Heyting Algebras: Duality Theory

107 Pages·2019·1.609 MB·English

by  Leo Esakia, Guram Bezhanishvili, Wesley H. Holliday

Trends in Logic 50
Leo Esakia
Heyting
Algebras
Duality Theory
Edited by Guram Bezhanishvili
Wesley H. Holliday
Translated by Anton Evseev
Trends in Logic
Volume 50
TRENDS IN LOGIC
Studia Logica Library
VOLUME 50
Editor-in-Chief
HeinrichWansing,DepartmentofPhilosophy,RuhrUniversityBochum,
Bochum,Germany
EditorialBoard
ArnonAvron,DepartmentofComputerScience,UniversityofTelAviv,TelAviv,Israel
KatalinBimbó,DepartmentofPhilosophy,UniversityofAlberta,Edmonton,AB,Canada
GiovannaCorsi,DepartmentofPhilosophy,UniversityofBologna,Bologna,Italy
JanuszCzelakowski,InstituteofMathematicsandInformatics,UniversityofOpole,
Opole,Poland
RobertoGiuntini,DepartmentofPhilosophy,UniversityofCagliari,Cagliari,Italy
RajeevGoré,AustralianNationalUniversity,Canberra,ACT,Australia
AndreasHerzig,IRIT,UniversityofToulouse,Toulouse,France
WesleyHolliday,UCBerkeley,Lafayette,CA,USA
AndrzejIndrzejczak,DepartmentofLogic,UniversityofLodz,Lódz,Poland
DanieleMundici,MathematicsandComputerScience,UniversityofFlorence,Firenze,Italy
SergeiOdintsov,SobolevInstituteofMathematics,Novosibirsk,Russia
EwaOrlowska,InstituteofTelecommunications,Warsaw,Poland
PeterSchroeder-Heister,Wilhelm-Schickard-Institut,UniversitätTübingen,Tübingen,
Baden-Württemberg,Germany
YdeVenema,ILLC,UniversiteitvanAmsterdam,
Amsterdam,Noord-Holland,TheNetherlands
AndreasWeiermann,VakgroepZuivereWiskundeenComputeralgebra,UniversityofGhent,
Ghent,Belgium
FrankWolter,DepartmentofComputing,UniversityofLiverpool,Liverpool,UK
MingXu,DepartmentofPhilosophy,WuhanUniversity,Wuhan,China
JacekMalinowski,InstituteofPhilosophyandSociology,PolishAcademyofSciences,
Warszawa,Poland
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RyszardWojcicki,InstituteofPhilosophyandSociology,PolishAcademyofSciences,
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Leo Esakia (Deceased)
Author
Guram Bezhanishvili Wesley H. Holliday
(cid:129)
Editors
Heyting Algebras
Duality Theory
123
Author Editors
LeoEsakia (Deceased) Guram Bezhanishvili
Tbilisi, Georgia Department ofMathematical Sciences
NewMexico State University
LasCruces, NM, USA
Wesley H.Holliday
Department ofPhilosophy andGroup in
Logicandthe Methodology of Science
University of California
Berkeley, CA, USA
Translated by
AntonEvseev (Deceased)
Birmingham, UK
ISSN 1572-6126 ISSN 2212-7313 (electronic)
Trends inLogic
ISBN978-3-030-12095-5 (hardcover) ISBN978-3-030-12096-2 (eBook)
ISBN978-3-030-12098-6 (softcover)
https://doi.org/10.1007/978-3-030-12096-2
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LeoEsakia(1934–2010)
Foreword
This translation of Leo Esakia’s book on Heyting algebras has been in the making
for a long time. The book was originally published in 1985 by the Georgian
publishinghouseMetsniereba(Science).ItwasthefirstvolumeofEsakia’splanned
two volume monograph on Heyting algebras. The book turned out to be popular
among Soviet logicians, and Esakia had begun work on the second volume, an
outline of which is presented in the Appendix of the first volume. But after the
breakoftheSovietUnion,thepublishinghouseceasedtoexist,andtheprojectdied
with it.
Logicians and mathematicians in the West were aware of the existence of the
book,andthereweremanyrequeststotranslatethebookintoEnglish.Butasfaras
we know, there was no formal contract to translate the book by any Western
publishing house.
Hilary Priestley was among the Western mathematicians interested in the book,
havingreceivedacopyfromAnnaRomanowska.In2003,HilaryenlistedaRussian
student, Anton Evseev, to translate the book into English. At the time, Anton was
an undergraduate studying Mathematics at the University of Oxford. His
hand-written translation was not widely circulated, but Hilary mentioned the
translation to Mai Gehrke, who in turn mentioned it to Nick Bezhanishvili and
David Gabelaia. After Esakia’s death in 2010, several tributes were planned: a
special issue of Studia Logica (Vol. 100, No. 1–2) dedicated to him appeared in
2012, and a special volume of Outstanding Contributions to Logic in his honor
appeared in 2014 (Leo Esakia on Duality in Modal and Intuitionistic Logics,
Springer). In addition, it was decided that the English translation of Esakia’s book
be edited for publication. Hilary scanned and emailed Anton’s hand-written
translationtoNickandDavid.WiththehelpofMamukaJibladze,NickandDavid
used some funds from Esakia’s last grant to hire staff from the A. Razmadze
Mathematical Institute to type up the translation. The first round of editing of the
translationoccurredinthesummerof2012byGuramBezhanishviliandthesecond
