closed under □.

For any ordinal

a

^with

a

+ 1 < u;u> + 1 define

7 ia

⁼

(ha, < a, Aa)

^by

hQ

^:=

a

+ 1, < Q := < n

h*

^{and Aa := {a n} (a + l)|a £ A ).

Let A Q := Th(H a) and A := A ^ . Any 7fa is a generated subframe of and Aa = [0]/ia- Wu,3+i looks as follows:

u>3 + 1 u>2 u> 1 0

The logic 6'.3 = (7./i will play a fundamental role in our investigation of incomplete subframe logics. The rooted G.3-frames are the irreflexive linear orders without infinite ascending chains. The set of finite rooted G.3-frames is, up to isomorphism, the set {T n*|n G ^u>}.

L e m m a 4.3.1 For all a < u>u>:

1. Hq is descriptive.

2. H i is subdirect irreducible if a = 0 or a a successor ordinal.

3. Aa = CiiAßlß < a ) if a is a limit ordinal.

4- A „ is a subframe logic.

5. F r(A a) = F r{G .3) if a ^ uju and a > u>.

P r o o f. 1. It is clear that Ha is refined. Suppose that U is an ultrafilter in W+. I f U contains a finite set then f| U ^ 0. Now suppose that U contains no finite set. Then a = ujß + m for a ß ^ 0 and ß £ u> + 1 and m £ u. But then it is easy to check that there exists ß\ < ß with uß\ £ a for any a £ U. Hence f| (/ / 0. 2. I f a = 0 then { 0 } is an opremum of Hq. I f a = ß + 1 then (a + 1) - {ß } is a opremum o f Ti+. 3. Suppose that a is a limit ordinal and 7 is a valuation of H a with 7(<£) / 0. Let 7(<j>) = a £ Aa. Then, by Claim I, there is a ß < a with ßr\a ^ 0. Define a valuation 71 on Hp by 71 (p ) := -f(p)r\hp.

Then 71{<£) # 0. 4. Let a £ Aa. We show that (H a)a is a A a-frame. Then it is shown that Aa is complete with respect to a class o f frames closed under subframes and therefore

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