LOGIC comma ALGEBRA comma AND COMPUTER SCIENCE \noindentBANACHLOGIC CENTER , ALGEBRA PUBLICATIONS , AND comma COMPUTER VOLUME SCIENCE 46 INSTITUTE OF MATHEMATICS \noindentPOLISH ACADEMYBANACH CENTER OF SCIENCES PUBLICATIONS , VOLUME 46 WARSZAWA 1 9 9 9 \ centerlineHELENA RASIOWA{LOGICINSTITUTE , ALGEBRA comma OF , .. ANDMATHEMATICS 1917 COMPUTER endash 1994 SCIENCE} to the power of 1 WIKTOR BARTOLBANACH CENTER PUBLICATIONS , VOLUME 46 \ centerlineInstitute of{ MathematicsPOLISH ACADEMY comma OF.. Warsaw SCIENCESINSTITUTE University} OF MATHEMATICS Banacha 2 comma 2 hyphen 97 Warszawa commaPOLISH ACADEMY Poland OF SCIENCES \ centerlineE hyphen mail{WARSZAWA : bartol at 1 mimuw 9 9 9 period} edu periodWARSZAWA pl 1 9 9 9 1 EWA .. OR L-suppress OWSKAHELENA RASIOWA , 1917 – 1994 \ centerlineInstitute of{ TelecommunicationsHELENA RASIOWA , \quad 1917WIKTOR−− $ BARTOL 1994 ˆ{ 1 }$ } Szachowa 1 comma 4 hyphen 894Institute Warszawa ofcomma Mathematics Poland , Warsaw University \ centerlineE hyphen mail{WIKTOR : orlowska BARTOL at itl period} Banacha waw period 2 , 2 pl - 97 Warszawa , Poland ANDRZEJ .. S K OWRON E - mail : bartol @ mimuw . edu . pl \ centerlineInstitute of{ MathematicsInstitute comma of Mathematics .. Warsaw UniversityEWA , \quad ORWarsawLOWSKA University } Banacha 2 comma 2 hyphen 97 WarszawaInstitute comma Poland of Telecommunications \ centerlineE hyphen mail{Banacha : skowron 2 at , mimuw 2 − 97 periodSzachowa Warszawa edu 1 period , 4 , - 894Poland pl Warszawa} , Poland 1 period .. Elements of a biography periodE - mailHelena : orlowska Rasiowa@ commaitl . waw mathematician . pl and logician comma passed \ centerlineaway on August{E − 9 commamail : 1 bartol994 period $@$ Her influenceANDRZEJ mimuw on the . S edu K shape OWRON . of pl mathematical} logic can hardly be overestimated period Institute of Mathematics , Warsaw University \ centerlineShe was born{EWA in Vienna\quad onOR June\L 20OWSKA commaBanacha} 1 9 21 , 7 2 to - 97very Warszawa patriotic , PolandPolish parents period As soon as Poland had regained its independenceE in - 1 mail 9 1 : 8 skowron after more@ thanmimuw a century . edu . pl of partitioned \ centerlinest ateless existence{ Institute1 . comma Elements of the Telecommunications whole of family a biography settled in . Warsaw}Helena period Rasiowa Helena , mathematician quoteright s father and logician was a high hyphen level , passed away on August 9 , 1 994 . Her influence on the shape of mathematical logic can \ centerlinerailway specialist{hardlySzachowa semicolon be overestimated 1 .. , his 4 − knowledge894 . Warszawa and experience , Poland in the} field led him to assume very important positionsShe was in the born railway in Vienna administration on June 20 period , 1 9 1 The 7 to girl very comma patriotic her Polishparents parents quoteright . As only soon child comma had\ centerline {asE Poland− mail had : regained orlowska its independence $@$ itl in 1 . 9 waw 1 8 after . pl more} than a century of partitioned good conditionsst ateless to grow existence up physically , the whole and mentally family settled period in.. And Warsaw comma . Helena indeed ’ comma s father she was exhibited a high many- \ centerlinedifferent skills{levelANDRZEJ and railway interests\ specialistquad commaS K ; varying OWRON his knowledge from} music and comma experience which in she the was field learning led him at to amusic assume school very simultaneouslyimportant with her positions normalstudies in the railway in a secondary administration school comma . The girl to business , her parents management ’ only child comma , had \ centerlinewhich she studied{goodInstitute conditions for more of than to Mathematics grow a semester up physically after , completing\ andquad mentallyWarsaw her secondary. University And , education indeed} , she period exhibited many But finally thedifferent most importantskills and interests of her interests , varying comma from asmusic the ,future which was she to was prove learning comma at tooka music the school lead period \ centerlineIn 1 938 comma{simultaneouslyBanacha the t ime 2 was , with 2 not− her97 very normal Warszawa propitious studies for , inPoland entering a secondary a} university school period , to business Even if management not many in , Europe werewhich convinced she studied that war for was more inevitable than a semester comma the after next completing year was her to prove secondary how mistaken education . But \ centerlinehline {finallyE − mail the most : skowronimportant of $ her @ interests $ mimuw , as . the edu future . pl was} to prove , took the lead . 1 sub ReprintedIn by 1 courtesy 938 , the from t ime Modern was not Logic very comma propitious vol periodfor entering 5 comma a university no period . Even 3 open if parenthesis not many July 1 995 closing1 . \quad parenthesisElementsin Europe comma were of pp a period convinced biography 231 thatendash war . 247Helena was semicolon inevitable Rasiowa circlecopyrt-c , the , next mathematician year Modern was to prove and how logician mistaken , passed awayLogic on Publishing August period 9 , 1 994 . Her influence on the shape of mathematical logic can hardly be overestimatedopen square bracket . 9 closing square bracket She was born in Vienna on June 20 , 1 9 1 7 to very patriotic Polish parents . As soon as Poland had regained1Reprinted itsby courtesy independence from Modern in Logic 1 9, 1 vol 8 . after 5 , no more . 3 ( thanJuly 1 a 995 century ) , pp . 231 of – partitioned st ateless247; existencecirclecopyrt ,− thec Modern whole Logic family Publishing settled . in Warsaw . Helena ’ s father was a high − l e v e l railway specialist ; \quad his knowledge and experience in the field led him to assume very important positions in the railway administration[9] . The girl , her parents ’ only child , had good conditions to grow up physically and mentally . \quad And , indeed , she exhibited many different skills and interests , varying from music , which she was learning at a music school simultaneously with her normal studies in a secondary school , to business management , which she studied for more than a semester after completing her secondary education . But finally the most important of her interests , as the future was to prove , took the lead .

In 1 938 , the t ime was not very propitious for entering a university . Even if not many in Europe were convinced that war was inevitable , the next year was to prove how mistaken

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$ 1 { Reprinted }$ by courtesy from Modern Logic , vol . 5 , no . 3 ( July 1 995 ) , pp . 231 −− $ 247 ; circlecopyrt −c $ Modern Logic Publishing .

\ [ [ 9 ] \ ] 1 0 .. W period BARTOL ET AL period \noindentthose of the1 majority 0 \quad wereW .period BARTOL Rasiowa ET ALhad to. interrupt her studies comma for no legal education was possible in Poland after 1 939 period Many people fled the country or at least they fled the \noindentbig towns commathose more of the subject majority to German were bombardments . Rasiowa and had terror to period interrupt .. So did her the studies Ra acute-s , family for comma no legal education wasalso possible because of the in fact Poland that most after high 1 hyphen 939 . ranking Many people administrative fled officials the country and members or at of least they fled the bigthetowns government1 , 0 more wereW being. subject BARTOL evacuated ET to AL German . towards Romaniabombardments period The and Ra terror acute-s quoteright . \quad sSo spent did a year the in Ra Lvov $period\acute{ s } $ familyAfter Soviet , those invasion of the in September majority were 1 939 . the Rasiowa town washad taken to interrupt over by her the studies Soviet Union, for no period legal education alsoThe life because of manywas of possible Poles the became in fact Poland endangered that after most 1 939comma high . Many so− eventuallyranking people fled the administrative the father country decided or at to least return officials they to fled and the members of theWarsaw government periodbig towns were , more being subject evacuated to German towards bombardments Romania and terror. The . Ra So $ did\acute the Ra{s ´s }family$ ’, s spent a year in Lvov . AfterLife was Soviet veryalso restricted invasion because in of Poland the in fact September during that most the Nazi 1 high 939 o - ccupation ranking the town administrative period was Nevertheless taken officials over comma and by members thethere Soviet were of the Union . Theenough life courageous ofgovernment many people Poles were to organizebecame being evacuated an endangered underground towards , life Romania so comma eventually . not The only Ra for thes´ ’ armed s spentfather conspiracy a year decided in Lvov to . return to Warsawagainst the. NazisAfter Sovietcomma invasion but also in for September the development 1 939 the of town all the was areas taken of aover nation by the quoteright Soviet Union s life which. The are vit al for itslife survival of many comma Poles education became endangered and comma , in so particular eventually comma the father higher decided education to return among to Warsawthem period .. Thus LifeHelena was Rasiowa very. restricted followed her studies in Poland in mathematics during comma the Nazi risking o her ccupation life comma . as Nevertheless did everybody , there were enoughwho dared courageous to conspireLife was duringpeople very restricted that to dark organize in period Poland period an during underground the Nazi o ccupation life , . not Nevertheless only for , there armed were conspiracy againstPolish mathematics theenough Nazis courageous acquired , but particular also people for to strength organize the developmentin an the underground pre hyphen of war life all years , not the comma only areas for .. armed mainly of a conspiracy nationaf hyphen ’ s life which are vitter theal emergence foragainst its of survival the the Nazis Polish , ,but School education also of for Mathematics the and development , in in 1 92particular of 1 all period the areas .. The , of higher namesa nation of education ’ Stefan s life which among are them . \quad Thus HelenaMazurkiewicz Rasiowavit and al for followedWac its l-suppress survival her , education sub studies aw Sierpi and in ,acute-n in mathematics particular ski comma , higher , who risking education were in her Warsawamong life them comma , . as or Thus did those everybody of Stefan Banachwho dared toHelena conspire Rasiowa duringfollowed her that studies dark in mathematicsperiod . , risking her life , as did everybody who and Hugo Steinhausdared to comma conspire who during were that in Lvov dark comma period . were well known to mathematicians all over the Polishworld periodmathematics ..Polish One of mathematics acquiredthe branches particular acquired which became particular strength important strength at in that in the the time pre pre besides -− warwar functional years years , mainly , \quad af - mainly a f − teranalysis the comma emergenceter the set emergencetheory of theand of topology Polish the Polish comma School School was of of logic Mathematics Mathematics comma with in researchers 1 in 92 1 1 92. 