On the content of lattices of logics. Part I

REPORTS ON M A T H E M A T IC A L LO GIC 13 ( 1981), 17—28 Wojciech D Z IK T he Silesian U niversity o f K atow ice O N T H E CONTENT O F LATTICES O F LO G ICS PART I. TH E REPRESENTATION TH EO R EM FOR LATTICES O F LOGICS F o r every consequence (closure operator) Cn on a set S the family '4 o f all Cn-closed sets, partially ordered by set inclusion, forms a complete lattice (%>, ę ) . Such a lattice will be called a lattice o f logics. Lattices like this will be investigated in this paper. If some assumptions are satisfied, then this lattice is distributive and even implicative. The m ost im portant result o f this part is the topological representation theorem for lattices o f logics. The problem o f distributivity o f a lattice (% , s ) , some properties of bases of # , dual spaces and sets irreducible in are also studied. If a lattice is distributive, then, since it has the zero element C n (0 ) and the unit element S, it can be conceived as a pseudo-Boolean algebra v , a —) in the sense of Rasiowa and Sikorski [6]. In this case one may try to establish the content o f lattice o f logics, i.e. the set £(%'’) of all propositional formulas which have the value S (unit) for every homomorphism o f algebra o f propositional language (built by means o f connectives ~ ) into the pseudo-Boolean algebra ( # , v , a , —). The representation theorem presented here serves to establish £ (^ ) , for some # , which will be published in the second p art of this paper. This paper is a slightly modified version o f a chapter from the author’s doctoral dissertation under the supervision o f Professor W. A. Pogorzelski, to whom I am indebted. 1. Consequences and closure systems Let S’ be a non-empty set. A mapping C n : 2s -*• 2s will be called a consequence or closure operator on S, if for all X , Y ^ S ( 1) (2) X ^ C n ( X ) = C n(C n(JT)), Jlfg y = > C n (Z )c C n (y ) Received October 24, 1979 (cf. e.g. [2]). A consequence Cn is said to be finitistic (or algebraic) if for each A 'ę S : (3) Cn(Af) = U ( C n ( r ) : Y ^ X , Y- finite} . A family # o f subsets of a non-empty set S is said to be a closure system on S if # is closed under arbitrary intersections, i.e. f) S ’ e & if Clearly, S e ( Theorem 1 (E. H. M oore, cf. [2]). a) Every consequence Cn on S defines the closure system on S : (4) V Cn = { X ^ S - . C n ( X ) = X} . b) Every closure system ^ on S defines the consequence on S: (5) Cn«(X) = 0 { Z e < g : X ^ Z } , f or X<=S. The m apping Cn -> < i Cn is a one-to-one correspondence between consequences and closure systems on the same set S. M oreover, = and Cn¥cn = Cn. The closure system <gCn will be investigated in this paper. Since there exists a bijective correspondence between consequences on S and closure systems on S given by (4) and (5), we shall use them interchangeably. Besides, in most cases we leave out the subscripts in <€Cn and C n,;. Hence, we will write (6 d, %>k for a closure systems given by consequences Cn, Cnd, Cnt , respectively. In the sequel closure systems and consequences will be con­ sidered on the same arbitrary, non-empty set S. Theorem 2 (Birkhoff [1], Tarski [7], cf. [2]). Every closure system on S constitutes a complete lattice under set inclusion, ( f €, ę ), where join and meet is defined as follows, fo r X j e i e /, (6) V = Cn( U Xj ) , iel in particular, X v Y = C n ( X iel kj / \ X t = H X, iel iel Y), X a Y = X n Y, fo r X, Y e t f . This lattice always contains the zero element 0 = C n (0 ) = i = s = and the unit element f)0 . Let us observe, th at a lattice of logics ('tf, ę > can also be conceived as an algebra (%>, v , a ) or as a “com plete” algebra ( # , V> A )- 2. Irreducible sets and basis of a closure system Let <£ be a closure system on a set 5 and X e <6. A" is called irreducible in %>, in symbols X e © 0(#), if for every (*) X = n .