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Addition of Initial Segments. I Commentationes Mathematicae Universitatis Carolinae Antonín Sochor Addition of initial segments. I. Commentationes Mathematicae Universitatis Carolinae, Vol. 29 (1988), No. 3, 501--517 Persistent URL: http://dml.cz/dmlcz/106665 Terms of use: © Charles University in Prague, Faculty of Mathematics and Physics, 1988 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE 29,3 (1988) ADDITION OF INITIAL SEGMENTS I Antonfn SOCHOR Abstract: In the alternative set theory, for every real initial seg­ ment RfiN there is either £c R with R= i<fr:(3oCG R,)n#£ f + ocior J « N~R with R=-taW $ ;(V^oc fcR )#+oC< £$ where R = {£ *R;( V<* 6 R)# + +0C+16R}. This result can be used in measure theory. More generally, we ex­ tend addition and subtraction to the system of all initial segments of N and we investigate properties of these operations. In particular, we describe the behaviour of these operations on all initial segments which are real classes. Further properties of these operations can be found in the following paper [S]. Key words: Alternative set theory, natural number, finite natural num­ ber, initial segment, real class, tf-semiset, S-semiset. Classification: Primary 03E70 Secondary 03H15 We use the notions usual in the alternative set theory (AST; see f.V]), in particular the symbols N and FN denote the class of all natural numbers and the class of all finite (in another terminology standard) natural numbers respectively. A class X is called a ST-semiset ( 6* -semiset resp.) if there is a sequence 4x ;n<*FNl with X= fKxn;nfFNj (X= Utx ;nfcFNl resp.). Complete subclasses of N are called initial segments and cuts are initial seg­ ments closed under the successor operation. The most important axiom of AST is the prolongation axiom i.e. the sta­ tement (VF)((Fnc(F)&dom(F)=FN)-^(Jf)(Fnc(f)8cF9f)). Let us recall that every initial segment which is simultaneously sf-semiset and 6*-semiset is a set (cf. § 5 ch. II [Vl; this statement is a consequence of the prolongation axiom) and that the sole cut which is a set is the empty set 0 (this assertion is implied by induction accepted for sets). The system of real classes defined in [C-V3 plays an important role in - 501 - AST. In this paper we need only one property of real classes (proved in the cited article), namely that every initial segment which is simultaneously a subclass of a set and real, is either a Tf-semiset or a 6T-semiset. However, let us remind at least that all sets, & -semisets and 6 -semisets are real classes and that for every property $(z,Z,,...,Z ) in which only real clas­ ses are quantified and for all real classes X,,...,X , the class U; #(x,X1,...,Xn)} is real, too. We use letters R, S, T and U to denote initial segments; the letters<s£ , ($ ... denote natural numbers and the letters k, n and m are reserved for variables running through finite natural numbers. Following a Zlato§' s idea, we define for every initial segment R R'=Ru{R> i.e. we put R =R for every nonempty cut and <*/= oc +1 for each oC« N. Evid­ ently R' is a nonempty initial segment and we have «*,€Re<sO+l6R'&R=«(#;(3oC6 R')#<cC$&(R=SsrR'=S')& &(R£S5iR'SS'). We are going to define addition and subtraction on the system of all initial segments; to avoid misunderstanding, we use for the operation of sub­ traction the symbol -r because our operation extends subtraction defined on natural numbers, however, it does not extend subtraction defined on the class of integers (see (2)) which operation is denoted by the symbol - . (Let us mention that the symbol - is used in AST also to denote the difference of classes X-Y=-{ x eX;x t Y}.) For every two initial segments R, S, we define their sum by R+S= C^;(Joe 6 R')(jp€S')<#<«+/$? and their difference by R -S=-{^ ;( V ft € S')#+/3 s R? = = {* ;(V/56S')(lo06R)/J^oo&^&oc-/3}- The class RTR plays an important role in our investigation and we are going to denote it by the symbol R+, i.e. we define (cf. (1) and (3c)) + R = •(•eR;(yoC£ R)oC +#+leRT. We say that an initial segment R is closed under the operation + iff (y<x,!36 R)oc +/36R. If a cut R is closed under the operation + , then R' is also closed under the - 502 - operation + , (because 0'= {OT is closed under + ).Let us mention thatiOi is closed under the operation + , however, it is no cut (and ioY= {0,U is not closed under + ). In the following we summarize some useful statements concerning the abo­ ve defined operations starting with the trivial ones: (1) a) R5R+S and R-rSSR because O&S' for every S and since oc e R —*-(©O<ac+l+0&oo+lcR'). D b) RCS~*RTS=0 because for fie S with ft £ R and every ^€ N we have * + ftz 0* R&/Ses'. O (2) The operations + and T defined above extend the arithmetical addi­ tion and subtraction and, moreover, for £ £ J e N we have In fact for every £ , ps N we have fi» ;#<£+?} -4<fr;(3<* *f )(3tf .6 J )*<* + ?? = ={*;C3*e §' )(3'? s J' )*< % + v} = § +$, for every £ ^ 6 € N we have §-$=<*;*<§-$» =-t+;*+S<?J = *(*<*£$)<# + + ir<§i=-C# ;(Vt «$')*+*<£» = §"*-? and the last statement is a trivial consequence of (lb). D (3) a) R+0=R=RxO and OT*R=0=0+. • b) (R+S)'=-V^;(3oC6 R')(3/3€ S')^.£oC+/3} (RrS)'-{^ ;(V/5 6 S')<#+/U R'? (assuming SfcR) (R+)'=-(/£;( VoCC. R')^+<tf€R'}. The statements are trivial consequences of the definitions, however, one has to distinguish whether the initial segments in question are sets or proper classes. D c) If S4=0, then R+S=-C#;(3oC€ R')(3/$€S)#£<*+/$J (R+S)'=-C.-^;(3oC6R')(3/3sS)^.4oC+/3+l} R-rS=-E<#»;(V/.ie S)^+/J+16R} (R-rS)'=-\^;(V/?6 S)^ + (3eR} + s ={^;(V/ie s)^+^+ieSl =s-rS n d) If R 4=0 4= S, then R+s=-C^;(3oce R)(3pe s)^^ac+/3+iK a (4) The operation + defined on the system of all initial segments is associative and commutative because of the associativity and commutativity - 503 - of the arithmetical addition (and because of (3b)). D (5) a) S&R+—> R+S=R=RvS because SSR implies by (3b) (V/3 6 S')(\feCeR') fi + <*,eR' and thus we get R+SS-C^JGOC 6 R')i^^oc}=R and furthermore (3b) implies also the formula (Voce RXVts (R+)')oc+reR. O b) R+cS-**RcR+S. We have (R+)'c S' and thus there is £ e S' with tl ^ (R+)' and by (3b) we get (Jote R')/3 + OC$ R'; this shows R'c (R+S)'. O c) (R+C S&R+ 0)~* R-rSCR. For A e S with j5^ R+ there is << € R <see (3c)) so that aC+jJ+l ^ R and *& being an element of R is no element of RTTS because /3+leS'. O d) If both R and S are cuts closed under the operation + , then R+S=RUS. According to (3b) we have (R+S)'=Ru S=(RuS)' because both R' and S' are closed under the opera­ tion + . D (6) (RTS)TT=RT(S+T). The formulae *6(R-rS)TT (Vr «V)#+f e(RrS) (VesV)(V(3e s')# + t?+/UR (VcTe(T+s)')^+cT€R -3»6(RT(T+S)) . are equivalent by (3b). D (7) If RCT abd ScU, then R+S ST+U and RTU cT-rS. a (8) a) If SSR, then R-rS is the greatest T with T+SSR. The formulae , (RTS)+S5R (V^€(R-rS)')(V^3€S')^+/JeR' (V^t(V(?6S')(^+peR')^->(V^cS')^+^eR')l - 504 - are equivalent; assuming * eTsR&r^R-r-s, we are able to find Ac S' with *t + B ^R and it is ir + /.?€ T+S because t+l6T'^t + |3<(r+l)+l3 . b) If R s^O, then R+S is the smallest U with U-rS2R. The formulae (R+S)-.S£R and (Voce R)(V(5 € S')(3oC6R')oC +/3<oC+/S are equivalent and the second one is valid (put So =oO+l); supposing <£^ U^6R+S&R*0 we can find oCe R' and 13 6 S' with #<oG + /* &o£4-0 and then 0C-I6R&0C-I ^ UvS- D (9) a) The formulae R is a cut R+FN=R R-rFN=R are equivalent. To prove this assertion it is sufficient to realize that FN is the smallest nonempty cut. D b) For every 2 , £ +FN is the smallest cut containing £ and £ -r FN is the maximal cut not containing £ . These statemen ts are trivial consequences of a) and (la),(2),(5d), (6) and (8). O At the end of this paper we are going to give some examples of cuts R, S such that (RTTS)+SCR, however, for every R, S if there is T with T+S=R, then (RvS)+S=R; similarly there are R, S with (R+S)*-SDR, however, for every R, S , if there is U with UvS=R, then (R+S)-rS=R.
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