1such The . as names\quad Jan L-suppress ofThe Stefan names ukasiewicz of Stefan Mazurkiewicz and Wac $ \ l { aw }$ S i e r p i $ \acute{n} $ ski , who were in Warsaw , or those of Stefan Banach comma Mazurkiewicz and Wac law Sierpin ´ ski , who were in Warsaw , or those of Stefan Banach and andStanis Hugo l-suppress SteinhausHugo sub Steinhaus aw , Le who s-acute , who were were niewski in in Lvov Lvov comma , were , Kazimierz were well well known Aj dukiewicz known to mathematicians to comma mathematicians Alfred all over Tarski the and world all others over period the worldRasiowa . became\quad. One stronglyOne of of the influenced the branches branches by which Polish became which logicians important became period importantStill at that in conspiracy time at besides that comma functional time she wrote besides analysis , functional analysis , set theory and topology , was logic , with researchers such as Jan \L ukasiewicz , her Master quoterightset theory sand thesis topology under , the was supervision logic , with of researchers Jan L-suppress such asukasiewicz JanLukasiewicz and Boles , Stanis suppress-llaw Le sub aw Soboci n-acuteS t a n i ski s period $ \s´lniewski{ aw , Kazimierz}$ Le Aj $ dukiewicz\acute{ s ,} Alfred$ niewski Tarski and , others Kazimierz . Aj dukiewicz , Alfred Tarski and others . History commaRasiowa however became comma turnedstrongly against influenced her once by Polish more periodlogicians .. In. Still 1 944 in comma conspiracy the Warsaw , she wrote Uprising broke Rasiowa became strongly influenced by Polish logicians . Still in conspiracy , she wrote out and commaher Master in consequence ’ s thesis comma under Warsaw the supervision was almost of Jan completelyLukasiewicz destroyed and Boles commalaw notSoboci onlyn ´ becauseski . of herwarfare Master commaHistory ’ sbut thesis also, however because under , turned of the the against systematic supervision her destruction once more of .Jan which In\L followed1 944ukasiewicz , the the Warsaw uprising and Uprising after Boles broke $ \ l { aw }$ Sobociit had been $ \acuteout put and down{n ,} in period$ consequence s k Rasiowa i . , quoteright Warsaw was s thesis almost was completely burned comma destroyed together , not with only the because whole of house period SheHistory , howeverwarfare , but , turned also because against of the systematic her once destruction more . \quad which followedIn 1 944 the uprising, the Warsaw after it Uprising broke outherself and survived , inhad consequence with been her put mother down ., in Rasiowa Warsaw a cellar ’ that was s thesis was almost coveredwas burned completely by the , together ruins of destroyed with a demolished the whole , not house only . She because of warfarebuilding period , butherself also survived because with her of mother the systematic in a cellar that destruction was covered by which the ruins followed of a demolished the uprising after itAfter had the been warbuilding comma put down . Polish . mathematics Rasiowa ’ comma s thesis .. asall was other burned areas of , life together comma began with to the recover whole it s house . She herselfinstitutions survived commaAfter its withthe moods war her ,and Polish mother its people mathematics in period a cellar Many , as had all that other been was killed areas covered comma of life , many by began the had to ruins diedrecover comma ofit s a many demolished had b uleft i l d the i n g country . institutions comma ,but its those moods who and remained its people considered . Many had their been duty killed to be , the many reconstruction had died , many had of Polish scienceleft the and country universities , but period those whoOne remainedof the important considered conditions their duty for this to be reconstruction the reconstruction of Afterwas to the gather warPolish together , Polishscience all thoseand mathematics universities who could . participate One , \quad of the inas important recreating all other conditions mathematics areas for this periodof reconstructionlife Rasiowa , began was to recover it s institutionshad in the meantimeto gather , its accepted together moods a all and teacher those its who quoteright people could participate s . position Many in had in a recreating secondary been killed mathematics school period , many . .. Rasiowa That had is died had where , many had leftshe was the discovered countryin the meantime by , Andrzej but accepted those Mostowski who a teacher and remained brought ’ s position back considered in to a the secondary University their school period duty . She to That rewrotebe is the where reconstruction she ofher Polish Master quoteright sciencewas discovered sand thesis by universities Andrzejin 1 945 Mostowskiand in . the One nextand of brought year the she backimportant started to the her University academic conditions career . She foras rewrote an this her reconstruction wasassist to ant gather atMaster the University together ’ s thesis of in all Warsaw 1 945 those and comma inwho the the could next institution year participate she she started remained her in linked academic recreating with career for the mathematics as an assist . Rasiowa hadrest in of her the lifeant meantime period at the University accepted of Warsaw a teacher , the institution ’ s position she remained in a linked secondary with for school the rest of . her\quad That is where sheAt this was University discoveredlife . she prepared by Andrzej and defended Mostowski her Ph and period brought D period back thesis to in 1 the 950 open University parenthesis . under She rewrote the herguidance Master of Prof ’ s periodAt thesis this .. University Andrzej in 1 945 Mostowski she and prepared in closing the and defendednext parenthesis year her period Ph she . D started .. . thesis The tit in her le 1 950of academic the ( under thesis the was career Algebraic as an treatmentassist antguidance at the of University Prof . Andrzej of Mostowski Warsaw , ) .the The institution tit le of the thesis she was remainedAlgebraic linked treatment with for the restof the of functional herof thelife calculi functional . of Lewis calculi and of Heyting Lewis and and Heyting it pointedand to it the pointed main field to the of mainher field of her future future researchresearch : .. algebraic : algebraic methods methods in logic in logicperiod . .. In In 1 1 956 956 she she made made her her second second academic academic de de - hyphen \ hspace ∗{\ f i l l }At this University she prepared and defended her Ph . D . thesis in 1 950 ( under the

\noindent guidance of Prof . \quad Andrzej Mostowski ) . \quad The tit le of the thesis was Algebraic treatment of the functional calculi of Lewis and Heyting and it pointed to the main field of her future research : \quad algebraic methods in logic . \quad In 1 956 she made her second academic de − HELENA RASIOWA comma 1 9 1 7 endash 1 9 94 .. 1 1 \ hspacegree ....∗{\ openf i l parenthesis l }HELENA equivalent RASIOWA to , habilit 1 9 1 ation 7 −− today1 9 closing 94 \quad parenthesis1 1 .... in the Institute of Mathematics of the Polish \noindentAcademy ofgree Sciences\ h f comma i l l ( where equivalent between 1to 954 habilit and 1 957 ation she held today the post ) \ ofh Associate f i l l in Pro the hyphen Institute of Mathematics of the Polish fessor jointly with an analogous position at the University comma becoming Professor in 1 957 \noindentand subsequentlyAcademy Full of Professor Sciences in 1 967 , periodwhere For between the degree 1 she 954 submittedHELENA and 1 RASIOWA 957 two papers she , 1 9 held 1: 7 .. – quotedblleft 1 9the 94 post1 1 Alge of hyphen Associate Pro − fessorbraic models jointlygree of axiomatic with ( equivalent an theories analogous to quotedblrighthabilit ation position today and quotedblleft ) at inthe the University Institute Constructive of Mathematics theories , becoming quotedblright of the Professor Polish comma in which 1 957 togetherand subsequently formedAcademy Full of Sciences Professor , where betweenin 1 967 1 954 . For and 1 the 957 degreeshe held the she post submitted of Associate two Pro papers - : \quad ‘ ‘ Alge − braica thesis models namedfessor Algebraic of jointly axiomatic with models an of analogoustheories e lementary position ’’ theories and at the ‘‘ and UniversityConstructive their applications , becoming theories period Professor ’’ in , 1 957which and together formed aHer thesis interests namedsubsequently were notAlgebraic limited Full Professor to models pure research in of 1 967 e period lementary. For Always the degree ready theories she to submitted cooperate and their two with papers oth applications hyphen : “ Alge - . ers and awarebraic of the models importance of axiomatic of a strong theories mathematical ” and “ Constructive community theories comma ” she , which participated together formed a Herininterests many formsthesis of were this named community notAlgebraic limited quoteright models to ofpure s e activities lementary research period theories . .. Always Since and their 1 964 ready applications comma to and cooperate. until her retirement with oth in− ers1 993 and comma aware sheHer of headed interests the the importance were section not of limited Foundations of to a pure strong of research Mathematics mathematical . Always and ready later community to of cooperate Mathematical , with she oth participated - ers inLogic many after forms itsand creation aware of this of in the 1970 community importance period .. of For ’ a s strongmore activities than mathematical 1 5 years . \ shequad community wasSince Dean , she of 1 the participated 964 Faculty , and in until many her retirement in 1of 993 Mathematics , sheforms headed comma of this Computerthe community section Science ’ s activities of and Foundations Mechanics . Since open 1 of964 parenthesisMathematics , and until 1 her 958 retirement and endash later 1 in 960 1 of 993comma Mathematical , she 1 962 endash 1Logic 966 comma after 1headed 968 its endash the creation section 1 978 closing of in Foundations 1 parenthesis 970 . of\quad periodMathematicsFor more and