Chvalina, Jan

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumJan ChvalinaIsomorphic algebraic pre-closures and equivalent set-systemsArchivum Mathematicum, Vol. 13 (1977), No. 3, 141--147Persistent URL: http://dml.cz/dmlcz/106970Terms of use:© Masaryk University, 1977Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCH. MATH. 3, SCRIPTA FAC. SCI. NAT. UJEP BRUNENSIS XIII: 141—148, 1977 ISOMORPHIC ALGEBRAIC PRE-CLOSURES AND EQUIVALENT SET-SYSTEMS JAN CHVALINA, Brno (Received December 28, 1976) Theorem 1.1 in [3] chap. II says that each closure system ^ o n a given set S defines a closure operation on this set and conversely each closure operation on S defines a closure system on this set, thus there is given a one-to-one correspondence between all closure operations on S and all closure systems on this set. In the mentioned theorem this correspondence is expressed explicitely. From here it follows that the system of all closure operations on a given set S can be mapped injectively into the system exp exp S such that two closures are isomorphic if and only if the corresponding set-systems are equivalent (in the sense of paper [6]) i.e. S?t, S?2 e exP exP S are equivalent if there exists a permutation/of the set S such that ^ 2 -= {/(X) : XeSft} orS^1 = {/(X) : Xe^2}. This equivalence is denoted by ~ . A natural question is whether the above described monorelational embedding is extendable onto a certain system of more general structures so called pre-closure operations. In paper [5] this problem is solved for topological closures and Cech's topologies. The aim of this paper is to show that there exists a system (closed with respect to isomorphisms) of the cardinality 2cafdS (for an infinite carrier set S) of algebraic pre-closure operations to which it is possible to extend the just mentioned embedding into (exp exp 5, ~ ) . Terms and notations concerning algebraic closure operations are taken from papers [2], [3]. Let S be a set, C be a map of exp S into itself and n be a positive integer. By Cn will be denoted the n-fold composition of C with itself. A mapping C: exp S -> -• exp S is a pre-closure if for any P c .Sand Q c S these conditions are satisfied: P c C(P) and P c Q implies C(P) c C(Q). A pre-closure C which satisfies C2(P) c C(P) for all P contained in S is a closure. A pre-closure (or closure) which satisfies the compactness condition, for any P c S and for any x e C(P) there is a finite set Q c P such that x e C(Q\ will be called algebraic. This condition is equivalent to the condition: C(P) = *= U {C(Q) *Q<=P, card Q < K0} for each P c S. By a pre-closure (closure) space we mean an ordered pair (S, C), where C is a pre-closure (closure) on the set S. A set P is said to be closed in the space (S, C) or 141 C-closed if C(P) = P. The system of all closures on the set S will be denoted by ^(S). Let pre-closure spaces (S t , Ct), (S2, C2) and a mapping f:S1->S2 be given. The mapping / i s said to be an isomorphism of the space (S l 5 Cx) onto the space (S2, C2) if / is bijective and f(Ct(P)) = C2(f(P)) for each set P c S t . If Sx -=- S2 == S, pre-closure spaces (S, Cx)9 (S, C2) are isomorphic, we say that pre-closures Ct, C2 are