Exploring Meta-Symmetry for Configurations in Closure Spaces (Developments of Language, Logic, Algebraic system and Computer Science)

Exploring Meta‐Symmetry

for

_{Configurations}

in Closure

_Spaces

Marcin J. Schroeder

Akita Intemational_{University,Akita,Japan}

[email protected] 1. Introduction

Thispaper continues researchpresentedearlierin which the_studyof_symmetrywasgeneralized

to _arbitrary closure spaces. [1] The present paper is exploring one of the _{consequences of this}

generalization.All_possibleclosurespaceson agiven set\mathrm{S}_(i.e.all_possibleclosure_operatorsonthis commonset_S)canbe associatedwithcorrespondingMoorefamilies \ovalbox{\tt\small REJECT} of closed subsets.Then each

of theseMoore families of closedsubsets\mathscr{M}canbe consideredaclosedsubfamilyof the powerset_ge

=2^{\mathrm{s}}ofS. For_{arbitrary family} ãofsubsets of\mathrm{S} we canfind the least Moorefamily,\ovalbox{\tt\small REJECT}of subsets

includingthis_{family (\mathscr{B}\subseteq a)}.Equivalently, for everyfamily \mathscr{R}of subsets of\mathrm{S}we canconsidera

larger family\ovalbox{\tt\small REJECT}withall intersections of subsets_belongingtothe_{original family} \mathscr{R}. Obviouslythis

will be aMoorefamily\ovalbox{\tt\small REJECT}consideredbefore,i.e. theleast Moorefamily includingtheoriginalone.

Weassumethattheintersectionof_everyemptyfamilyis entire_powersetof S.

Thus,thesetof all closurespaces definedon\mathrm{S},orall closureoperatorsdefinedon\mathrm{S}definesone

specific closure operatorfonthe_powerset _\wp.Now,in theearlierpaperatheoryofsymmetriesfor

configurationsofclosed subsets ofanarbitraryclosure_spaceisdeveloped.Inthis paperweapplythis

generalizationof_symmetrytothe_particularcaseof theclosure space definedonthe powerset _\wp of theset\mathrm{S}_bythe_{closure operator which}extends anyfamilyof subsetstothe least Moore_family, i.e.

ourclosure_operatorfon _\wp.Theclosurespace definedbythisclosureoperatoronthepowersetof\mathrm{S}

is calledameta‐closure_{space. ‘Ihe closed}subsets ofameta‐closure spaceare_directlyandbijectively

correspondingto closure_operators on S. The _symmetryof_{configurations} of closure _{operators (or}

spaces)defined this_{way is called here“meta‐symmetry”.}

2. From

_{Erlangen Program}

toGeneral

_Concept

of

_Symmetry

Spectacularsuccessof the_conceptof symmetryinmathematics,where it became understood as

invariancewith_respecttoaclass of transformations(inthe consequenceof theErlangen Programof

Felix Klein _published in 1872 _[2]) and _following this success rise to the fundamental role of

symmetryand_{symmetrybreaking}in_physicsand_physicalsciences_generatedinterestinthis_concept

among representatives of other disciplines as far from physics as those in the humanities. The

immense_popularityofthe book_{“Symmetry”published by}Hermann_Weylin 1952_greatlycontributed

tothis wide _spreadof interest. _[3]Weyl demonstrated theuniversal character ofsymmetryas atool

for the_studyofshuctures not_onlyinmathematics and natural_sciences,but also inart.

Ahalf_{centuryearlier before symmeay}and invariance with_respecttotransforrnations becamethe

focus of intellectual intercourse across the wide _range of non‐scientific disciplines, structural

(synchromc) studies started to beconsidered as an alternative to those with historical _(diachronic)

methodology.The_oppositionof_synchronicanddiachronic_perspectiveswasintroducedbyFerdinand

published byhis _disciplesin 1916 _[4]). Howeverthe need for this distinctionwasrecognizedeven

earlier in the works_{ofpsychologists.}

Structuralism as aphilosophical direction can be understoodas the _response to the calls for a

methodologyfor_disciplines in which theuseofmeasures and numbersdominatingnatural sciences

seemed_impossibleorineffective._Explosionof enthusiastic interest in structuralism and in its tool of symmetry studies initiatedbythe _{Erlangen Program}didnotlast_longoutside of science. _Onlyquite

recentlystructuralism andstructural realism becamethe_subjectofrevived interest.