^ = > X e & . X is called finitely irreducible in in symbols X e if X # S and condition (*) holds for every finite family or, equivalently, for every Y , Z e W : X = Y r\Z= > => X = Y or X = Z. X is complete in <6, X e ©cpi(^), if A' is a maximal set in Corollary 3. a) S epl« O E S 0W £ ® i( < f ) , b) S i » 0(^ ) (since S = f ) 0 ) . Let ^ be a closure system of a consequence Cn on S. Let L ^ C n , X ) = {Ye<&: C n W s Y, Y t Y and V e C n (K u { ¢ )), for all ¢ ^ } . It is so called the family of W-relative Lindenbaum extensions. The family o f all relative Lindenbaum extensions, L,(Cn), is defined as follows: Z e L r(Cn) iff Z e z7 (C n , X ), for some X ^ S , *F e S . R e m a r k : We write “ <=” instead o f “ £ and Theorem 4. L / C n) = © o tffc) • Proof. Suppose th at Y e Lv (C n, X ) for some X ^ S and V* e S, and, on the contrary, let Y = H %i an(i Y ^ i e l , for some It follows that e f) X t = Y, iel iel which is impossible. On the other hand, if Y e SqO^), then Y c f) { Z e t f : Y<=Z}. Hence, O f Z e # : Y c Z } \ Y / 0 . Let us take an element V' from this set. Clearly, C n ( 0 ) £ y and ¥ e C n( Y u {¢)), if 4>$Y. It follows that Y e L v ( C n ,0 ) . q.e.d. By means o f Theorem 4 one may show that S Cpi(#) = ®0(^ ) is not generally true. The natural question arise: when S 0(# ) / ©iC^) ? The family o f sets 9i is called downward directed if for every X , Y e 91 there exists Z e 5R such that Z ^ X n Y. Theorem 5. I f < # , s ) is a distributive lattice and L = f) downward directed and L $ 5R / 0 , then L e S 1(c^’)\© 0(<^’). where M s S j W is Proof. It suffices to show that f) ®iC<0 for some downward directed 9 ł£ © ,(# ). Assume that f) = X n Y, i.e. X n Y ^ Z for every Z e 9 ł . By distributivity, Z = ( X a Y ) v Z = ( X v Z ) a ( Y v Z) . Since Z e i B , ^ ) , then: (*) J f ę Z or Y ^ Z for every Z e 9 ł . Suppose now, that X # f) 9ł # Y, i.e. there are Z , , Z 2 e SR such that <Pe X\ Z L and W e Y \ Z 2, for some ¢ , ¥ e S. But the family 91 is downward directed; so there exists T e 5R such that T ę Z , n Z 2 and ¢ , ^ ¢ 7 . Thus X £ T and Y £ T for r e i R , which contradicts (*). q.e.d. Let The set L l e ^ is called a proper extension o f L if L, □ ! . Theorem 6. I f L e S 1(<^’)\© 0(<^’), then L has no minimal proper extension in <€. Proof. Let ££©(,("<?). It is easy to check that: (*) L = f) {Z e L c Z } . Now, assume that X=>L. Then there exists Z € # such that Z=>L and A '£Z . Otherwise, by means o f (*) we would get Z. = f] {Z e L c z Z j ^ f) {Z e -SfsZ} = X, which is impossible. Thus, we have X n Z<^X. Moreover, L<=X n Z, since L = X n Z together with L e © X #) implies L = X or L = Z , which is impossible. Therefore, we have shown th at for every X tdL there is Z e'g' such that Lc= X n Z<= X, which means that L has no minimal proper extension, q.e.d. R e m a rk . It is an immediate consequence of the theorem above that S4 propositional modal logic o f Lewis has no minimal proper extension (closed under modus ponens, substitution and necessitation) which was proved in a separate paper by G. E. Hughes, cf. [4]. Since S4 is Hallden-complete, S4 £ © [(# ). Moreover, S4 has the finite model property, but has no finite adequate model, hence S4 $ S e(#). A non-empty family © £ # is a basis o f a closure system ^ if W = { f) !2: i.e. if for every X e V there exists £ ^£ © such that X = f) 2 (cf. [2]). Corollary 7. L et S B s^. 