laterthan of 1 Mathematical 5 years she Logic was after Dean its of the Faculty Strongly attachedcreation to in the 1 970Polish . Mathematical For more than Society 1 5 years comma she was she Deanwas its of Secretary the Faculty open parenthesis 1 955 endash 1 957\noindent closing parenthesisofof Mathematics , Computer , Computer Science Science and Mechanics and Mechanics ( 1 958 – 1 960 ( 1 , 1958 962−− – 11 966 960 , 1 968 , 1 – 962 −− 1 966 , 1 968 −− 1 978 ) . and then Vice1 978 hyphen ) . President .... open parenthesis 1 958 endash 1 959 closing parenthesis semicolon .... in the Warsaw\noindent divisionStronglyStrongly of the Society attached attached she to was the to Polish the Mathematical Polish Mathematical Society , she was Society its Secretary , she ( 1 was 955 – its 1 957 Secretary ) ( 1 955 −− 1 957 ) elected Presidentand then twice Vice open - President parenthesis ( 1 957958 endash– 1 959 1 ) 959 ; in and the 1 Warsaw 963 endash division 1 965 of closing the Society parenthesis she was period In the Association\noindent forandelected Symbolic then President Logic Vice − twicePresident ( 1 957 – 1\ 959h f i and l l ( 1 963 1 958 – 1 965−− )1 . In 959 the ) Association ; \ h f i l l forin Symbolic the Warsaw division of the Society she was she was a CouncilLogic member open parenthesis 1 958 endash 1 960 closing parenthesis and a member of the Executive Committee\noindent forelectedshe was a President Council member twice ( 1 958 ( 1 – 957 1 960−− ) and1 959a member and of1 the 963 Executive−− 1 965 Committee ) . In for the Association for Symbolic Logic European AffairsEuropean open Affairs parenthesis ( 1 972 1 972 – 1 endash 977 ) .1 She977 closing also was parenthesis Alternate period Assessor She ( also 1 972 was – 1Alternate 975 ) and Assessor open parenthesis\noindent 1 972sheAssessor endash was a 1 975Council closing member parenthesis ( 1 and 958 Assessor−− 1 960 ) and a member of the Executive Committee for open parenthesis( 1 975 1 975 – 1 979 endash ) in 1 the 979 Division closing of parenthesis Logic , Methodology in the Division and ofPhilosophy Logic comma of Science Methodology of the Inter and Philosophy of\noindent Science of theEuropean- nationalInter hyphen Union Affairs of History ( 1 972 and Philosophy−− 1 977 of ) Sciences . She . also On the was other Alternate hand , she contributed Assessor ( 1 972 −− 1 975 ) and Assessor national Unionto the of History development and Philosophy of mathematics of Sciences in Poland period , On both the as other a member hand comma of the she Committee contributed on \noindentto the development(Mathematics 1 975 of−− mathematics1 of 979 the Polish ) in in Poland Academy theDivision comma of Sciences both of as since aLogic member 1 96 , 1 of Methodology until the Committee her very last and on days Philosophy , and of Science of the Inter − nationalMathematics Unionas of a the member of Polish History , Academy and for and 20 of years Philosophy Sciences Chairperson since 1 of 96 , Sciences 1 of until the her Group very . On on last Education the days other comma and hand and Research , she in contributed toas the a member developmentMathematics comma and of offor mathematicsthe 20 yearsMinistry Chairperson of Science in Poland comma and Higher ,of both the Education Group as a on member .Education Since 1 of 972 and the she Research was Committee on in the on MathematicsMathematicsScientific of of the the Ministry Council Polish of of Science the Academy Institute and Higher of of Sciences Computer Education Science since period of 1 Since the 96 Polish 1 972 until sheAcademy was her on of very the Sciences last , days , and asScientific a member Councilguiding , and of it the s for works Institute 20 as years Chairperson of Computer Chairperson in Science 1 972 – of 1 , the 983 of Polish . the Last Academy Group but not on of least Sciences Education , in the comma recent and years Research in Mathematicsguiding it sshe works of greatly asthe Chairperson contributed Ministry in toof 1 972 the Science endash foundation 1 and 983 of period Higher the Polish .. Last Education Society but not of least Logic . Since comma and Philosophy 1 in 972 the recent she of was years on the Scientificshe greatly contributedScience Council . to of the the foundation Institute of the of Polish Computer Society of Science Logic and of Philosophy the Polish Academy of Sciences , guidingof Science it period sNo works description as Chairperson of Rasiowa ’ s in activities 1 972 would−− 1 be 983 complete . \quad withoutLast a mention but not of her least deep , in the recent years sheNo descriptiongreatlyinvolvement contributed of Rasiowa in quoteright the to development the s activities foundation of the would research ofbe complete the j ournals Polish without to which Society a mention she was of of attached Logic her and . The Philosophy o fdeep Sci involvement encemost . in cherished the development by her , of theFundamenta research j Informaticaeournals to which( she FI was ) , attached was established period in 1 The most ..977 cherished , mostly by due her to comma her efforts .. Fundamenta , and she was Informaticae its Editor - in.. open- Chief parenthesis until her death FI closing . Even parenthesis when comma No.. was description establishedher in iofllness .. 1 Rasiowa 977 began comma to ’ take s activities the better of her would , she be never complete ceased to without control the a preparation mention of of her deepmostly involvement due tothe her consecutive efforts in comma the issues development and of the she j was ournal its of Editor . In the particular hyphen research in , she hyphen actedj ournals Chief as Editor until to her of which a death special period she anniver was Even attached when her . Thei llness most began\quad- sary to take issuecherished the commemorating better by of her her comma the , publication\quad she neverFundamenta of ceased the 20 to th control Informaticae volume the of preparation FI in 1\quad 994 of . the(FI), Moreover \quad was established in \quad 1 977 , mostlyconsecutive due, issues to making her of the once efforts j ournal more proof period , and of In her she particular inexhaustible was its comma Editor energy she acted ,− shein as was− Editor anChief active of a until special Collecting anniver her Editor death hyphen . Even when her isary llness issue commemoratingbeganwith Studia to take Logica the the publication( since better 1 974 of of the ) andher 20 th Associate , volume she never of Editor FI in ceased with 1 994 the period toJournal control .. Moreover of Approximate the comma preparation of the consecutivemaking onceReasoning more issues proofsince of of her the 1 986inexhaustible j . ournal energy . In comma particular she was , an she active acted Collecting as Editor Editor of a special anniver − sarywith Studiaissue Logica commemoratingProfessor open parenthesis Rasiowa the educated since publication 1 974generations closing of parenthesis of the students 20 and th and Associate volume researchers ofEditor ; FI in withparticular in 1the 994 Journal , she . \quad of Approx-Moreover , imatemaking oncesupervised more proof over 20 of Ph her . D inexhaustible . theses . Her lectures energy were , knownshe was all over an active the world Collecting . She Editor withReasoning Studia sincewas Logica visiting 1 986 period (professor since at 1 1 974 4 universities ) and Associate , ranging from Editor Bahia Blanca with in the Argentina Journal to Moscow of Approximate ReasoningProfessor Rasiowa sincein the Sovieteducated 1 986 Union .generations , passing of through students Campinas and researchers in Brasil semicolon , UNAM in [ Universidad particular comma Nacional she supervised overde Mexico 20 Ph period] in Mexico D period , Rome theses in Italy period and .. Oxford Her lectures University were knownin England all over . the In the world USA period she .. She was Professorvisiting professor Rasiowawas hosted at 1 educated 4 by universities universities generations comma such asranging Princeton of from students , Bahia University Blanca and of in Chicagoresearchers Argentina , University to Moscow ; in of particular , she supervisedin the Soviet over Union 20 comma Ph . passing D . theses through . Campinas\quad Her in Brasil lectures comma were UNAM known open square all over bracket the Universidad world . \quad She was Nacionalvisiting professor at 1 4 universities , ranging from Bahia Blanca in Argentina to Moscow de Mexico closing square bracket in Mexico comma Rome in Italy and Oxford University in England period .. In the USA\noindent in the Soviet Union , passing through Campinas in Brasil , UNAM [ Universidad Nacional she was hosted by universities such as Princeton comma University of Chicago comma University of \noindent de Mexico ] in Mexico , Rome in Italy and Oxford University in England . \quad In the USA she was hosted by universities such as Princeton , University of Chicago , University of 1 2 .. W period BARTOL ET AL period \noindentCalifornia1 at 2 Berkeley\quad commaW . BARTOL and the ETUniversity AL . of North Carolina at Charlotte period Moreover comma she gave invited lectures at 46 universities and research establishments abroad comma in some of \noindentthem moreCalifornia than once period at Berkeley , and the University of North Carolina at Charlotte . Moreover , she gaveShe contributed invited lectures to mathematical at 46 literature universities with three and important research books andestablishments more than abroad , in some of them1 0 publications more than1 2 periodW once . BARTOL .. . The ET topics AL . covered by her work include proof theory and deductive logic commaCalifornia algebraicat methods Berkeley in logic, and and the algebras University related of North to logics Carolina comma at Charlotte classical and . Moreover nonclassical , she Shelogics contributed commagave algorithmic invited to mathematical lectures and approximation at 46 universities literature logics and comma research with artificial three establishments intelligence important abroad period books , She in some worked and of more them on a thannew 1monograph 0 publications tomore her than very . once last\quad days . The period topics .. The intendedcovered title by was her Algebraic work include Analysis .. proof of Non theory hyphen and deductive logicClassical , algebraicFirst OrderShe contributed Logics methods period to in mathematical logic and literature algebras with related three important to logics books and , classical more than 1 and nonclassical logics2 period , .. algorithmic Algebraic0 publications logic andperiod. The approximation .. topics The study covered of the by logics relationship her work , include artificial between proof logic theory intelligence and algebra and deductive comma . She logic origi worked hyphen on a new monographnated .. by .. to, algebraic the her .. work very methods .. of last George in logicdays ..Boole and . \ algebrasquad .. andThe .. related continued intended to logics .. by , title.. classical A period was and .. AlgebraicLindenbaum nonclassical logics commaAnalysis , .. A\ periodquad o f Non − ..Classical Tarski comma Firstalgorithmic Order and approximation Logics . logics , artificial intelligence . She worked on a new monograph M period Hto period her very Stone last comma days . R period The intended McKinsey title comma was Algebraic A period Analysis Mostowski and of Non L period - Classical Henkin emdash to mention2 . \quad only aAlgebraicFirst few emdash Order Logicslogic. . \quad The study of the relationship between logic and algebra , origi − natedwas the\quad main researchby2 .