round in the summer of 2017 by Guram and Wesley Holliday.
vii
viii Foreword
At last the final product is before your eyes. The initial work by Anton Evseev
madeavaluablecontributiontowardbringingLeoEsakia’sclassicmonographtoa
wider audience, and we were very grateful when Anton willingly agreed that the
translation could form the basis of a version edited for publication. It is with great
sadness that we report Anton’s untimely death in February 2017 at the age of 33.
The mathematical community has been robbed of an exceptionally talented col-
league, and we regret that Anton himself will not see this English version of
Heyting Algebras in print.
Although many of the main results of Esakia’s book have already made their
way into the mathematical literature, there is no better way to see them developed
than through Esakia’s concise and lucid presentation. We hope the publication of
this translation will make Esakia’s intellectual legacy accessible to a wider
audience.
Guram Bezhanishvili
Nick Bezhanishvili
David Gabelaia
Mai Gehrke
Wesley H. Holliday
Mamuka Jibladze
Hilary Priestley
Contents
1 Preliminary Notions and Necessary Facts. . . . . . . . . . . . . . . . . . . . . 1
1.1 Universal Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Ordered Sets and Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Heyting Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Heyting Algebras and Closure Algebras. . . . . . . . . . . . . . . . . . . . . . 15
2.1 Heyting Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Closure Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Modal Systems and Superintuitionistic Logics . . . . . . . . . . . . . . . 21
2.4 Filters and Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Skeletal Closure Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Duality Theory: Hybrids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 The Hybrid of Topology (Stone) and Order (Kripke) . . . . . . . . . . 41
3.2 Fundamental Properties of Hybrids . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 The Category of Hybrids and Hybrid Maps . . . . . . . . . . . . . . . . . 54
3.4 The Category of Heyting Algebras and the Category of Strict
Hybrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 Grzegorczyk Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Appendix. .... .... .... .... ..... .... .... .... .... .... ..... .... 77
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 91
ix
’
Editors Note
TheoriginaltitleofEsakia’sbook,translatedintoEnglish,was‘HeytingAlgebrasI.
Duality Theory’. As explained in the Foreword, volume II did not materialize, so
wehaveremoved‘I’fromthetitle.TheplannedcontentsofvolumeIIarediscussed
in detail in the Appendix.
Esakia was unhappy that a large number of typos and mathematical mistakes
were introduced intheRussianversionofhis book,whichhecould notcorrect,as
hewasnevergivenanopportunitytodoafinalproofreading.Wetookthelibertyto
make the corrections without flagging them, as doing so would distract the reader.
We have, however, added specially marked editorial footnotes when we felt that
furtherexplanationwasinorder.Inaddition,wefilledingapsinsomeproofs.Two
casesarenoteworthy.First,weeditedproofsattheendofChap.2,wherewedrew
from the paper “Scattered and hereditarily irresolvable spaces in modal logic” by
GuramBezhanishviliandPatrickJ.Morandi,ArchiveforMathematicalLogic,Vol.
49, 2010, pp. 343–365, as well as further communication between Guram and
Patrick. Second, Julia Ilin pointed out a gap in Esakia’s original proof of what is
nowTheorem5.13,sowereplaceditbyanalgebraicversionoftheprooffrompp.
158–9ofG.Boolos,TheLogicofProvability(CambridgeUniversityPress),1993.
Since the original Russian publication of Esakia’s book, some of the terminol-
ogy in the area has changed. In particular, several objects and results are now
namedafterEsakia:e.g.,Esakiaspaces,Esakia’slemma,theBlok-Esakiatheorem,
etc.Attheendofthisnote,weprovideatablecomparingsomeofEsakia’soriginal
terminologywithmodernterminology.Insomecases,wehaveoptednottogivethe
mostdirecttranslationofaRussianterm,optinginsteadforamorenaturalEnglish
substitute (e.g., we use the term ‘skeletal Heyting algebra’ instead of the more
directly translated ‘stencil Heyting algebra’).
Throughout we have made light changes to notation for readability and con-
sistency. Parentheses are often omitted after functions and inverses, but they are
oftenaddedaroundmeetsandconjunctions.Insomecases,weusemodernsymbols
in place of the original symbols (e.g., for sum and product) or adopt modern
notational practices (e.g., in the notation for functors in Sect. 3.3 of Chap. 3).
However, we retain Esakia’s convention that bold typeface signals a definition.
xi