33 is a basis o f ^ iff X = f] { Z e S : J ę Z } fo r all X e W . Lemma 8. I f S then the following conditions are equivalent : i) SB is a basis o f (6. ii) I f fo r all Z e 93: K ę Z => A 'sZ , then X ^ Y , where X , Y e ^ . iii) I f *¥ $ Y , then there is Z e ® such that Y ^ Z and V $ Z . An easy proof is omitted. Every closure system has a basis, e.g. (or ^ { S 1}) is a basis o f <6. We are interested in finding the smallest basis o f a given closure system <€. The following corollary will be useful. Corollary 9. i) / / - ¾ is a basis o f <ś, then 930(%?) <=93. ii) I f © <=©' and © is a basis o f 'if, then so is S '. iii) I f © o ^ ) ls a basis o f %>, then it is the smallest basis o f W. The following theorem, connected with the name o f A. Lindenbaum, is well known. Theorem 10 (cf. [1], [2]). I f Cn is a finitistic consequence and V $ X, then Lv (C n, X ) =£ 0 . Hence, i f Cn is finitistic, then © 0(^ ) is a basis o f V>. In fact, it is the smallest basis o f (€. R e m a rk . F or some non-finitistic consequence Cn the smallest basis of # also exists, as will be shown later. Lemma 11. //"93 is a basis o f <6, then 0 A", = f) {Z e 93: 3 X , £ Z } , where X ,e teT teT. teT Theorem 12. A closure system <€ has the smallest basis i f and only / / © 0(c&) is a basis of%>. Proof. In view o f Corollary 9, (iii), it suffices to prove Let S be the smallest basis and assume, on the contrary, that S # ©0C<0, which means that Y $ SB0(^ ), for some 7 e © . Hence, there exists {X,}leT, X , e ^ ( t e T) such that Y = f] X, and Y<=.Xt IB T for all t e T. But for every Z e S , if there is t e T such that X t ^ Z then Y c Z . Thus, by Lemma 11, Y = f) {Z e © : 3 X , ę Z j 2 f | { Z e ® : Y c Z j . Therefore reT (*) y= n { Z e S :K c Z } It suffices to show th at ©\{X} is a basis of # , contradicting the assumption. We apply Lemma 8. Let Xe<tf, & e S and J . Since 93 is a basis, there exists Z e © such that A'£ Z and <P $ Z. There are two cases: a ) Z ^ , i.e. Z e © \{y } , b) Z = Y, i.e. X ę Y and ¢ ¢ 7 . In this case, by (*), there exists Z e 5$ such that F c Z and $ £ Z . Hence, from both cases a) and b) and Lemma 8, it follows that © \{y} is a basis of Contradiction, q.e.d. By language (of propositional calculus) we mean an algebra <5, <5j, .. ., (5n), where <5;, /< /;, is a finitary connective and 5 is a set o f formulas. Form ulas are formed as usual by means of infinite set o f propositional variables, A t = {p,},£T, and connectives, i.e. A tę 5 and if , ..., <Pk e S, then <>;($], \ . , 4>k) e S, where <5; is a A>ary connective. A pair 9)1 = J ) is called a matrix of a language S, if M = <M , f , is an algebra similar to ( S , <5,, ..., <S„), M # 0 and T c M (T is a set o f designated values). The set of all homomorphisms o f ( S , <5,, ..., S„) into M = ( M , f , ...,/„> will be denoted by Horn (5, M). By i)J( we mean the matrix consequence o f matrix 9R, in the sense of Łoś and Suszko [5], i.e. for every X ^ S , $ e S ¢ 6 9 ^ ( ^ ) iff V (i>(Ar)ęr=> i ' ( $ ) e r ) . t)cHom(S,M) Lemma 13 (R. Suszko). For every matrix o f the closure system o f S, {u-1 ( r ) : v e Horn (S', M )} is a basis Lemma 14 (cf. [2] Theorem . II.8). J f S $ ©, © is a basis o f Z , s Z 2 =>Z[ = Z 2, then © = ©cp,(^). and fo r all Z j , Z 2 e © : Lemma 15. Let 9)t = <M , T ) be a matrix o f a language { S , <5, , ..., <5„). Suppose that there is an unary connective <5 e {<5,, , <5„} such that the corresponding function f 6 in the matrix sJJt satisfies the following