isomorphic and we write Cx s C2. A system 58f of pre-closures on a set S is said to be closed with respect to closure isomorphisms if C e f , Ct € ^(S), C ^ Ct implies that Q e # . A pre closure C will be called n-iterable if n is the least positive integer such that Cn is a closure. Let T be a non-empty subset of S. A decomposition of the set T determines a decomposition in the set S (in the sense of [1] chap. I). This decomposition will be denoted by Tin accordance with [1]. A kernel of a set P c S (denoted by [P]) in the decomposition T (where T c S, T ^ id) is a union of all the blocks of T which are subsets of the set P. In what follows we suppose that S is an infinite set. Consider a system of triads of the form {T9 T9 A}9 where Fis a non-void subset of S satisfying conditions card T = 2, card (S - T) = X0, T is a decomposition of T such that XeT implies card X < K0 and A is a finite subset of S, lineary ordered, disjoint with T. We assign to every such a triad a mapping C: exp S -• exp S which is defined as follows: Let A = {ax, a2, ..., an} and the ordering of A be given by the n-tuple of indices {1, 2 , . . . , n}. For X c S such that Xn ^ c {aj and [X] = 0 (i.e. X does not contain as a subset any element of T)9 we put C(X) = X. If X n A = {ah,..., afJ, /fc :g «, then C(X) = Xu {afl + 1 , . . . , aik + 1} u X0, where we put an+1 = an and X0 = {a0} if [X] T* 0 and X0 = 0 otherwise. A triad {T9 T9 A} corresponding to the mapping C will be denoted by {TC9 TC9AC} or {TC,TC9 (AC9 <)} if ti is necessary to express the ordering of the set Ac. Finally, denote by <rfk(S) the system of mappings C of exp S into itself such that Ac = {a0, ax,..., ak} and put s/(S) = Lemma 1. ^ (S) is a system of algebraic pre-closures on S, closed with respect to closure-isomorphisms, such that jrf(S) n ^(S) = 0 and to each positive integer k there exists in sf(S) a k-iterable pre-closure. Proof. Let n be an arbitrary positive integer, Css/n(S). From the above construction it follows immediately that C is a pre-closure on S. Let P c S be a non-void set. If P n Ac c {an} and [P] = 0 (in the decomposition Tc) then the same holds for each finite Q c P, hence C(P) = U ( c ( 0 ) • 0 c ^ card Q < K0}. Let P n A = {ah9..., aik}9 ik g # and [P]c =5-= 0. Then by the construction of (S, C) we have C(P) = P u {a0, ah + l9..., a^+J. Let jce C(P) be an arbitrary point. If x — a09 then x e C(Y)9 where F e Tc is a finite subset of P. If x = ay, jefl! 4- 1,...?4 + 1}» t^1 1 ^eC{dj-i}, where aj^.1eP. If moreover xeP , 142 then clearly x e C{x}. In all the possible cases considered with respect to the set P we get in the similar way that x e C(P) is followed by x e C(Q) for a suitable finite subset Q a P. Thus (5, C) is an algebraic pre-c!osure space. Since for arbitrary XeTc there holds C(X) = Xu {a0} and C2(X) = Xu {a09 at} where ax # a09 a1 $ X, the pre-closure C is not a closure and we get s/(S) n ^(S) = 0. Let D be a pre-closure isomorphic to C, / : (S, V) -> (S9 D) the corresponding closure-iso-morphism. Put r =/(F c ) , T = {/(X) : Xe Tc}9 b( =/(a f) , / = 0, 1, ..., n and B = {b, : / = 0, 1, ..., n}. Since/is a permutation of the set 5, we have that there exists a closure, say Cl9 which corresponds to the triad {T, T9 B} in the above sense. Let P c S be an arbitrary set. If P n B c_ {bn} and P does not contain any element of Fi.e. [Pf = 0, then Q(P) =P. T h e n / ' V ) n A c {att} and [f""1^)]^ = 0 thus C(f-'(P)) = /" 1 (P ) . Hence D(P) =D(f(r\P))) =/(C ( / - J (P ) ) ) = = P = C!