F U N D AM E NTA nnoindentMATHEMATICAEF U N D AM E NTA 1 65 open parenthesis 2 0 0 closing parenthesis nnoindentNonreflectingMATHEMATICAE stationary subsets of P sub kappa lambda by n centerline f1 65 ( 2 0 0 ) g Yoshihiro A b eF open U N D parenthesis AM E NTA Yokohama closing parenthesis Abstract period We explore the possibility of forcing nonreflecting stationary sets of P sub kappa lambda n centerline f NonreflectingMATHEMATICAE stationary subsets of $ P f nkappa g nlambda $ g period 1 65 ( 2 0 0 ) We also present a P sub kappa lambda generalization of Kanamori quoteright s weakly normal filters comma n centerline fby g Nonreflecting stationary subsets of Pκλ which induces by stationary reflection period Yoshihiro A b e ( Yokohama ) n centerline1 period .. Introductionf Yoshihiro period A b .. e Throughout ( Yokohama .. this ) g .. paper .. kappa .. denotes .. a .. regular .. un hyphen Abstract . We explore the possibility of forcing nonreflecting stationary sets of Pκλ. countable cardinal and .... lambda a cardinal .... greater equal kappa period .... For any such pair .... open We also present a Pκλ generalization of Kanamori ' s weakly normal filters , which induces parenthesisn hspace ∗fn kappaf i l l commag Abstract lambda . closing We explore parenthesis the comma possibility P sub kappa of lambda forcing nonreflecting stationary sets of $ P f nkappa stationaryg nlambda reflection. $ denotes .... the .... set .... open1 . brace Introduction x subset lambda . Throughout : bar x bar less this kappa paper closingκ bracedenotes period .... a For .... x in P sub kappa lambdaregular comma un - .... let .... kappa sub x = bar x cap kappa bar comma nnoindentP sub kappaWe sub also x x = present open brace a s $ subset P f x : n barkappa s barg less nlambda kappa sub$ x closing generalization brace comma and of x-hatwideKanamori ' s weakly normal filters , which induces countable cardinal and λ a cardinal ≥ κ. For any such pair (κ, λ);Pκλ = open brace y in P sub kappa lambda : x subset y closing brace period denotes the set fx ⊂ λ : j x j < κg: For x 2 Pκλ, let κx = j x \ κ j; nnoindentWe say Xstationary subset P sub kappa reflection lambda i . s unbounded if X cap x-hatwide negationslash-equal varnothing for any x in P sub kappa lambda period Let FSF sub kappa comma lambda n hspace ∗fn f i l l g1 . nquad Introduction . nquad Throughout nquad t h i s nquad paper nquad be the filter generated by open braceP x x-hatwide= fs ⊂ x :j :s xj< in κ Pg sub; and kappax = f lambday 2 P λ closing: x ⊂ y braceg: period .. Every filter κx x b κ on$ n Pkappa sub kappa$ lambdanquad idenotes s assumednquad a nquad r e g u l a r nquad un − to be fine comma thatWe say i s commaX ⊂ Pκ extendingλ i s unbounded FSF subif kappaX \ x comma6= ? for lambda any x 2 periodPκλ. Let .. IfF F SF i sκ,λ a filterbe the comma F nnoindent countable cardinal and n h f i l l $ nlambdab $ a c a r d i n a l n h f i l l $ ngeq nkappa to the power of plusfilter denotes generated the set by fxb : x 2 Pκλg: Every filter on Pκλ i s assumed to be fine , that i s . $ n h f i l l For any such pair n h f i l l $ ( +nkappa , nlambda ),P f nkappa g open brace X subset, extending P subF SFkappaκ,λ: lambdaIf F i : sP a filter sub kappa;F denotes lambda the minus set X element-negationslash F closing bracenlambda period$ We say X subset P sub kappa lambda i s c los ed if union of sub alpha less delta x sub alpha in X for any subsetnnoindent hyphendenotes increasingn chainh f i l l thefXn⊂hP f iκ lλ l: Psκ eλ t −nXelementh f i l l $− nfnegationslashFx nsubsetg: nlambda : nmidangbracketleftx nmid x sub< alphan barkappa alpha lessng delta. right $ n angbracketh f i l l For in Xn h with f i l l delta$ less x kappan in semicolonP f n Xkappa i s a g We say X ⊂ P λ i s c los ed if S x 2 X for any ⊂ − increasing chain cnlambda lub if it is closed, $ andn h unbounded f i l l l e t periodn h f i lκ .... l We$ nkappa f α<δx gα = nmid x ncap nkappa nmid hx j α < δi in X with δ < κ; X i s a c lub if it is closed and unbounded . We , $say S subset P subα kappa lambda is stationary if S cap X negationslash-equal varnothing for any club X period say S ⊂ P λ is stationary if S \ X 6= for any club X: Let CF denote the club .. Let CF sub kappa commaκ lambda denote ? κ,λ filter on P λ generated by the club subsets of P λ. n beginthe