\quad subject Algebraicthe of\ Helenaquad logicwork Rasiowa . The\quad period studyo One f of George the of therelationship crucial\quad moments betweenBoole in\ logicquad the and deand hyphen algebra\quad , continued \quad by \quad A. \quad Lindenbaum , \quad A. \quad Tarski , Mvelopment . H . Stone oforigi algebraic - , nated R . study McKinsey by of logic the was , work A the . introduction Mostowski of George of and Boole Lindenbaum L . and Henkin quoteright continued−−− sto and by mention Tarski A . quoteright Lin- only a s few −−− wasmethod the for main treatingdenbaum research equivalence , A .subject Tarski classes ,of of M formulas Helena . H . Stone as Rasiowa elements , R . McKinsey of . anOne abstract ,of A .the Mostowski algebraic crucial and moments L . Henkin in the de − velopmentsystem period of— This to algebraic mention provided only a study link a few between of — was logic theories the main was based researchthe on introduction classical subject logic of Helena and of Boolean RasiowaLindenbaum alge . One hyphen ’ of s the and Tarski ’ s methodbras period forcrucial An treating analogous moments equivalence correspondence in the de - velopment classesbetween intuitionistic of algebraic formulas logic study as and of elements pseudo logic was hyphen the of introduction ancomplemented abstract of algebraic systemlattices was. ThisLindenbaum established provided by ’ s Stone and aTarski link and Tarski between’ s method comma theoriesfor while treating the equivalencebased application on classesof classical algebraic of formulas methods logic as elements and Boolean alge − brasto modal . An logics analogousof an was abstract originated correspondence algebraic by McKinsey system and . between This Tarski provided open intuitionistic parenthesis a link between 1 944 logictheories endash 1and based 948 pseudoclosing on classical parenthesis− complemented period Anotherlattices important waslogic established and Boolean alge by - Stone bras . An and analogous Tarski correspondence , while the between application intuitionistic of logic algebraic and methods toidea modal leading logicspseudo to the emergence - was complemented originated of the lattices field by of wasalgebraic McKinsey established logic and was by Tarski theStone treatment and ( 1Tarski 944 of formulas ,−− while1 the 948 application ) . Another important ideaas algebraic leadingof functions algebraic to the in methods certain emergence algebras to modal of comma logics the field wasinitiated originated of by algebraic L-suppress by McKinsey ukasiewicz logic and Tarski was and Postthe ( 1 944 with treatment – their 1 948 of formulas asgeneralization algebraic) . of Anotherfunctions truth tables important inperiod certain idea leading algebras to the emergence , initiated of the by field\L of algebraicukasiewicz logicand was the Post with their generalizationHelena Rasiowatreatment became of of truth very formulas active tables as in algebraic these . areas functions in the in early certain fifties algebras period ,.. initiated Her research byLukasiewicz and work on algebraicPost with logic their was generalization aimed at finding of trutha precise tables description . for the mathematical Helenastructure Rasiowa of formalizedHelena became Rasiowa logical very systems became active period very in active .. these Her in early these areas papers areas in in contain the the early early several fifties examples fifties . Her researchof . \quad workHer research workalgebras on associated algebraicon algebraic with logic logic logical was was systems aimed aimed asat wellfinding at as finding algebraica precise a description proofs precise of some for description the of their mathematical prop hyphen for structure the mathematical of structureerties period offormalized .. Jointly formalized logical with Roman systems logical Sikorski . Hersystems she early presented papers . \quad contain the firstHer several algebraic early examples proof papers of algebrasthe contain G dieresis-o associated several del examples of algebrascompleteness associatedwith theorem logical for systems withclassical logicalas predicate well as algebraic systems logic open proofs parenthesisas well of some as open of algebraic their square prop bracket - erties proofs 4 closing . Jointly of square some with bracket of their closing prop − parenthesise r t i e s . period\quadRoman ....Jointly Next Sikorski comma she with presentedshe Roman algebraically the Sikorski first proved algebraic she proof presented of the Go ¨thedel first algebraic proof of the G $ \analogousddot{o} theorems$completeness d e l for intuitionistic theorem for and classical modal predicate logics open logic parenthesis ( [ 4 ] ) . open Next square , she algebraically bracket 5 closing proved square bracket closing parenthesisanalogous period theorems In the papers for intuitionistic open square and bracket modal 9 comma logics ( 1 [ 35 comma] ) . In the 1 4 paperscomma [ 1 9 6 , comma1 3 , 1 4 , \noindent1 8 commacompleteness1 2 6 1 , comma 22 comma theorem 23 commafor classical 24 comma predicate 25 comma 26 logic closing ( square [ 4 bracket] ) . \ sheh f turned i l l Next to other , she algebraically proved nonclassical logics1 8 comma , 2 1 , 22 applying , 23 , 24 with , 25 success , 26 ] she turned to other nonclassical logics , applying with success \noindentthe algebraicanalogousthe tools algebraic she had theorems tools developed she had for period developed intuitionistic .. All .these All investigations these and investigations modal led her logics to led a synthesisher ( to [ a 5 synthesis of ] ) . In of the papers [ 9 , 1 3 , 1 4 , 1 6 , the results hithertothe results obtained hitherto semicolon obtained she ; shedeveloped developed a general a general framework framework for an for algebraic an algebraic presen presen hyphen - \noindenttation of propositional1tation 8 , 2 of1 propositional and , 22 first , order 23 and ,logics first 24 period order, 25 logics In , 26particular . ] In she particular comma turned she, she to established established other elegant nonclassical elegant formal formal logics , applying with success techniques fortechniques associating for classes associating of algebras classes with of algebras logical systems with logical and for systems providing and their for providing their \noindentalgebraic semanticsthealgebraic algebraic period semantics .. In tools the. following In she the had following comma developed ,we we shall shall briefly . briefly\quad outline outlineAll the thesethe basic basic elements investigations elements of herof her led her to a synthesis of themethod results for anmethod hitherto algebraization for an obtainedalgebraization of logics period ; of she logics developed . a general framework for an algebraic presen − tationRasiowa of quoteright propositionalRasiowa s first ’ s monograph first and monograph first The MathematicsThe order Mathematics logics of Metamathematics of . Metamathematics In particular open[ square 3 , 1 she ] , written bracket established jointly 3 1 closing elegant square formal brackettechniques commawith written for R associating . jointly Sikorski , contains classes a comprehensive of algebras survey with of algebraic logical theories systems of logical andcalculi for providing . their algebraicwith R period semanticsAlgebraization Sikorski comma . is\quad applied containsIn to a theclassical comprehensive following , intuitionistic survey , we of , modal shall algebraic and briefly theories positive of outline logics logical . Thecalculi the book period basic elements of her methodAlgebraization forfo an cuses is algebraizationapplied on metalogical to classical theorems comma of logics intuitionistic on the . predicate comma calculi modal of and these positive logics logics and on period an inves The - book fo cuses on metalogicaltigation of elementarytheorems on theories the predicate based calculi on them of these. The logics main and idea on underlying an inves hyphen the concept of Rasiowatigation ’ of s elementaryassociating first monograph theories algebraic based systems The onMathematics them with period logical The calculi of main Metamathematicsis the idea following underlying : the Let [ conceptL 3be 1 a ] first of , writtenorder jointly withassociating R . Sikorski algebraic systems , contains with logical a comprehensive calculi is the following survey : .. Let of L algebraic be a first order theories of logical calculi . Algebraization is applied to classical , intuitionistic , modal and positive logics . The book fo cuses on metalogical theorems on the predicate calculi of these logics and on an inves − tigation of elementary theories based on them . The main idea underlying the concept of associating algebraic systems with logical calculi is the following : \quad Let $ L $ be a first order HELENA RASIOWA comma 1 9 1 7 endash 1 9 94 .. 1 3 \ hspacelanguage∗{\ suchf i l l that}HELENA V is its RASIOWA set of individual , 1 9 1 variables 7 −− 1 comma 9 94 \ orquad comma1 3 and comma right arrow are the binary propositional \noindentconnectiveslanguage of L comma such logicalnot that is a $ unary V $ propositional is its set connective of individual comma and variables exists and forall $ are , the\vee quantifiers, \wedgeof L period, Let\rightarrow open parenthesis$ A are comma the cup binary comma propositional cap comma right arrow comma minus comma union of sub comma intersection of closing parenthesis be a complete algebra suchHELENA that cup RASIOWA comma , 1cap 9 1 comma 7 – 1 9 94 and1 right 3 arrow are binary\noindent connectiveslanguage such that ofV $is L its set , of individual\neg $ variables is a unary, ∨, ∧, → propositionalare the binary propositional connective , and $ \ exists $ andoperations $ \ f o r commaconnectives a l l $ minus are of isL, the a¬ unaryis quantifiers a unary operation propositional and union connective of and intersection , and ∃ and of∀ areare infinitary the quantifiers operations of L. the common of $L .