(P). Now, let P n i? = {b{1, ..., b,.J be valid with 4 =" n and [P] = r = U Xj9 where X,, X2,..., Xr are elements of T. Then/"1 (P) n A = {*,,,..., a<J, w h e r e / ^ ) = bil9 ...9f(aik) = bik and/_1(Xy), j = 1, 2 , . . . , rare all the elements of Tc contained i n / "1 ^ ) - We have D(P) = D(f(f'1(P))) = f(C(f"'(P))) = = f(/"1(/>)u{a0,al.1 + 1 , . . . , a , k + 1}) = P u { b 0 , b f l + 1 , . . . , b i k + 1} =C , (P) . We get in this way that for each subset P of S there holds D(P) = Ci(P), thus D = Ct and we have that s#(S) is closed with respect to closure isomorphisms. Let k be a positive integer. Consider arbitrary Ces#k„1 (S). ThenC*+1(7") = Fu {a09al9..., ak„t} = C*+1(F)andCw(F) = Fu {a0, al9 ..., am_J c F u {a0, a,, ..., aw} = = Cm + 1(^) for m < k. Since C*(P) = C* + 1(P) for each P c 5, i.e. C* is a closure and Cw is not if m < k, we have that C is a k-iterable pre-closure, q.e.d. Now, put 3T(S) = %(S) u s#(S) and define a mapping F of /^"(S) into exp exp S by F(C) = {X: X c 5, C(X) = J } u U { I : I c . 5 , C*(X) * 5, C*+1 (X) = S} fc_i for every Ce,r(S). Lemma 2. 2>t C1? C2 e ^ (5 ) , Ct ^ C2. Then it holds F(Cx) ^ F(C2). Proof. Let {Tl9 Tl9 (Al9 <t)}9 {T29 Tl9 (Al9 <2) be triads corresponding to C!, C2 respectively, where Cx, C2 are arbitrary different pre-closures from $t(S)» Consider all possible cases: (1) Tt = T2, Tx = T29 (Al9 <t) # (A29 <2), (2) Tx =T29 Tx # T29 (3) T1 * T2. Let the case (1) occur. Suppose A1 ^ A2 and put P = I \ u ^ t = T2 u Al^ Q = F! u A! u A[2 = F2 u A! u A2. Since Tt n Ax = 7^ n A2 = 0, it holds P T* Q. Further C^P) = P thus PeF(Ct) and C2(P) = 0, C1(P) = C2(g) = 143 2> « Q # S for card (S - T{) \> K0, i = 1, 2. Thus P$F(C2). Let ^ = A_. <i * <2* ? u t {#o»ai> •••»*»} = ^i» {&o>*i> •••-*«} = Al2. There exists a pair of indices ij such that at~bj9i # j. Let / e {0, 1, 2, ...,/*} be the greatest non-negative integer with the property at * 6 p where y < /. Put P = {ai9 ai+l9...9 an}. There is Ct(P) = P but C2(P) # P and C2(P) # 5 for every fc. Hence Pe F(CX)9 P$F(C2) and we have F(Ct) # F(C2) in the case (1). Let the case (2) occur. There exists a block Xe Tt which does not belong to T2. If [X]2 = 0 , then the set X is C2-closed thus XeF(C2) whereas X$F(CX). If [X]2 # 0, there exists a block Fe T2 contained as a subset in X. Then [Y]t = 0 and thus Y is a enclosed set i.e. YeF(Cx) but Y£F(C2), thus F(CX) ^ F(C2) again. Suppose now that (3) holds, i.e. Txj*T2. Admit first that (Al9 <x) = *= (^2» <i)» Without loss of generality it can be supposed Tx — T2 ^ 0. Let xe Tx - Tl9 XE Tt be such a block that J C G ! Let [X]2 = 0. Then X<£ F(Cx), Xe F(C2) since C2(X) = X. If [X]2 # 0, then there exists a set Ye T2 with Y c X. Then [Y]x = 0 thus C^Y) = Yand we have Ye F(CX)9 Y$ F(C2). Now, consider the case (Ai9 <x) # (A29 <2). If Ax = Af2, <x # < 2 , then we get in the same way as in the case (1) that there exists