condition fo r x e M : f\x)sT Then S cpi(#iji) = iff x£T v e H om (5, M ) and, hence, ©cpi(#si) is a basis of Proof. It suffices to show th at the assumptions o f Lemma 14 hold. L etu, w e Horn (S ,A f) and u_1( r ) s w~i (T). If then ¢ ( ¢ ) ^ 1 , hence v(6$) = f >{ v { $ ) ) e T , thus <5$ e v~1( T ) ^ w ' ^ T ) , i.e. w ( 8 $ ) e T . But if Q e w ' ^ T ) , then w(<P)eT, hence w(54>) = / ^ ( 0 ) ) ^ 7 1. Contradiction. Consequently v ~ \ T ) = w ~ 1(T). Moreover, from the properties o f / 3 it follows that we cannot have v ( S ) ^ T , for any v = H orn(5 , M) . q.e.d. Corollary 16. a) I f the assumptions o f the lemma above are fulfilled by a matrix 9Jt then has the smallest basis. b) There are infinitely many non-finitistic consequences (not necessary on the same language S ) the closure systems o f which have the smallest bases. 3. Distributive law in lattice of logics A lattice <«g\ c > is distributive if for all'A', Y, Ze<&, X a ( Y v Z ) = ( X a Y ) v ( X a Z ) , or Xw (YaZ) = (Xv Y)a(XvZ); in the other words C n A 'n C n ( K u Z ) = C n ( X n ( Y kj Z)), or C n ( A 'u ( K n Z ) ) = C n (;r u X ) n C n ( r u Z ) . R e m a rk . We write (A, a) and C n(a) instead o f (A u {a}) and Cn({a}). Theorem 17 (cf. [3]). Let <6 be a closure system which has a basis following conditions are equivalent: (1) ( # , £ > is a distributive lattice, (2) C n (A , a) n C n (A , P) ę C n (A u (Cn(a) n C n (/?))), fo r A ^ S , a , /? e S, (3) C n(a) n Cn(/?) ę Z => a e Z or P e Z , fo r every Z e ©, a , P e S, (4) X n r s Z => X g Z or Y ę Z , fo r every Z e S , X , Y e <#. Then the Proof. Clearly, (1)=^(2). Now, assume (2), and suppose that (3) does not hold, i.e. C n(a) n C n (j5 )s Z a n d a, P $ Z , for s o m e Z e ©, a , /? e S. Then, by (2), Z £ C n ( Z , a) n n C n ( Z ,/? )ę C n (Z u (Cn(a) n Cn (/?))) = Z. Contradiction. It is quite easy to check that (3) implies (4). Finally, we assume (4). Since (A 'a Y ) v ( X a T ) ^ X a ( Y v T ) holds in every lattice, so, in view of Lemma 8 ii), it suffices to show th at for every Z e S , if ( X a Y ) v ( X a T ) £ Z then X a ( Y v T ) s Z , where X, Y, Te<£. B u tif ( ^ A Y ) v ( X a T ) < = Z , for Z e ©, then X n K g Z and X n T ę Z . Hence, by (4), ( X ^ Z or Y ^ Z ) and ( X ^ Z or J ę Z ) . Consequently, X z Z o r ( J ' u 7 ’) s Z , thus, X a ( Y v T ) = X n C n ( Y u T ) ^ Z . Hence (1) is proved, q.e.d. R e m a rk . In particular, the theorem above holds for every closure system of finitistic consequence (cf. Theorem 10). A complete lattice ( K , < ) is said to be join-infinite-distributive, if the join-infinitedistributive law holds in (K , < ) , i.e. for all X , Y , e K , t e T , X a \ / Y, = V ( X a Yt). 1eT teT Recall that, as usual, ^ is a closure system o f a consequence Cn on S. Lemma 18. ( f ś , c ) a finitistic. is a join-infinite-distributive lattice, i f a consequence Cn is Let us turn to the implicative lattices and pseudo-Boolean algebras. Let ( K , be a lattice, X , Y e K. An element Z e K is called the pseudo-complement o f X relative to Y, if Z is the greatest element such that X a Z < Y. If it exists, it is denoted by the symbol X -± Y. If, for every X , Y e K , Y exists, then ( K , < ) is called an implicative lattice (relatively pseudo-complemented lattice in [6]). Following Rasiowa and Sikorski [6], every implicative lattice (K , < ) with the zero element 0, is called the pseudo-Boolean algebra. Hence, every pseudo-Boolean algebra can be conceived as { K , v , a , , —), where — X = X-> 0. Recall, that in every lattice ( * ? , £ ) there is the zero element, namely 0 = C