clubf a l filteri g n ∗g on P sub kappaκ lambda generated by the club subsetsκ of P sub kappa lambda period All the notions defined above for P λ can b e naturally translated into P Allf the nkappa notions definedf x gg abovex for = P subnf kappas lambdansubset can b e naturallyxκ : translatednmid intos nmid < nkappa f x g ng , and P nwidehatx if κ i sf regularxg = uncountablenf y . Forn in instanceP ;Xf n⊂kappaP x ig s unbounded nlambda if for: any x nsubset P sub kappa subκx x x ifx kappa sub x i s regular uncountable period .. Forκx instance comma X subset P sub y ng . y 2 P x there is z 2 X such that y ⊂ z; and F SF denotes the kappa sub x x i s unboundedκx if κx;x nendf a l i g n ∗g filter on P x generated by fs \ P x : s 2 P xg which i s a κ − complete filter on for any y in P sub kappa subκx x x there is z inb X suchκx that y subsetκx z comma andx FSF sub kappa sub x comma P x: x denotes the κx In the next section for certain large T ⊂ P κ+ we force a stationary set Wefilter say on $ P X sub kappansubset sub x xP generatedf nkappa by openg brace nlambda s-hatwide$ capi s P unbounded subκ kappa sub if x x $X : s in Pn subcap kappanwidehatfxg nnot= n varnothingS ⊂ P κ+ $such for that anyS \ P $ xx i sn nonstationaryin P f for nkappa any x 2gT: nlambda . $ Let $ FSF f nkappa sub x x closing brace whichκ i s a kappa sub xκx hyphen complete As the counterpart , in the third section , we present a generalization of w line − e , filternlambda on P subg$ kappa sub x x period akly normal filters on regular cardinals due to Kanamori and show that the existence beIn the the next filter section generated for certain large by T $ subsetnf P n subwidehat kappaf kappaxg to: the x powern in of plusP we forcef nkappa a stationaryg nlambda set of such filters gives the reflection of stationary set s of P λ. ng S subset. $ Pn subquad kappaEvery kappa filter to the power on $ of P plusf such nkappa that Sg cap nlambda P subκ kappa$ subi s x assumed x i s nonstationary for 2000 Mathematics Subject Classification : Primary 3 E 35 , 3 E 55 . anyto x be in T fine period , that i s , extending $ FSF f nkappa , nlambda g . $ nquad I f Research partially supported by \ Grant - in - Aid for Scientific Research ( C ) , The Min $F$As the isafilter counterpart comma $, in the F^ thirdf section+ g$ comma denotes we present the set a generalization of - istry of Education , Science , Sports and Culture of Japan 9640299 " . w line-e akly normal filters on regular cardinals due to Kanamori and show that n beginthe existencef a l i g n ∗g of such filters gives the reflection of stationary set s of P sub kappa lambda period [55] nf2000X Mathematicsnsubset SubjectP Classificationf nkappa g : Primary nlambda 3 E 35:P comma 3f E n 55kappa periodg nlambda − X element−negationslash F Researchng . partially supported by quotedblleft Grant hyphen in hyphen Aid for Scientific Research open paren- thesisnendf Ca lclosing i g n ∗g parenthesis comma The Min hyphen istry of Education comma Science comma Sports and Culture of Japan 9640299 quotedblright period n hspaceopen square∗fn f i bracket l l gWe 5 say 5 closing $ X squarensubset bracket P f nkappa g nlambda $ isclosedif $ nbigcup f nalpha < n delta g x f nalpha g n in X $ f o r any $ nsubset − $ increasing chain nnoindent $ n langle x f nalpha g nmid nalpha < n delta nrangle $ in $ X $ with $ n delta < nkappa ; X$ i s a c lub if it is closed and unbounded . n h f i l l We nnoindent say $ S nsubset P f nkappa g nlambda $ is stationary if $ S ncap X nnot= n varnothing $ for any club $X . $ nquad Let $ CF f nkappa , nlambda g$ denote the club filter on $ P f nkappa g nlambda $ generated by the club subsets of $ P f nkappa g nlambda . $ n hspace ∗fn f i l l g All the notions defined above for $ P f nkappa g nlambda $ can b e naturally translated into nnoindent $ P f nkappa f x gg x $ i f $ nkappa f x g$ i s regular uncountable . nquad For instance $ , X nsubset P f nkappa f x gg x $ i s unbounded if f o r any $ y n in P f nkappa f x gg x$ there is $z n in X $ such that $ y nsubset z , $ and $ FSF f nkappa f x g , x g$ denotes the nnoindent f i l t e r on $ P f nkappa f x gg x $ generated by $ nf nwidehatf s g ncap P f nkappa f x gg x : s n in P f nkappa f x gg x ng $ which i s a $ nkappa f x g − $ complete f i l t e r on $ P f nkappa f x gg x .