a s e t P c S with P e F(CX)9 P $ F(C2). Let Ax # A2, Ax = {a09ai9 . . . ,a„}, _42 = {b09 bl9..., * m } . If there exists ate € At — A2 with / < n then C2{a,} = {aj for card X = 2 whenever XG T2 and thus {aj e F(C2) but {aj £ F(Cj). Let at e Ax — A2 implies i = n. If there exists bj eA2 - Ax with j < m, then similarly as above {bj} e F(CJ but {£,} £ F(C2). If bje A2 ~ ^ implies j = m, and a{e Ax — ^ 2 is followed by i = n, then we have {a„-l9 an} e F(Cj) and {#„-!, an} $ F(C2). Now, consider the case A A — A2 = = {aw}. Then there exist positive integers i9j with i <n9 j < n9 i ^ j such that bm-x = at, fem = a j and thus {6W-1? ftw} is a C2-closed set, i.e. {bm„l9 bm} e F(C2) but {6W-!, bm} $F(Ct). Therefore F(Ct) # F(C2) in the case (3) and we have that the mapping F restricted onto the system sf(S) is injective. Lemma 3. Let Ci9C2e F(S). Then Ct £ C2 if and only if F(Ct) ~ F(C2). Proof. If Ci9 C2 are closure operations i.e. Ci9C2e <£(S)9 then F(CX)9 F(C2) are systems of all Ct-closed, C2-closed sets respectively, thus Cx £ C2 if and only if F(CX) ~ F(C2). This is the well-known assertion. Suppose that Ct e jtf(S) — - #(S), C2esf(S) - #(S) are isomorphic pre-closures. Denote by { F ^ T j , (^l1} <i)}, {Tl9T29(Al9 <2)} triads corresponding to Ci9C2 respectively. It was shown in the proof of lemma 1 that if Cx £ C2, then there exists a permutation / o f the set S such that T2 = / (F 1 ) , -T2 - {/(-*) - Xe Tx} and (^2, <2) is order-isomorphic to (Ai9 <t) with the isomorphism / . Let Ax = {a0,ai9 ...9 an}> A2 = {b0, * x , . . . , ftn}. Let P be a C2-closed set and put Q = / " * (P). The following three cases are possible: (1) P n _42 c {&„} and [P n F2]2 = 0, 144 (2) bt<=P r\A29 i < n implies bi+teP n A2 and [P n r 2 ] 2 « 0, (3) [P n T2]2 # 0 and A2 a P. Considering the properties of the mapping/we get that Q c {aH} and [Q n Tt]x = = 0 in the case (1). Similarly, at e Q n At9 i < n implies a l+1 e Q n At and [Q r\ Tt]t = 0 if the case (2) occurs and [Q n Tt]t ¥* ®, At cz Q in the case (3). Thus Q =rx(P) is a Crclosed set. Now, let P e F(C2) be such a set that C2(P) # # 5, C^+1(P) = 5, where H » - l . Then P = S - {*0, ft1$..., **} hence / "HP ) = S - { a 0 , a 1 , . . . , a J k } and thus f'1(P)eF(Ct). Therefore F(C2) c c {/(X): Xe F(Ct)}. If P is a set of the form P =/(£?), where g is a suitable set belonging to F(Ct)9 we get, similarly as above, considering all possible cases with respect to the set Q that the inclusion F(C2) z> {/(X) : XeF(Ct)} holds. Thus we have F(Ct) ~ F(C2). Now, suppose F(Ct) ~ F(C2) for Ct9C2e st(S). There exists a permutation/ of the set S such that F(C2) = {/(X) : Xe F(Ct)}. Put J ~f(Tt)9 T = {/(X) : Xe e f j . Let (Al, <) be a chain such t h a t / : (A^, <j) -» (A[, <) is an order-iso-morphism of At onto .A. Denote by C a pre-closure on S determined by {T9 T9 (A, <)}. As it was shown in the proof of lemma 1, the pre-closure Cis isomorphic to the pre-closure Ct. Further, it is easy to see that F(C) = {/(X) :XeF(Ct)}> thus we have F(C) = {/(X) : Xe F(Ct)} = F(C2) and by lemma 2 it holds C = Cl9 hence pre-closures Ct, C2 are isomorphic. Finally, let Ct e s/(S)9 C2e<g(S)9 i.e. Ct non s C2. We are going to show that F(C t) non ~ P(C2). Admit on the contrary that there exists a permutation/of the set S with F(C2) = = {/(X) :XeF(Ct)}. Let .A, = {a0, at,..., a„}, Since Cj(S - {a09at}) = = 5 - {a j , Cf(5 - {a0, at}) = C,(5 - {aj) = S thus 5 - {a09 at}e F(Ct)9 we have S - {/(a0), /(a t)} e F(C2). Further C ^ u .Ax) = Tt u ^ l 5 hence the set f(Tt) Kjf(At) is C2-closed. Then also f(Tt) u {f(a2)9 ...,/(*„)} == [S -- {/(tfoX/fai)}] n- ( / ( r j u / ( ^ i ) ) is a C2-closed set. From here Tt u {a2, . . . , «„} = / ~ 1 / ( r 1 <j {a2,..., a„}) e F(Ct)9 which contradicts the definition of F(Ct). Hence F(Ct) non ~ F(C2). The proof is complete. Lemma 4. Let C l5 C2eF(S)9 Ct ± C2. TAen F(Cx) # P(C2). Proof. If C l5 C2 e<^(S), then ^ ( C J , P(C2) are systems of all closed sets in corresponding closure spaces, thus Ct # C2 implies F(Ct) ^ F(C2). If C l 9 C2 e e ^ (S) , Ct * C2, then F(Cx) ^ F(C2) by lemma 2. If Ct e tf(S)9 C2 e <$(S)9 then with respect to lemma 3 F(Ct) = F(C2) is followed by Ct ^ C2 which is a contradiction, thus F(Ct) ^ F(C2). LemmaS. It holds: cards/(S) = 2cards,card [^(S)j^ ]-= c a r d S W ^ e ^ ( S ) / ^ implies card $* ^ card 5. Proof. Let r c 5 be a set of an infinite cardinality tn. The system of all such 145 decompositions of the set T blocks of which have finitely many elements has the cardinality m. Denote by & the system of triads {T, T, A} satisfying the above conditions, namely N0 = card T, card (S - T) = K0, Xe Timplies 2 = card X < < K0, 2 = card A < K0 and A n T = 0. Clearly, card «T = 2cards because there is at least 2cards different sets T satisfying the just mentioned conditions. On the other hand card 3F = 2cardS. card S . K0 = 2cardS. From the equality card & = = card*fi/(S) it follows the first assertion. Consider the decomposition «af(S)/ = . Denoting by & a decomposition of #" such that {Tx, Tl,Al}e^ and {T2, T2, A2} e & belong to the same block of #" if there exists a permutation/of the set S with T2 =f(Tt), T2 = {/(X) : Xe Tt}, A2 ~f(At), we have card j*(.S)/£ = = card *F « card S . card S . K0 = cdrd S. Let 3T € s/(S)/^. Denote by # the corresponding block of P. Let {T, T, A} e ^ . Consider these two possible cases: (1) card (S - T) = card S, (2) K0 = card (S - T) < card S. In case (1) we chose an arbitrary element ae T and assign to every element xe S — (T v A) a triad {Tx, Tx, Ax}, where Tx ~fx(T), Tx = {fx(X) : Xe T}, Ax = A and fx is a permutation of the set S defined by :fx(s) = s for s e S, x ?- S # a and/*(a) = x, /-(*) = AT. Evidently card S ^ card ^ . Let case (2) occur. We construct other triads from {T, T, A} in the following way. Let X, Ye T, X ^ Y, aeX, beY. Put Tx = T, A, = A, Xt = (X - {A}) u {£}, y t = (Y - {6}) u {a} and finally Tt = ( r - {X, K}) u {Xl9 Yx}. If C, Cx are corresponding pre-closures, then it holds C(X%) = X! ?