$ Let $(S AT , \cup , \cap , \rightarrow , − , \bigcup { , } domain of whichLet (A, is the∪, ∩ class, →, − of, all, nonempty) be a complete subsets algebra of A period such thatBy a∪ realization, ∩, and → ofare the binary language S T \bigcapL in a nonempty)operations $ beset J a and, − completeis in athe unary algebra operationalgebra A we and such meanand any that mappingare $ infinitary\cup R which operations, assigns\cap a the common, $ and domain $ \rightarrow $ arefunction binary f Rof : J which to the is power the class of V of minus-arrowright all nonempty subsets J to each of A. functionBy a realization symbol f of theL and language a functionL in P a sub R : J to the power of Vnonempty arrowright-minus set J and A to in the algebra A we mean any mapping R which assigns a \noindent operations $V , − $ is a unary operation and $ \bigcupV $ and $ \bigcap $ any predicatefunction symbolfR P of: J L period−→ J to The each realization function R symbol is thenf extendedof L and to a function all the termsPR : andJ −→ all A to arethe infinitary formulasany of L predicate operations period Let symbol v in the JP toof common theL. The power realization of V be aR valuationis then extended of individual to all variables the terms in anda set all J period Then we V domaindefine : of whichthe formulas is the of L. classLet v ∈ ofJ allbe a nonempty valuation of subsets individual ofvariables $ A in a . set $J. ByThen a we realization of the language $x subL $ R open in a parenthesis nonempty v closing set $ parenthesis J $ and = v in open the parenthesis algebra x closing $A$ parenthesis we mean for any any mapping individual variable $R$ xwhich comma assigns a Line 1 f open parenthesis t sub 1 comma period perioddefine period : comma t sub m closing parenthesis R open parenthesis v\noindent closing parenthesisfunction = f R open $f parenthesis R : t sub Jˆ 1{ RV open}\ parenthesislongrightarrow v closing parenthesisJ $ tocomma each period function period period symbol x (v) = v(x) for any individual variable x, comma$ f $ t subof mR $L$ open parenthesis anda function v closingR parenthesis $P { R closing} : parenthesis J ˆ{ V comma}\longrightarrow Line 2 P open parenthesisA $ t to sub 1 comma period period period comma t sub n closing parenthesis R open parenthesis v closing parenthesis = P sub R f(t1, ..., tm)R(v) = fR(t1R(v), ..., tmR(v)), open\noindent parenthesisany t subpredicate 1 R open symbol parenthesis $ v P closing $ of parenthesis $ L . comma $ The period realization period period $ comma R $ t issub then nR open extended to all the terms and all P (t , ..., t )R(v) = P (t (v), ..., t (v)), parenthesis v closing parenthesis closing parenthesis comma1 Linen 3 openR parenthesis1R FnR or G closing parenthesis R open parenthesis\noindent v closingthe formulas parenthesis of = F $Lsub R open . $ parenthesis Let $v v( closingF ∨\Gin) parenthesisR(v)J = F ˆ{R( cupVv) ∪} G$G subR( bev) R, opena valuation parenthesis of v closing individual variables in a set $ J . $ Then we parenthesis comma Line 4 open parenthesis F and G closing parenthesis(F ∧ G)R(v R) = openFR(v parenthesis) ∩ GR(v), v closing parenthesis = F sub R open parenthesis v closing parenthesis cap G sub R open parenthesis v closing parenthesis comma Line 5 open (F → G)R(v) = FR(v) → GR(v), parenthesis\ begin { a l F i g right n ∗} arrow G closing parenthesis R open parenthesis v closing parenthesis = F sub R open parenthesis v (¬F )R(v) = −F (v), closingde f i parenthesis ne : right arrow G sub R open parenthesis v closing parenthesis comma LineR 6 open parenthesis logicalnot \end{ a l i g n ∗} [ V F closing parenthesis R open parenthesis(∃xF )R(v) =v closing{FR( parenthesisvj): j ∈ J =}, minuswherev Fj ∈ subJ Rand openvj(x parenthesis) = j, v closing parenthesis comma Line 7 open parenthesis exists xF closing parenthesis R open parenthesis v closing parenthesis = union of open v (y) = v(y)forally 6= x, brace\ centerline F sub R open{ $ parenthesisx { R } v sub( j v closing ) parenthesis = v : j ( in J xclosingj )$ brace foranyindividualvariable comma where v sub j in J to the power $x , $ } T V of V and v sub j open parenthesis(∀xF x)R closing(v) = parenthesis{FR(vj): =j j∈ commaJ}, where Linev 8j v∈ subJ jis open as above parenthesis . y closing parenthesis = v open parenthesisThe y algebraic closing parenthesis semantics for of the all yclassical negationslash-equal predicate logic x commais determined by the class of real- \ [ \ begin { a l i g n e d } f ( t { 1 } , . . . , t { m } ) R ( v ) = open parenthesisizations forall of thexF closing language parenthesisL in Boolean R open algebras parenthesis . We v say closing that parenthesis a formula is = true intersection in a of open brace F f R ( t { 1 R } ( v ) , . . . , t { mR } ( v ) ) , \\ sub R open parenthesisrealization v subR whenever j closingF parenthesis(v) = 1 for : j every in J closingvaluation bracev, where comma 1 is where the unit v sub element j in J ofto the power of V is P ( t { 1 } , . .R . , t { n } ) R ( v ) = P { R } ( as above periodthe underlying Boolean algebra . A formula F is valid in the classical predicate logic if t { 1 R } ( v ) , . . . , t { nR } ( v ) ) , \\ The algebraicand semantics only if F ofis the true classical in all predicate realizations logic in is any determined Boolean algebra by the class . Similarly of , the semantics (F \vee G ) R ( v ) = F { R } ( v ) \cup G { R } ( realizationsof intuitionisticthe language logicL in Boolean is given byalgebras realizations period in .. Heyting We say that algebras a formula , positive is true first in order a logic is v ) , \\ realization Rassociated whenever with F sub realizations R open parenthesis in the class v closing of relatively parenthesis pseduocomplemented = 1 for every valuation lattices . v Modal comma where 1 is (F \wedge G ) R ( v ) = F { R } ( v ) \cap G { R } ( the unit elementlogic of , also considered in the book , is semantically determined by the class of realizations v ) , \\ the underlyingin what Boolean is known algebra as period topological .. A formula Boolean F algebras is validin , i the . e classical . Boolean predicate algebras logic endowed if with a (F \rightarrow G ) R ( v ) = F { R } ( v ) \rightarrow and only if Funary is true operation in all realizations behaving like in any an Booleaninterior operator algebra period . Similarly comma the semantics of G { R } ( v ) , \\ intuitionistic logicIn the is given next bymonograph realizations , An in Algebraic Heyting algebrasApproach comma to Non positive - classical first Logics order( logic [ 49 is ] ) Helena ( \neg F ) R ( v ) = − F { R } ( v ) , \\ associated withRasiowa realizations develops in a the general class ofalgebraization relatively pseduocomplemented theory for propositional lattices logical period systems Modal . Within ( \ exists xF ) R ( v ) = \bigcup \{ F { R } ( v { j } ) logic commathe also framework considered of in this the theory book comma she presents is semantically algebraizations determined of a number by the class of nonclassical of realizations logics : j \ in J \} , where v { j }\ in J ˆ{ V } and v { j } ( x ) in what is knownsuch as as : topological classical Boolean and positive algebras implicative comma ilogics period , logics e period weaker Boolean than algebras positive endowed implicative with a = j , \\ unary operationlogic behaving , minimal like logic an interior , positive operator logic period with semi - negation , constructive logic with strong v { j } ( y ) =v ( y ) for all y \not= x , \end{ a l i g n e d }\ ] In the next monographnegation , and comma many An - valued Algebraic Post Approach logics . The to Non fundamental hyphen classical idea behind Logics algebraization open parenthesis is the open square bracket 49 closingfollowing square : bracket Let S = closing (L, C) parenthesis be a propositional Helena logic , where L is a propositional language that Rasiowa developsincludes a general implication algebraization as a binary theory connective for propositional and C is a consequencelogical systems operation period Within determined \ centerline { $ ( \ f o r a l l xF ) R ( v ) = \bigcap \{ F { R } ( v { j } the frameworkby theof this axioms theory and she inference presents rules algebraizations . As usual , of for a a number set X of of formulas nonclassical we denote logics by C(X) the ) : j \ in J \} , $ where $ v { j }\ in J ˆ{ V }$ is as above . } such as : .. classicalleast set and of formulas positive which implicative contains logics the comma axioms logics and the weaker set X thanand positive is closed implicative with respect logic comma .. minimal logic comma .. positive logic with semi hyphen negation comma .. constructive logic with strongThe algebraic semantics of the classical predicate logic is determined by the class of realizationsnegation comma of and the many language hyphen valued $ L Post $ logics in Boolean period The algebras fundamental . \ ideaquad behindWe say algebraization that a formulais the is true in a following : Let S = open parenthesis L comma C closing parenthesis be a propositional logic comma where L is a propositional\noindent languagerealization that $ R $ whenever $ F { R } ( v ) = 1$ for every valuation $ vincludes , $ implication where 1 as is a binary the unitconnective element and C of is a consequence operation determined by the axioms and inference rules period As usual comma for a set X of formulas we denote by C open parenthesis X closing\noindent parenthesisthe theunderlying Boolean algebra . \quad A formula $ F $ is valid in the classical predicate logic if andleast only set of if formulas $ F which $ is contains true the in axioms all realizations and the set X and in is any closed Boolean with respect algebra . Similarly , the semantics of intuitionistic logic is given by realizations in Heyting algebras , positive first order logic is associated with realizations in the class of relatively pseduocomplemented lattices . Modal logic , also considered in the book , is semantically determined by the class of realizations in what is known as topological Boolean algebras , i . e . Boolean algebras endowed with a unary operation behaving like an interior operator .