Myowndiagnosticof thedisappointment, of the lostpopularityandof the decline of interestin

structuralism outsideofmathematics and_physicalscienceswasthat itsmethodologyderivedfromthe

Erlangen Programof Kleinwasheavily dependentoncoordinatization,_verynaturalin_geometryand

physics,but of limited_meaning outside these_{disciplines. [1]} Transfornations ofageometric_space canbeeasilydefined with theuseof coordinates ofpoints. Without coordinatestheonlyalternative was to consider fmite, discrete, or combinatorial cases in which all possible permutations are

considered. This is _why structuralism in _{anthropology, [5]} or in developmental psychology, [6]

engaged only extremely simpleformsof_symmetrydescribedintermsof the_{Klein group.}

This issuewasconvoluted with_quitefrequent misunderstandingof thecoreidea of symmetry in

Klein’s _{Erlangen Program.} In some extent this confusion can be “blamed” on _Klein, because his

original expositionexcluded_completelythecaseoftopological_{groups of}transformationsrelegatedto

the referencetothe works of_SophusLee.Thus,evenifthe group of all rotations aroundapointisnot

discrete,its_{topological properties}werenot_investigatedor evenmentioned.

Ofcourse_ulamingKleinwould Ueananachronism,asgeneral topologydidnotexistatthat time and Lee_justreferredto _continuityfor metric_spaces. Exclusion ofthe _continuityof_groups suchas

that of rotations from considerationin 1872wasnot_problematic.Afterall,Klein_{clearly required}the

introductorystepof theselection of thegroupof transformations which defines thegeneralconceptof

geometry (forKlein it wasprojective geometry). Subgroups of this _group selectedby appropriate

conditions defined_particular forms of_geometry,or whatWeyl called symmetriesof the space. [2]

Onlyafter_appropriategroup of transformations chaxacterizin\mathrm{g}the choice ofgeometrywas selected,

symmetriesof_{particular configurations}of_{geometric objects}were studied intermsof_subgroups of transformations for which_{configurations}wereinvariant. Therewas noreasontoinvolveinthis entire

“symmetric group”, i.e. group of all permutations. For instance, Euclidean geometry was

characterized _by the _subgroupof all Euclidean isometries _{(transformations} that preserve Euclidean distance of_{points). Only} in the next _{step after establishing} this _symmetry ofthe _{space, specific}

configurations ofpointscould beconsidered.

The omission of the fact that for the _study of _symmetry of _{given configuration} not all transformations for which this _{configuration} is invariant are important and actually _many such

transformations have tobe excluded_{had grave consequences. Many}recentbooks_popularizing the conceptof_{symmetry promote}nonsensical views suchas_symmetryofaconfigurationis any function

which makes this_{configuration}invariant andall thesefunctions form its_“groupof_{symmetries”.}Of

coursethis kills the very idea ofKlein’sErlangen Programin which it is _necessaryto startfrom the distinction ofaspecific_{group of}transfotmationsdefiningthecontextof_{study (for}instance_geometry,

topology,orwhatever itis)and thentoconsideraGalois conmection betweensubgroupsof this_group

and _{configurations} of_{points (or possibly} other _objects, such as lines) ordered by inclusion. The

group (group of alltransformations),but this is onlyvery specialcaserarely interesting outside of

combinatorics.

The_problemstudied in myearlier workwashowtoeliminate the need for coordinatizationof the

set _{(in geometry} entire _plane or space) in order to defme transformations and to _distinguish

appropriategroups and their subgroups. This goal wasachieved Uy afommulation ofthe theoryof

symmetryintermsofanarbitraryclosure space. To make this paperself‐containedabrief_summary

of the earlier paper willbe included.

3.