n (0 ) = f ) V . Theorem (i) (¾% (ii) (¾ , (iii) ( # , (iv) 19. ę> ę) 91) s) The following conditions (i) or (ii) imply (iii) and (iv): is a distributive lattice and Cn is the finitistic consequence, is a distributive lattice and has a basis S s © ^ ) , is a join-infinite-distributive lattice, is an implicative lattice. F or the p roof the following theorem is needed. Theorem 20 (Ward, cf. [1]). A complete lattice ( K , < ) is implicative i f and only i f it is join-infinite-distributive. In this case, we have X—*Y = \ J {Z e K: I a Z ^ Y}. Proof of Theorem 19. By Theorem 10 and Corollary 3 a), i) implies ii). To prove ii)^ iii), it is enough to show th a t \ /( A 'A X a \ J Yj, since the converse is generally iel true. We use Lemma 8 ii). Let \ J (X iel a K ,)ęZ , for Z e ® . Then X n Y , ^ Z , for every iel i e l . Since ( % , is'distributive, it follows, by Theorem 17, (4), that J f ę Z or K .S Z for every i e l . Hence, X ^ Z or C n (U K ,)ęZ , consequently (A 'a \ / y () £ Z . By Theorem 20, iel iel iii)<s>iv). q.e.d. Corollary 21. I f , ę ) is a distributive lattice and V has a basis © S © , ^ ) then (f&, is a (complete) pseudo-Boolean algebra. In this algebra (¥>, v , a , - ^ , - ) we have X—» Y = Cn (U { Z e # : X n Z s K}) or shortly, X Y = {a e 5: Cn(a) n X ę Y). By £(¾1) we denote the set of all propositional formulas (constructed by means of connectives + ~ ) which holds in a pseudo-Boolean algebra (W , v , a —), i.e. a e £ ( # ) iff h(x) = S, for every hom om orphism h of algebra of propositional language into algebra ( # , v , a , —). Corollary 22. I f ^ has a basis © £ © ,( # ) and , c ) is distributive, then £(¾7) contains all intuitionistically derivable (or intuitionistically valid) propositional formulas. 4. Dual spaces and the representation theorem If S' is a non-empty set, Cn a consequence (closure operator) on S and # a closure system on S, then the pairs <S , C n) and (S ’, # ) are called closure spaces. Because of the one-to-one correspondence between consequences and closure systems we may identify ( S , C n) and <S, # ) as the same closure space, where = &Ca or Cn = Cn^. In the paper [2] o f D. J. Brown and R. Suszko the notions of Galois connections and a dual space for a given closure space are presented (after O. Ore and C. J. Everett). F or our purpose only the case of, so called, © -natural dual spaces ([2], p. 17) will be used. These spaces will be called in this paper © -dual spaces. Let © be a basis o f a closure system <6. The closure space (© , Cnd) will be called TB-dual space fo r the space ( S , C n) if Cnrf is an operator Cnd: 2® -* 2®, defined as follows Cn\ U ) = { Z e © : f | t f s Z } , for i / s © Clearly, Cnd satisfies the conditions (1)-(3), hence it is a consequence (closure operator) on ©. Observe, th at CnJ(0 ) = 0 , if S $ ©. Let <gd be the closure system o f Cnd, i.e. <6* = { { /£ © : Cnd(U) = U). In view o f Theorem 1, (f€d, £ > is a complete lattice. Lemma 23. I f © is a basis o f <6, then in © -dual space ( S , ^ ) , fo r every F e © , we have'. F e ^ d iff F = { Z e © : X ^ . Z } fo r some Xe<€. Since Fe%>d iff F = { Z e © : f) F ^ Z ) , the p roof o f the lemma above is immediate. A closure operator (consequence) Cn on a set X is called topological, if Cn (0 ) = 0 and Cn(i4 u B) = C n (Ą u C n(fl), for A , B ^ X . Then Cn satisfies well known four conditions of topological closure (due to K. Kuratowski). A closure system c6 on a set X is topological, i f 0 e and A u B e W for A , B e W . In both cases m entioned above ( X , C n) and (A', # ) will be called topological spaces. A