In the next monograph , An Algebraic Approach to Non − classical Logics ( [ 49 ] ) Helena Rasiowa develops a general algebraization theory for propositional logical systems . Within the framework of this theory she presents algebraizations of a number of nonclassical logics such as : \quad classical and positive implicative logics , logics weaker than positive implicative l o g i c , \quad minimal logic , \quad positive logic with semi − negation , \quad constructive logic with strong negation , and many − valued Post logics . The fundamental idea behind algebraization is the following : Let $S = ( L , C )$ bea propositional logic , where $L$ is a propositional language that includes implication as a binary connective and $ C $ is a consequence operation determined

\noindent by the axioms and inference rules . As usual , for a set $ X $ of formulas we denote by $ C ( X ) $ the least set of formulas which contains the axioms and the set $ X $ and is closed with respect 14 .. W period BARTOL ET AL period \noindentto the inference14 \quad rules periodW . BARTOL In particular ET AL comma . C open parenthesis varnothing closing parenthesis is the set of all the theorems of S period An algebra \noindentassociatedto with the S is inferenceany algebra of rules a type .similar In particular to that of the algebra $ , of C formulas ( of\ varnothing S ) $ is the set of all the theorems of $Sand endowed . $ Analgebra with a distinguished element 1 period Elements of such an algebra are interpreted associatedas truth values14 with commaW . $BARTOL whereas S $ ET is 1 AL represents any . algebra the value of true a type period similar .. Let V be to the that set of of propositional the algebra of formulas of $ Svariables $ ofto L period the inference By a valuation rules . In in particularan algebra,C A( associated∅) is the setwith of S all we the mean theorems any mapping of S. An algebra andv in endowed A to theassociated power with of a Vwith distinguished periodS is Every any algebra formula element of F a typeof the 1 similar language . Elements to that L induces of ofthe a algebrasuch mapping an of formulasF algebra sub A : of A areS toand the interpreted power of V minus-arrowrightas truth valuesendowed A defined , with whereas a distinguished 1 represents element 1 the. Elements value of truesuch an . algebra\quad areLet interpreted $ V $ as truthbe the set of propositional variablesas follows : ofvalues $ , L whereas . $ 1 represents By a valuation the value true in an. algebra Let V be the $ setA $ of propositional associated variables with $ S $ we mean any mapping p A open parenthesisof L. By a v valuation closing parenthesis in an algebra = vA openassociated parenthesis with S pwe closing mean parenthesis any mapping for any propositional variable \noindent $ v V \ in A ˆ{ V } . $ Every formula $ F $ of theV language $ L $ induces a mapping p comma v ∈ A . Every formula F of the language L induces a mapping FA : A −→ A defined as $ Fopen{ parenthesisA } follows: hash A : ˆ{ F closingV }\ parenthesislongrightarrow A open parenthesisA $ d v e closing f i n e d parenthesis = hash open parenthesis F sub Aas open f o parenthesis l l o w s : v closing parenthesispA closing(v) = v parenthesis(p) for any propositionalfor every unary variable propositionalp, connective hash of L comma open parenthesis F hash(#F G) closingA(v) =parenthesis #(FA(v)) for A every open parenthesisunary propositional v closing connective parenthesis # = of FL, sub A open parenthesis \ centerline { $p A ( v ) = v ( p )$ foranypropositionalvariable v closing parenthesis hash G( subF # AG) openA(v) parenthesis = FA(v)#G vA( closingv) for every parenthesis binary for connective every binary # of connectiveL. hash of L period $ p , $ } V A formula FA is formula said to beF is true said in to A be if and true only in A ifif F and sub only A open if FA parenthesis(v) = 1 for v every closingv ∈ parenthesisA . = 1 for every v in A to the power of V periodAny propositional logic S determines thus a class of S− algebras , where an S− algebra is \ centerline { $ ( \# F ) A ( v ) = \#(F { A } ( v ) ) $ Any propositionalan algebra logicA Sassociated determines with thusS asuch class that of S for hyphen every algebras formula commaF ∈ C( where∅) wean have S hyphenFA(v) = algebra 1 is V foran every algebra unary Afor associated every propositionalv ∈ withA . SThe such completeness that connective for every theorem formula $ \ states# F $ in Cthat o open f if $S parenthesis Lis consistent , $ varnothing} , then the closing set parenthesis we have F sub A openC(∅ parenthesis) is equal to v the closing set of parenthesis all formulas = which 1 are true in each S− algebra . \ centerlinefor every v in{ A$Helena to ( the F Rasiowapower\# of paid V periodG very ) The much A completeness attention ( v to theorem many ) -= valuedstates F that Post{ A if logics S} is consistent( . She v introduced comma ) \#G then the set{ A } (C v open )parenthesis $several for generalizations varnothing every binary closing of the connective parenthesis standard ism− equal $valued\# to $ the Post o set f logics of $ all L ; formulas she . considered $ which} the are proposi true in each- S hyphen + algebra period tional and predicate versions of ω − valued Post logics and used them in her reasoning on \noindentHelena RasiowaAformulaprograms paid very . The $F$ much following attention is section said to many describes to be hyphen true some valued ofin these $A$Post applications logics if period and in more She only introduced detail if . $F { A } ( vseveral ) =generalizations 1$3 . forevery Mathematical of the standard $v mfoundations hyphen\ in valuedA of ˆ{ computer PostV } logics. science $ semicolon . sheHelena considered Rasiowa the greatlyproposi hyphen tional and predicatecontributed versions to the of development omega to the of power research of plusin Poland hyphen on valued applications Post logics of logical and methods used them in in the her reasoning on\ hspace ∗{\ ffoundations i l l }Any propositional of computer science logic . She $ was S $ one determines of the first to thusrealize a the class great importance of $ S − $ algebras , where an $ Sprograms− $ period algebraof mathematical The following i s logic section for describescomputer some science of these — and applications at the same in more time detail she clearly period saw the 3 period .. Mathematicalsignificance of foundations computer science of computer for the science development period of .. logicHelena itself Rasiowa . In greatly the last 20 years of \noindentcontributedanher to algebra the scientific development activity $ A of$ she research foassociated cused in her Poland efforts with on on applications the $ realization S $ such of logical of this that mainmethods for idea every through formula papers $ F \ inin theC( foundations, seminars\ varnothing of computer and research science ) projects $ period we . have .. Many She was $of herF one students{ ofA the} first and( to collaborators realize v ) the great= who attended 1 $ her importance oflectures mathematical or seminars logic , those for computer who wrote science their emdash Ph . D . and theses at the under same her time supervision she , are now \noindentclearly sawf othecontinuing r significance every the $ work of v computer she\ in initiated scienceA ˆ .{ forV There} the development. remains $ The no doubt completeness of logic today itself that period she theorem was .. In right states in her that if $ Sthe $ last is 20 consistent yearsappreciation of her scientific of , the then significance activity the set she of fo the cused field her . Among efforts the on authors the realization of important of this research results main idea throughon logical papers and algebraiccomma seminars methods and in computer research projects science , period the names .. Many of her of students her students can be and found \noindentcollaborators$quite who C often attended ( .\ varnothing Moreover her lectures , some or seminars of ) these $ is comma important equal those results to who the have wrote set appeared their of Ph all in period aformulas j ournal D period which which theses are true in each $ Sunder− her$ supervision algebrawould not comma have . existed .. are without now continuing her dedication the work and she to initiated which she period had been .. There Editor remains - in - Chiefno doubt todayfor that many she yearswas right , i .in e . herFundamenta appreciation Informaticae of the significance. of the field period Among Helenathe authors Rasiowa of importantThe paid form very of research this much short results article attention on makes logical difficult to and many algebraic a complete− valued methods presentation Post in computer logics of her achieve . She - ments introduced severalscience comma generalizationsin the the field names of applications of her of students the of standard logical can be and found algebraic $ quite m often− methods$ period valued in ..computer Moreover Post science logics comma . some ; she of these considered the proposi − important resultsShe is have the author appeared of more in athan j ournal 30 papers which would, two lecture not have notes existed ( [ 63 without ] , [ 72 ] her ) and an unfinished \noindentdedicationtional andmonograph to which and in predicate she which had she been relates versions Editor algebraic hyphen of methodsin $ \ hyphenomega of nonclassical Chiefˆ{ + for} manylogics − $ withyears valued applications comma Post i period in logics e period and used them in her reasoning on Fundamentaprograms .the The foundations following of computer section science describes . She some was able of to these write eight applications chapters before in she more was detail . Informaticaetaken period to hospital . 3 .The\quad form ofMathematical this short article foundations makes difficult of a complete computer presentation science of .her\quad achieveHelena hyphen Rasiowa greatly contributedments in the field to of the applications development of logical of and research algebraic inmethods Poland in computer on applications science period of logical methods inShe the is the foundations author of more of than computer 30 papers science comma two . \ lecturequad notesShe was open one parenthesis of the open first square to bracket realize 63 closing the great squareimportance bracket comma of mathematical open square bracket logic 72 closingfor computer square bracket science closing−−− parenthesisand at and the an same unfinished time she clearlymonograph saw in whichthe significance she relates algebraic of computermethods of nonclassical science for logics the with development applications of logic itself . \quad In thein the last foundations 20 years of computer of her science scientific period .. activity She was able she to write fo cused eight chapters her efforts before she on the realization of this mainwas taken idea to through hospital period papers , seminars and research projects . \quad Many of her students and collaborators who attended her lectures or seminars , those who wrote their Ph . D . theses under her supervision , \quad are now continuing the work she initiated . \quad There remains no doubt today that she was right in her appreciation of the significance of the field . Among the authors of important research results on logical and algebraic methods in computer science , the names of her students can be found quite often . \quad Moreover , some of these important results have appeared in a j ournal which would not have existed without her dedication and to which she had been Editor − in − Chief for many years , i . e . Fundamenta Informaticae .

The form of this short article makes difficult a complete presentation of her achieve − ments in the field of applications of logical and algebraic methods in computer science .

