Table Of ContentStudies in Computational Intelligence 878
Hung T. Nguyen
Vladik Kreinovich Editors
Algebraic
Techniques and Their
Use in Describing
and Processing
Uncertainty
To the Memory of Professor Elbert
A. Walker
Studies in Computational Intelligence
Volume 878
Series Editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
The series “Studies in Computational Intelligence” (SCI) publishes new develop-
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Hung T. Nguyen Vladik Kreinovich
(cid:129)
Editors
Algebraic Techniques
and Their Use in Describing
and Processing Uncertainty
To the Memory of Professor Elbert A. Walker
123
Editors
Hung T.Nguyen VladikKreinovich
Department ofMathematical Sciences Department ofComputer Science
NewMexico State University TheUniversity of Texasat ElPaso
LasCruces, NM, USA ElPaso, TX,USA
Faculty of Economics
ChiangMai University
ChiangMai, Thailand
ISSN 1860-949X ISSN 1860-9503 (electronic)
Studies in Computational Intelligence
ISBN978-3-030-38564-4 ISBN978-3-030-38565-1 (eBook)
https://doi.org/10.1007/978-3-030-38565-1
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Preface
In 2018, many of us were saddened by the death of Elbert Walker (1930–2018), a
renowned algebraist who, in addition of doing research in algebra, was one of the
leaders in applying algebraic methods to fuzzy techniques. This edited book con-
tains papers by his colleagues dedicated to him and his work.
Most papers in the memory volumes are technical, although some are philo-
sophical or personal. The technical papers cover many aspects of Elbert Walker’s
research; the variety of their topics reflects the variety of his research and explains
whyhewassomissed.Thesepapersalsoshowthathisideasandresultscontinueto
inspire, and we hope that this volume will inspire even more related results.
Enjoy!
Las Cruces, USA Hung T. Nguyen
El Paso, USA Vladik Kreinovich
Co-editors of the Elbert Walker Memory volume
v
Contents
Specker Algebras: A Survey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
G. Bezhanishvili, P. J. Morandi and B. Olberding
Least Square Approximations and Linear Values of Cooperative
Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Ulrich Faigle and Michel Grabisch
Elementary Divisor Domains as Endomorphism Rings . . . . . . . . . . . . . 33
László Fuchs
A Topos View of the Type-2 Fuzzy Truth Value Algebra . . . . . . . . . . . 41
John Harding and Carol Walker
ASymmetry-BasedExplanationoftheMainIdeaBehindChubanov’s
Linear Programming Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Olga Kosheleva, Vladik Kreinovich and Thongchai Dumrongpokaphan
Why Bohmian Approach to Quantum Econometrics:
An Algebraic Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Vladik Kreinovich, Olga Kosheleva and Songsak Sriboonchitta
Direct Decompositions of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Adolf Mader and Otto Mutzbauer
Generators of Fuzzy Logical Operations . . . . . . . . . . . . . . . . . . . . . . . . 89
Mirko Navara and Milan Petrík
An Enjoyable Journey of Research Collaboration
with Elbert A. Walker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Hung T. Nguyen
In Memory of Dr. Elbert A. Walker . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Nadipuram R. Prasad
vii
viii Contents
Duality via Truth for Some Fuzzy Modal Logic. . . . . . . . . . . . . . . . . . . 129
Anna Maria Radzikowska and Etienne E. Kerre
Me and Elbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Fred Richman
Fuzzy Bigraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Apostolos Syropoulos
Specker Algebras: A Survey
G.Bezhanishvili,P.J.MorandiandB.Olberding
DedicatedtothememoryofElbertWalker
Abstract ForacommutativeringRwithidentity,aSpeckerR-algebraisacommu-
tativeunitalR-algebrageneratedbyaBooleanalgebraofidempotents,eachnonzero
element of which is faithful. Such algebras have arisen in the study of (cid:2)-groups,
idempotent-generated rings,Boolean powersofcommutative rings,Pierceduality,
and rings of continuous real-valued functions. We trace the origin of this notion
fromearlystudiesofsubgroupsofboundedinteger-valuedfunctionstoavarietyof
current contexts involving ring-theoretic, topological, and homological aspects of
idempotent-generatedalgebras.