lattice ( K , < ) will be called a lattice o f sets, if K ^ 2 X, for a set X, < is E , join and meet is set-theoretical union u and intersection n , respectively, and the empty set 0 is the zero element of < ). Theorem 24. Assume, that a closure system <€ has a basis © s © / # ) . Then the following conditions are equivalent: i) (%>, £ ) is a distributive lattice, ii) © -dual space ( S , C n d) is a topological space, iii) < ? , £ ) is a lattice o f sets. F o r the p ro o f we need the following theorem. Theorem 25 (Brown, Suszko [2] Th. 3, p. 17). Let (© , Cnd) be the © - dual space for the closure space (S , C n). Then (a) Cnd(0 ) = 0 i f f S t B, (b) C n^F , u F2) = Cni(Fi ) u Cn\ F 2), fo r F , , F2£ © , iff fo r every Z e ©, X , Y e V: I n F s Z => I c Z or Y ę Z . Proof of Theorem 24. i) ^»ii). Assume i). Since S’£ SjC ^), then, by Theorem 17,(4), and Theorem 25 we easily get ii). ii)=>iii). Since <€* is a closure system, U a W = U n W, for U, W e V d. M oreover, by ii), U v W = C n \U u W ) = Cnd(t/) u Cnd( W) = U u W, and Cnd(0 ) = 0 , i.e. 0 e c£d. iii)=^i). Assume iii). It follows, that U v W = (J u W, for U, W e <gd. Hence, Cnd(A u B ) = Cnd(Cn“(A) u Cnd(fl)) = Cn“(A) v Cnd(B) = Cnd(A) u Cnd(B), for Thus, by Theorem 25, (b) and Theorem 17, (4), (& , is a distributive lattice, q.e.d. R e m a r k . If Cn is a finitistic consequence, and ( # , £ > is distributive, then, by the theorem above and Theorem 10, we have © 0(# )-d u al topological space, i.e. the smallest S -d u a l space. If S 0(#) / © i(# ) (cf. Theorem 5), we may get, in some cases, a lot of (infinitely many) S -d u a l topological spaces, where S 0( # ) £ © £ © ( # ) . It is easy to check that, for © '£ © " £ © ,( # ) <©' Cnd> is a (dense) topological subspace o f <©", Cnd>. We will use the standard topological notions o f T 0-space, 7^-space, compact space, topological basis (of open sets) o f a topological space, dense-in-itself space etc. (cf. e.g. [6]). Theorem 26. Let (%>, ę ) be a distributive lattice and a closure system with a basis 5 0 ^ 8 ,(¾ 1). Then the SB-dual topological space ( S , C n d) has the following properties: (1) ( S , C n d) is T0-space, (2) ( S , Cnd) is a compact space iff the consequence Cn is compact (cf. Brown, Suszko [2], Th. 5, p. 17), (3) { { Z e S : a ^ Z } : a e 5} is a topological basis o f open sets o f the space < S , Cnd>, (4) The following conditions are equivalent: a) <©, Cnd> is T r space, b) © = ©cpl(<f). (5) I f n (S \{ Z )) = C n (0 ), fo r every Z e S , then <(©, Cnd) is dense-in-itself Recall, that a consequence Cn on a set S is called compact, if C n (^ ) = S implies Cn ( A0) = S, for some finite A 0 ^ A . Proof. (1) Let Z v , Z 2 e ©, Z , * Z 2, e.g. Z , £ Z 2 . Then Z 2 i Cnd({Z,}), hence Cn'({Z,}) # Cnd({Z2}). (3) Let U be an open set in < S , Cnd), i.e. U = S \F , for some F e md. By Lemma 23, U = © \{ Z e S : J ę Z } , for some X e Since X, Ze<€, we have U = S \ F = © \{ Z e S : V a e Z } = U { Z e © :a £ Z } . neX aeX (4) <(©, Cnd) is a T, - space iff { Z e S : Z 0 g Z } = {Z0}, for every Z 0 € ©. Hence, in view o f Lemma 14, we get a) o b). It is well known th at in every topological space <X , C ), where C is a topological closure operator, the interior operator, Int, is defined. It is also known, that the family 0 o f all open sets of a topological space constitutes a complete lattice <0, £ > , where arbitrary join and finite meet are set-theoretical: \ J Ut = (J £/,-, £/, a U2 = £/, n U2, but infinite iel iel meet is defined as follows / \ £/; = In t(f) £/,-). for £/,- e ¢, / e / . Hence, a complete