\noindent She is the author of more than 30 papers , two lecture notes ( [ 63 ] , [ 72 ] ) and an unfinished monograph in which she relates algebraic methods of nonclassical logics with applications in the foundations of computer science . \quad She was able to write eight chapters before she was taken to hospital . HELENA RASIOWA comma 1 9 1 7 endash 1 9 94 .. 1 5 \ hspaceHer contribution∗{\ f i l l }HELENA to theoretical RASIOWA computer , 1 science 9 1 7 stems−− 1 from 9 her94 conviction\quad 1 that 5 there are deep relations between methods of algebra and logic on one side and essential prob hyphen Herlems contribution of foundations toof computer theoretical science oncomputer the other science period Among stems these from problems her she conviction clearly that there aredistinguished deep relations inference methods between characteristic methods of of computer algebra science and logic and its applicationson one side period and essential prob − lemsThis ofconviction foundations of hers had of been computer supported science by her results on the on many otherHELENA hyphen . Among RASIOWA valued these and , 1 9 nonclas 1 7 problems – 1 9 94hyphen1 she 5 clearly distinguishedsical logics commaHer inference especially contribution on methods applications to theoretical characteristic of variouscomputer generalizations science of stemscomputer of from Posther algebrasscience conviction to and logics that its there applications . Thisof programs convictionare and deep approximation of relations hers had between logics been period methods supported of algebra by andher logic results on one on side many and− essentialvalued prob and - nonclas − sicalHer investigations logicslems , ofon especially foundations logic and algebraic of on computer applications methods science in computer on of the various other science . Among cangeneralizations be these di hyphen problems of she Post clearly algebras to logics ofvided programs into twodistinguished and main approximation streams inference period methods The logics first characteristic includes . many of hyphencomputer valued science algorithmic and its applications logics and .their This applicationsconviction to investigation of hers of had programs been supported open parenthesis by her results open on square many -bracket valued 42and endash nonclas 44 - sical closing logics square bracket commaHer investigations open square, especially bracket on on 5 1logic applications closing and square of algebraic various bracket generalizations comma methods .. open in of square Post computer bracket algebras 53 science to endash logics 64 ofcanclosing programs be disquare− bracket commavided .. into openand square two approximation main bracket streams 69 endash logics . . 72 The closing first square includes bracket closing many parenthesis− valued comma algorithmic while the logicssec hyphen and their ond is concernedHer with investigations approximation on logic logics and in their algebraic relation methods with generalizations in computer science of Post can be di - vided \noindentalgebras openapplicationsinto parenthesis two main open streams to square investigation . The bracket first includes 73 endash of many programs 97 closing - valued ( square algorithmic [ 42 bracket−− 44 logics closing ] and , parenthesis [ their 5 1 ] , period\quad [ 53 −− 64 ] , \quad [ 69 −− 72 ] ) , while the sec − ondIn a is series concerned ofapplications papers on with algorithmic to investigation approximation logic comma of programs logics Rasiowa ( [ in presents 42 their – 44 the ] , relation [ results 5 1 ] of, her with [ 53 research – generalizations 64 ] , [ 69 – of Post algebrason generalizations (72 [ ] 73 ) of ,−− while classical97 the ] algorithmic )sec . - ond is logic concerned introduced with by approximation A period Salwicki logics to in complex their relation with algorithms commageneralizations which include of Post programs algebras with ( [ 73 stacks – 97 ] or ) .coroutines comma programs with recursive Incoroutines a series comma ofIn papers a as series well asof on papers procedures algorithmic on algorithmic and recursive logic logic procedures , , Rasiowa Rasiowa period presents presents .. 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In [ 74 ] . first During this pe − rioCat d hyphen , Rasiowaorder Ho open approximation worked parenthesis with Vietnamlogics several are closing constructed mathematicians parenthesis . They comma are based , and such onAndrzej approximation as George Skowron Epstein open operators parenthesis ( in University Warsaw University closingthe parenthesis sense of Z . period Pawlak , which are applied to sets and relations defined by first order formulas \noindentThe papersof. open In Charlotte [ square 75 ] these bracket , logics North 73 were endash Carolina extended 97 closing to ) the ,square Wiktor case bracket of a Marek chain correspond of ( equivalence State to the University period relations between which of Kentucky1 984 and 1 ) , Nguyen 994 period In opendetermine square lower bracket and 74 upper closing approximations square bracket . first order \noindentapproximationCat logics−TheHo paper are ( constructedVietnam [ 84 ] introduces ) period , and aThey generalization Andrzej are based Skowron on of Post approximation algebras ( Warsaw , calledoperators University semi in - the Post ) lattices . sense of Z period( algebras Pawlak ) . comma The primitive which are Post applied constants to sets are and elements relations of defined a poset by rather first than order of formulas a chain period, TheIn papers open squarecomplete [ 73 bracket−− lattice97 75 ] closing or correspond connected square semilattice bracket to the these as period it logics had been werebetween usual extended in 1 generalized 984 to the and case 1 Post 994 of algebras a chain . In of [ equivalence 74 ] first order relationsapproximation which logics are constructed . They are based on approximation operators in the sensedetermine of Z lower . Pawlak and upper , approximations which are applied period to sets and relations defined by first order formulas . InThe [ paper75 ] open these square logics bracket were 84 closing extended square to bracket the introducescase of a a generalization chain of equivalence of Post algebras relations comma called which semidetermine hyphen Post lower lattices and upper approximations . open parenthesis algebras closing parenthesis period .. The primitive Post constants are elements of a poset rather than\ hspace of a∗{\ chainf i comma l l }The paper [ 84 ] introduces a generalization of Post algebras , called semi − Post lattices complete lattice or connected semilattice as it had been usual in generalized Post algebras \noindent ( algebras ) . \quad The primitive Post constants are elements of a poset rather than of a chain , complete lattice or connected semilattice as it had been usual in generalized Post algebras 1 6 .. W period BARTOL ET AL period \noindentopen parenthesis1 6 \quad e periodW g . period BARTOL in papers ET AL by . Dwinger comma Traczyk comma Speed comma Rasiowa comma Cat hyphen Ho closing parenthesis period Several characterizations \noindentof sublattices( e of . areduct g . in ofa papers semi hyphen by Dwinger Post algebra , Traczyk which are semi, Speed hyphen , Post Rasiowa subalgebras , Cat are− givenHo ) comma . Several characterizations ofas sublattices well as different of characterizations a reduct of of a semi semi hyphen− Post Post homomorphisms algebra which period are These semi make− Post use of subalgebras a are given , astheorem well stating as1 different 6 thatW . every BARTOL characterizations semi ET hyphen AL . Post algebra of is semi a semi− hyphenPost Post homomorphisms product of generalized . These Post make use of a theoremalgebras in stating the( e .sense g . that in of papers Cat every hyphen by Dwinger semi Ho period− , TraczykPost algebra , Speed , isRasiowa a semi , Cat− -Post Ho ) . product Several characteri- of generalized Post algebrasIn open square inzations the bracket of sense sublattices 82 closing of Cat of square a− reductHo bracket . of a semiRasiowa - Post presents algebra an which algebraic are semi approach - Post to subalgebras approximate reasoning comma based onare given , as well as different characterizations of semi - Post homomorphisms . These make Inmodified [ 82 ] information Rasiowause of a theorem presents systems stating of Danaan that algebraic Scott every period semi approach - .. Post Semi algebra hyphen to is approximate a Post semi algebras- Post product reasoning built over of generalized Scott , based quoteright on s in hyphenmodified informationPost algebras in systems the sense of of Cat Dana - Ho Scott . . \quad Semi − Post algebras built over Scott ’ s in − formationformation systems systemsIn [ are 82 usedare ] Rasiowa as used tools presents as for tools an analysis an algebraic for of an the approach analysis properties to of of approximate approximate the properties reasoning , of based approximate on reasonings periodmodified information systems of Dana Scott . Semi - Post algebras built over Scott ’ s in - \noindentApproximatereasoningsformation reasonings systems are . also are treated used as in tools .. open for an square analysis bracket of the 76 properties closing square of approximate bracket comma .. open square bracket 77 closingreasonings square bracket . comma .. where they are based on a Approximatedecreasing sequence reasoningsApproximate of equivalence are reasonings also relations are treated also comma treated in which\ inquad define [ 76[ a ] 76 ,sequence [] 77 , ]\ ,ofquad closure where[ theyoperators 77 ] are , based\ periodquad on where they are based on a decreasingThe main resulta sequence decreasing consists sequencein of a characterizationequivalence of equivalence relationsof those relations sets , X ,which which which define are define intersections a sequence a sequence of of closure operators of closure operators . Thethe main family result of.all The closures main consists result period consists in a in characterization a characterization of ofthose those sets X setswhich are $ X intersections $ which of are intersections of In open squarethe bracket family of 83 all closing closures square . bracket methods of approximate reasoning related to a selection strategy are studied\noindent periodtheIn family [ 83 ] methods of all of closures approximate . reasoning related to a selection strategy are studied . Here again commaHere again semi , hyphen semi - Post Post algebras algebras are are used used as a tool for investigations period. The .. properties The properties of of Inapproximate [ 83 ] methodsapproximate deductions of aredeductions approximate expressed are in expressed a reasoning first order in logica related first introduced order to logic ain introduced the selection paper period in the strategy .. paper Its . are Its studied . Heresemantics again issemantics defined , semi via− is semi definedPost hyphen algebrasvia semi Post - algebras Post are algebras used and a asand representation a a representation tool for theorem investigations theorem period . . \quad The properties of approximateThe paper open deductions squareThe bracket paper are [86 86 closing expressed ] deals square with bracket in approximation a first .. deals orderwith logics approximation of logic different introduced types logicsT, ofwhere different inT theis types a paper T comma . \quad I t s wheresemantics T is a well is defined - founded via poset semi . Such− Post logics algebras were introduced and a by representation Rasiowa in [ 83 ] andtheorem they stem . well hyphenfrom founded theposet idea that period a set .. Such of objects logics to were be recognizedintroduced in by a Rasiowa processin ofopen approximate square bracket reasoning 83 closingis square bracket\ hspace and∗{\ theyfapproximated i l stem l }The from paper by a family[ 86 of] covering\quad deals sets and with by their approximation intersection , and logics it relies on of the different notion types $Tthe idea ,$ that whereof a rough set of sets $T$ objects in the to isa sense be recognized of Z . Pawlak in a .process The approach of approximate is axiomatic