· · ·
Keywords (cid:2)-group Singularelement Speckeralgebra Idempotent-generated
· ·
algebra Stonespace Ringsofcontinuousfunctions
· · ·
MathematicalSubjectClassification(2010) 06F20 06E15 16G30 54H10
1 Introduction
Ourgoalinthisarticleistogivesomehistoricalandpracticalmotivationforaclass
ofcommutativealgebrasknownasSpeckeralgebrasthatareencounteredinvarious
guises in contexts such as Baer-Specker theory, hyperarchimedean (cid:2)-groups and
B
G.Bezhanishvili·P.J.Morandi( )·B.Olberding
DepartmentofMathematicalSciences,NewMexicoStateUniversity,
LasCrucesNM88003,USA
e-mail:[email protected]
G.Bezhanishvili
e-mail:[email protected]
B.Olberding
e-mail:[email protected]
©SpringerNatureSwitzerlandAG2020 1
H.T.NguyenandV.Kreinovich(eds.),AlgebraicTechniquesandTheirUseinDescribing
andProcessingUncertainty,StudiesinComputationalIntelligence878,
https://doi.org/10.1007/978-3-030-38565-1_1
2 G.Bezhanishvilietal.
vectorlattices,ringsofcontinuousfunctions,Booleanpowersofcommutativerings,
andPiercesheaves.
Beforeoutliningoursurveyofthistopic,wewishtoacknowledgefirstourdebt
toElbertWalkerasfriend,mentor,colleague,andresearcher.Whileourtopicdoes
not directly build on Elbert’s work, it cuts across several of his interests in a path
thatwouldbeperfectlyfamiliartohim,fromabeliangrouptheoryandringtheoryto
foundations,categorytheory,andhomologicalalgebra.WhenwearrivedatNMSU
at different points during the years 1990–2002, Elbert was beginning his second
researchcareerinstatisticsandfuzzylogic.Characteristically,ratherthandistancing
himfromthefoundationsandalgebraresearchgroupsatNMSU,hisnewresearch
only deepened his impact on these groups, resulting in new collaborations among
the seminar members, as well as new directions for research that incorporated the
algebraicapproachestofuzzylogicpioneeredbyElbertandCarolWalker.Infact,
itwasnotunusualtohearElbertspeakinthealgebraseminaroneyearonabelian
p-groups,thenextontype2fuzzysets,andthenextonlocalizationofcategories.
It is in this same spirit of appreciation of multiple perspectives that we hope to
presentthetopicsinthisarticle.OurapproachtothetopicdrawsheavilyonElbert’s
influenceandpersuasiveviewpointonmathematics.AsElbertwouldoftenremind
us, his interest was not only in the proof of a theorem but why the theorem was
true. The topic of this survey grew from just such a desire to understand a similar
“why” for certain algebraic-topological dualities such as Gelfand-Naimark-Stone
duality,Kakutani-Krein-Yosidaduality,deVriesduality,andPierceduality.While
thesedualitiesremainfarinthebackgroundofthisarticle,connectionstothemcan
beseenmoreexplicitlyin[7–10]andwillbefurtherelaboratedoninfuturearticles.
For the present article, we want instead to focus on a rather simply defined class
of commutative algebras, that of Specker algebras, which plays a key role in our
algebraicapproachtothesedualities.
In Sect.2 we discuss the Specker group concept, which originates with Speck-
er’s work on subgroups of the Baer-Specker group. We trace this concept through
Nöbeling’s results on freeness and into Conrad’s work, which gives an axiomatic
descriptionofSpeckergroupsbylocatingthemwithintheclassof(cid:2)-groups(lattice-
orderedgroups).Theculminationofthislineofresearchistherepresentationofa
Specker(cid:2)-groupasagroupofZ-valuedcontinuousfunctionsonalocallycompact
Hausdorffzero-dimensionalspace(seeTheorem2.8).Althoughwemakenoclaim
oforiginalityfortheresultsthemselves,wegiveself-containedproofsofmostofthe
resultsinSect.2,emphasizingaduality-theoreticapproachinthespiritofStone.
Using the fact from Sect.2 that multiplication is always present on Specker
(cid:2)-groups, we shift focus in Sect.3 from groups to rings, and from rings to alge-
bras.WedefineSpeckerR-algebrasforanarbitrarycommutativeringwithidentity,
andwediscussseveraloftheirproperties.Inthistreatment,wefollow[6],butwe
alsopointoutadditionalconnectionsandapplicationsfrom[7, 10].
Inthefinalsectionofthepaper,weconsiderSpeckerR-algebrasinthetwoclassical
cases,R=Zand R=R.We use the former case and the work of Bergman [4] to
revisit the freeness results for Specker (cid:2)-groups discussed in Sect.2. In the case