* Xx u {a0} = C^X^ and C ^ Ct. Since card T = card T = = card S, we get again card S = card ^ . Hence card S = card $", q.e.d. As in § 4 of [5] we use, for the sake of brevity, the following notions. If P, Q are sets and g, a binary relations on P, Q respectively, then the mapping/: P -+ Q is called an embedding of the monorelational system (P, Q) into the monorelational system (Q, a) if/ is injective and for every pair of elements aeP, b e Q it holds aqb if and only if f(a) af(b). We summarize the obtained results in the following theorem. Notice that we have proved in fact a stronger assertion because the below described system of pre-closures was explicitely constructed. Theorem. Let S be an infinite set. There exist a system &"(S) of pre-closures on S containing #(S), closed with respect to closure-isomorphisms, and a mapping F of &~(S) into exp exp S, such that it holds: 1° Each element of ^(S)—#(S) is an algebraic pre-closure on S and to every positive integer n there exists an n-iterable pre-closure contained in &~(S) —- ^(S). 2° card [P(S) - V(S)] = 2cardS, card [(«T(S) - ^(S))/^] = card S and® e e [F(S) - «(5)]/S implies card X = card S. 3° F : &~(S) -• exp exp S is an embedding of the monorelational system (&~(S), = ) into the monorelational system (exp exp S, ~ ) and for every closure Ce(i(S) ** holds F(C) = { I c S : C(X) = X}. 146 Proof follows from lemmas 1,3,4 and 5. The paper [5], mentioned in the introduction, contains the following incorrect­ness. The system &A(P\ defined in § 3 p. 108 —109 is not a system of A-topologies and thus final system 3~(P) does not contain any A-topology. All lemmas and especially the main theorem of the paper [5] are valid, however for their proofs it is necessary to change the definition of 2TA(P) as follows: Denote by s4\(P) a system of all A-topologies on P satisfying the following condition. There exists a pair Xx, X2 c P of non-void sets with Xt u X2 # P, card (Xt n X2) = 1 such that if X c P then uX = Xu Y, where (i) Y = 0 if X n Xt = 0 = X n X2, (ii) Y = Xi9/e{l,2} if XnXf * 0 and X n Xj = 0 for fe {1, 2} , j # /, (iii) Y = X! u X2 i f Xn Xx * 0 * Xn X2. To every A-topology u from the system s/t(P) there is assigned a pair of sets Xj, X2 with the above described properties. We shall denote these sets by Lt(u), L2(u) respectively. Put $~A(P) = {uesfx(P) : cardL^w) ^ 2, cardL2(w) = 2, card (Lx(u) n L2(u)) = 1 and /^(w) u L2(w) ?-P}. It was proved by Vladimir Tichy that all assertions concerning STA(P) from paper [5] are true after the above change of the definition of the system & A(P). R E F E R E N C E S [1] O. Borůvka: Foundations ofthe Theory of Groupoids and Groups, VEB DVW, Beгlin 1974. [2] S. Burris: Representation theorems for closure spacesf Colloq. Math. 19 (1968), 187—193. [3] S. Burris: Closure homomorphisms, Journal of Algebra 15 (1970), 68—71. [4] Р. M. Cohn: Universal Algebra, Harper and Row, New Үork 1965. [5] J. Chvalina: On homeomorphic topologies and equivalent seťsystems, Arch. Math. (Brno) 2, ХII(1976), 107—115. [6] F. Neuman and M. Sekanina: EquivaІent systems ofsets andhomeomorphic topologies, Czeeh. Math. Journ. 15 (90) (1965), 323—328. J. Chvalina, 662 95 Brno, Janáčkovo nám. 2a Czechoslovakia 147 

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Isomorphic algebraic pre-closures and equivalent set-systems

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Isomorphic algebraic pre-closures and equivalent set-systems

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Institute of Mathematics AS CR, v. v. i.

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