lattice iel ie / o f open sets can be written as ( <9, f)> (T)> where f)° ^ i = In t(f) i€I iel M oreover, <0, ę ) is an implicative lattice — for every U, W e <9 the relative com ­ plement, U W, always exists: U W = (J { Ve( 9: U n V ^ W j . This lattice has 0 as the zero element. Hence, we can consider the pseudo-Boolean algebra of open sets (0 , u , n , = ) . It can be shown that U W = In t^ A ^ f/J u W) and = U — Int(Jf\£/), for every sets £/, W open in a topological space ( X , C ), cf. [6]. Recall, that a complete lattice , e ) can be conceived as (*6, \J , / \ ) , where \J X-, = C n( IJ A",-), f \ = H iel iel iel iel and, if the closure system % has a basis ©£©,(<<?) and < # , ^ ) is distributive, then the pseudo-Boolean algebra (¾1. v , a , —) can be considered. The representation theorem. Theorem 27. Assume, that a closure system on a set S has a basis © S © , ^ ) and ^ ) is a distributive lattice. Then i) The complete lattice (%?, \J , / \ ) is (completely) isomorphic to the complete lattice ( Vd, n , U ° ) ° f °Pen sets ° f topological space ( © , CnJ> © -dual fo r the closure space ( S , C n), where 0 d = {©\F: F e ^ } ii) The pseudo-Boolean algebra ('6 , v , a , , - ) / 5 isomorphic to the pseudo-Boolean algebra ((9d, u , n , =>, = ) o f open subsets o f topological space (© , Cnd) ©-c/mo/ fo r the space ( S , C n). Proof. Define the mapping <p: # -*■ <9d as follows: qo(X) = [Z e ©: X g Z } , for X e This m apping has the following properties: (a) X z Y < x p ( X ) < = ( p ( Y ) , for X , Y . If Xę, Y and Zeq>( X) , i.e. X<£Z for Z e ©, then Y £ Z , hence Z e ( p ( Y ) . Conversely, let X £ Y Then, by Lemma 8 (ii) there exists Z e © such that X s Z and X £ Z , hence Z e q>(X)\cp(Y), i.e. (b) ę is “one-to-one” and “onto” If X =£ Y, then, e.g. X £ Y, therefore, by the p ro o f o f (a), ę ( X ) £ ( p ( Y ) . Now, let U e Q*. Hence U = © \F, for some F e % J. Thus, by Lemma 23, there exists X s ^ such that U = © \F = © \{ Z e © : X<=Z} = <p(X). (c) < p (V Xd — U<p(*i)- This is very easy to check. (d) <p(f\ X t) = n ° <P(Xi) = Intd( f) q>(Xi)), where Intd is the interior operator defined ie I in the iel iel space < © ,C n d> as usually. Since Intd(F) = ©\Cnd(© \F ) and S \ f)ę(X,) iel = U(®\<K^<)) = {Z e then, by Lemma 11 we have *e I iel = H 3 I jC Z } , n(®\n<p(^i)) iel which, by the definition o f Cnd proves (d). Consequently, p art (i) of the theorem ie I is proved. It follows immediately from (a)-(d). th at the conditions (e)-(g) hold (e) ę(X-^> Y) = ę ( X ) => ę ( Y ) , (f) < ? ( - * ) = ^ t p ( X) , (g) (p(S) = © , ę ( C n ( 0 ) ) = 0 . Hence, the proof of (ii) is completed, q.e.d. = Corollary. I f the assumptions o f the representation theorem are satisfied, then E(%>) fo r pseudo-Boolean algebras v , a ) and ( 0 d, u , n , = ). £ ( 0 dj , References [1] G . B i r k h o f f : L attice theory, Am er. M ath. Soc., P rovidence, 1973. [2] D . J. B r o w n , R. S u s z k o : A bstract logics, D issertaliones M ath. C II, 1973. [3] W. D z ik , R. S u s z k o : On distributivity o f closure system s, B ull, o f the Section o f Logic, vol. 6, N o . 2, 1977, pp. 64-66. [4] G. E. H u g h e s : M odal system s with no m inim al proper extensions, R ep o rts on M ath. Logic, N o. 6, 1976, pp. 93-98. [5] J. Ł o ś , R. S u s z k o : R em a rks on sentential logics, ln d ag a tio n e s M athem atice 20, 1958, pp. ,177-189. [6] H . R a s i o w a , R . S i k o r s k i : The M athem atics o f M etam athem atics, P W N , W arszaw a 1970. [7] A . T a r s k i : Logic, Sem antics, M etam athem atics, O xford, 1956. View publication stats