reasoning ; a completeness is theorem approximatedis provedby a family algebraically of covering , using sets plain and by semi their - Post intersection algebras comma . Such and algebras it relies had on first the appeared \noindentnotion of roughwin e l literaturel sets− founded in the in sense [ 89 poset ] of . Z Their period . \ importancequad PawlakSuch period within logics The the approach class were of introduced all is axiomatic semi - Post semicolon by algebras Rasiowa a is completeness due in [ 83 ] and they stem from thetheorem idea is that provedto their a algebraically simplicityset of objects and comma strong using to analogies be plain recognized semi with hyphen Post algebras in Post a algebras ,process but also period due of to ..approximate theirSuch importance algebras reasoninghad first is approximatedappeared inin literature the by investigation a infamily open square of approximation covering bracket 89 sets closing logics and . square by The bracket their main result period intersection of Their the paper importance , refers and withinto it the relies the class on of the allnotion semi hyphen of roughrepresentability Post algebras sets in of these the sensealgebras of . Every Z . element Pawlak is . uniquely The approach represented is in aaxiomatic normal form ; . a completeness theoremis due to istheir proved simplicityIn [ 88algebraically ] an and epistemic strong analogies logic , is using designed with Post plain which algebras semi formalizes comma− Post approximating but algebrasalso due to reasonings their . \quad perSuch - algebras had first appearedimportance in informed theliterature investigation by groups in of of agents [ approximation 89 who] . perceive Their logics importance reality period via .. The perception within main result operators the ofclass the , paper which of are all their semi − Post algebras isrefers due to to the theirindividual representability simplicity attributes of these , and and algebras knowledge strong period operators analogies Everyelement , which with are is Postuniquely attributes algebras represented of subsets , in but of agents also due to their importancea normal formwhen in period the arriving investigation at a consensus of . It approximation is assumed that the logics set of agents . \quad is aThe poset mainT = result(T, ≤) of the paper refersIn open to square theordered bracket representability with 88 respect closing tothe square sharpness of bracket these of an algebrasperception epistemic and . logic Every the is ability designed element to distinguish which is formalizesuniquely ob - j ects approximating represented in reasoningsa normal per form hyphen. An axiomatization. of the logic introduced is given , and a completeness theorem is proved , formed by groupstogether of agents with several who perceive other metalogical reality via theorems perception . operators This logic comma is free which of the are para their - Inindividual [ 88 ] an attributesdoxes epistemic which comma appear logic and in knowledge other is designed epistemic operators logics which comma . The formalizes which research are initiated attributes approximating in of [ 88 subsets ] was of continued reasonings agents per − formedwhen arriving by groupsin [at 92 a ] consensus and of [ agents 93 ] period . who It is perceive assumed that reality the set via of agents perception is a poset operatorsT = open parenthesis , which T are comma their lessindividual or equal closing attributes parenthesisThe problem , and knowledge of axiomatizing operators fuzzy sets is , one which of the are most attributes interesting and of topical subsets of agents ordered withproblems respect to in the the sharpness theory of theseof perception sets . and In the [ 94 ability ] Rasiowa to distinguish introduces ob the hyphen notion of an LT − \noindentj ects periodwhenfuzzy An axiomatization arriving set , which is at a of modificationa the consensus logic introduced of L .− Itfuzzy is is given sets assumed in comma the sense and that of a completeness Goguen the set ( 1 of 967 theorem agents ) . This is is a poset T $ =proved ( comma Tnew together , approach\ leq with was several based) $ other on the metalogical theory of semi theorems - Post period algebras .. This , which logic made is free possible of the the para hyphen ordereddoxes which withdevelopment appear respect in other of to an epistemic the axiomatic sharpness logics theory period of of algebras The perception research of LT initiated− andsets ( the [in 94 open ability] ) and square a representation to bracket distinguish 88 closing square ob − bracketj ects was . continued Antheorem axiomatization. LT − fuzzy of sets the are endowed logic introducedwith a rich structure is given and the , classical and a completeness fuzzy sets theorem is provedin open, square together bracket with 92 closing several square other bracket metalogical and open square theorems bracket 93 closing . \quad squareThis bracket logic period is free of the para − The problem of axiomatizing fuzzy sets is one of the most interesting and topical \noindentproblems indoxes the theory which of these appear sets in period other .... In epistemic open square logics bracket 94 . Theclosing research square bracket initiated Rasiowa inintroduces [ 88 ] was continued the notion of an LT hyphen \noindentfuzzy set commain [ which92 ] is and a modification [ 93 ] . of L hyphen fuzzy sets in the sense of Goguen open parenthesis 1 967 closing parenthesis period .. This \ hspacenew approach∗{\ f i l wasl }The based problem on the theory of axiomatizing of semi hyphen Post fuzzy algebras sets comma is one which of made the mostpossible interesting the and topical development of an axiomatic theory of algebras of LT hyphen sets open parenthesis open square bracket 94 closing square\noindent bracketproblems closing parenthesis in the and theory a representation of these sets . \ h f i l l In [ 94 ] Rasiowa introduces the notion of an $ LTtheorem− period$ LT hyphen fuzzy sets are endowed with a rich structure and the classical fuzzy sets \noindent fuzzy set , which is a modification of $ L − $ fuzzy sets in the sense of Goguen ( 1 967 ) . \quad This new approach was based on the theory of semi − Post algebras , which made possible the

\noindent development of an axiomatic theory of algebras of $ LT − $ sets ( [ 94 ] ) and a representation theorem $ . LT − $ fuzzy sets are endowed with a rich structure and the classical fuzzy sets HELENA RASIOWA comma 1 9 1 7 endash 1 9 94 .. 1 7 \ hspaceof L period∗{\ f iZadeh l l }HELENA open parenthesis RASIOWA 1 , 965 1 closing9 1 7 parenthesis−− 1 9 94 appear\quad here1 as 7 a particular case period .. The results show the numerous \noindentadvantagesof of thisL . approach Zadeh ( semicolon 1 965 in ) particular appear here comma as it leadsa particular to a solution case of the . axiomatization\quad The results show the numerous advantagesproblem for LT of hyphen this fuzzy approach sets period ; in The particular papers open square , it bracket leads 94 to endash a solution 97 closing squareof the bracket axiomatization introduce andproblem develop forthe new $ approach LT − to$ fuzzy sets . The papers [HELENA 94 −− RASIOWA97 ] introduce , 1 9 1 7 – 1 9 94 and1 develop 7 the new approach to fuzzyfuzzy and and rough roughof L sets . Zadeh sets comma ( ,1 based 965 based ) on appear semion semi hyphenhere as− Post aPost particular algebras algebras case period . LT $ The hyphen. results LT fuzzy show− logics$ the fuzzy formalizenumerous logics approximate formalize approximate reasoningsreasonings appliedadvantages applied to notions of to this notions whichapproach are which ; not in particular totally are determined not , it leads totally to period a solution determined of the axiomatization . problem In open squarefor LT bracket− fuzzy 90 closing sets . The square papers bracket [ 94 a– 97theory ] introduce of algebras and of develop order the omega new plus approach omega to to fuzzy the power of * is developedIn [ 90 and ] a appliedand theory rough to a of sets construction algebras , based on of semi order - Post $ algebras\omega.LT −+fuzzy\omega logics formalizeˆ{ ∗ }$ approximate is developed and applied to a construction ofof an an approximation approximationreasonings logic applied logic period to notions. ..\ Postquad which algebrasPost are of algebras not order totally omega determined of plus order omega . $ to\omega the power+ of *\ ..omega are particularlyˆ{ ∗ }$ ∗ interesting\quad are particularlyIn [ 90 ] a theory interesting of algebras of order ω + ω is developed and applied to a construction ∗ becausebecause of of theof the anfact approximation fact that comma that in , logic spite in . spite of their Post of infinite algebras their order of infinite order commaω + theyω order preserveare , particularly they more analogiespreserve interesting with more analogies with PostPost algebrasbecause of finite of of finite orders the fact than orders that any , in other than spite known of any their generalization other infinite known order of , thegeneralization they latter preserve algebras more period of analogies the latter with algebras . BesidesBesides comma , theyPost they algebras have have proved provedof finite very very orders useful useful than in applicationsany in other applications known to approximation generalization to approximation logics of the of infinite latter algebras logics . of infinite typetype comma, allowingBesides allowing both, they both have lower lower proved and and upper very upperuseful approximations approximations in applications of sets to being approximation of recognizedsets being logics period recognized of The infinite theory type . The theory ofof Post Post algebras algebras, allowing of order of both omega order lower plus and $ omega upper\omega approximationsto the+ power\omega of of* contains setsˆ{ being ∗ a }set recognized$ representation contains . 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C period comma May 23 hyphen 25 comma 1 9 90 comma IEEE Computer Society Press comma 42 endash 47 period open square bracket 9 1 closing square bracket .. On approximation logics comma .. a survey comma Jahrbuch 1 990 comma Kurt G o-dieresis del Gessellschaft comma Vienna comma Austria comma 1 990 comma 63 endash 87 period open square bracket 92 closing square bracket .. Mechanical proof systems for logic of reaching consensus by groups of intelligent agents comma Intern period Journ period of Approximate Reasoning 5 comma No period 4 open parenthesis 1 99 1 closing parenthesis comma 41 5 endash 432 period open square bracket 93 closing square bracket .. open parenthesis with W period Marek closing parenthesis .. Me- chanical proof systems for logic II comma .. consensus programs and their processing comma Journ period of Intelligent Information Systems 2 comma No period 2 open parenthesis 1 992 closing parenthesis comma 149 endash 1 64 period open square bracket 94 closing square bracket .. open parenthesis with Nguyen Cat Ho closing parenthesis LT hyphen fuzzy sets comma Intern period .. Journ period on Fuzzy Sets and Systems 47 comma No period 3 open parenthesis 1 992 closing parenthesis comma 233 endash 339 period open square bracket 95 closing square bracket .. Towards fuzzy logic comma in : L period A period Zadeh and J period Kacprzyk open parenthesis editors closing parenthesis comma Fuzzy Logic for Manage hyphen ment of Uncertainty comma open parenthesis New York comma John Wiley ampersand Sons comma 1 99 2 closing parenthesis comma 5 endash 25 period open square bracket 96 closing square bracket .. open parenthesis with Nguyen Cat Ho closing parenthesis LT hyphen fuzzy logics comma in : L period A period Zadeh and J period Kacprzyk open parenthesis editors closing parenthesis comma Fuzzy Logic for Management of Uncertainty comma open parenthesis New York comma John Wiley ampersand Sons comma 1 99 2 closing parenthesis comma 1 2 1 endash 1 39 period open square bracket 97 closing square bracket .. Axiomatization and completeness of uncountably valued approxima- tion logic comma Studia Log hyphen ica 53 comma No period 1 open parenthesis 1 994 closing parenthesis comma 1 37 endash 1 60 period