_{Revisiting Concept}

of

_Symmetry

inGeneral Closure

_Spaces

The_followingnotationand_{terminological}conventions will be used_throughoutthetext:

Greek letters suchas $\phi$, _{$\varphi$, \mathrm{e}},etc.indicate functionsonthe elements ofagivenset \mathrm{S}and with the

values _belongingtoasetT. Small Latin letterssuchas_fg, h,etc.indicatefunctions definedonthe

subsetsofagivensetand with thevalues whicharesubsetsofthisset. Thedoubleuseof the_symbol

$\varphi$^{-1}(\mathrm{A})

,asthe set ofvalues for the inverse functionof $\varphi$, andas an inverse imageofasetAwith

respectto function $\varphi$whichmaynothave inverse,shouldnotcause_problems. The_composition of

functions iswrittenasajuxtapositionof theirsymbols,unless thefact oftheuseofacompositionof

functions is contrasted with _constructing function _images. The _symbol \cong indicates a bijective

correspondenceorisomorphism.Throughoutthe_paper,partiallyorderedsetsareoften called_posets.

The_purposeofthese_{preliminaries}isto _{specify terminological}and notational_conventions,not to

present the introduction to the _subject which canbe found elsewhere [7] These preliminaries are

followed_by the brief_presentation of main result of the earlier_paperon _symmetry in an arbitrary

closure space.[1]

Definition 3.1 Let _fbe afunction from the_powerset _ofa setS to _itselfwhich _satisfies the

followingtwoconditions:

(1) $\nu$ A\subseteq S:A\subseteq f(A),

(2) VA,B\subseteq S:A\subseteq B\Rightarrow f(A)\subseteq f(B), (3) VA\subseteq S.\cdot ff(A)=f(f(A))=f(A).

Then_fiscalleda closure_operator(ortransitive closure_operator) onS. Theset _ofall closure operatorson thesetS is indicated_{by I(S).}Aset _equippedwith a closure_{operator will be called}a

closurespace<S,f>.

The thirdconditionscanbereplaced byacondition: which is easiertouseinproofs,but whichin

combination withothertwo_givesexactlythesame_concept:

(3^{*}) VA,B\subseteq S:A\subseteq f(B)\Rightarrow f(A)\subseteq \mathrm{i}f(B).

The strongerform ofthiscondition_{VA,B\subseteq S:A\subseteq f(B) iff f(A)\subseteq f(B)} can be used instead_ofall

three conditions to _define a transitive _{operator, but this}factdoes not have asignificant practical importance.

Definition 3.2 Let_fbeaclosure operatoron asetS. The subsets A_{ofS satisfying}the condition

f(A)=A, calledf‐closedsetsform a Moorefamily f‐Cl, i.e. it is closed with respect to arbitrary

intersectionsand includes thesetS_(whichcanbeconsidered theintersection_{ofthe emptysubfamily}

of subsets). EveryMoore_family ne_defines a transitive _operatorf(A) =\cap[M\in\ovalbox{\tt\small REJECT}:A\subseteq Mj. Set

theoretical inclusion_defimesa_partialorderon _f‐Clwith respecttowhichitisacompletelattice. To

Letfandgbe operatorson asetS. The relation_{defmed byf\leq^{r}g if VA\subseteq S:f(A)\subseteq g(A)is}apartial

orderonI(S), with respectto which it isa complete lattice. Thispartialordercorresponds to the

inverse_ofthe inclusion_oftheMoore_{families ofclosed}subsets

Definition 3.3_Letfbea_{closure operator}on asetS,gaclosure operatoronsetT, and $\varphi$ bea

function from S to T The_function $\varphi$ is (fg)‐continuous _iỷ VA\underline{c}S: $\varphi$ f(A)\underline{\subseteq}g $\varphi$(A). We will write

continuous,ifno confusionis_likely.

Proposition3.1 _{Continuity ofthefunction} $\varphi$asdefinedabove is_equivalentto: \triangleright B_ag‐Cl.

$\varphi$^{J}(B)-\in f-C.

Definition 3.4 Let_fbeaclosure operatoron aset_S,gaclosure operatoronsetT,and $\varphi$ bea

function from StoT. The_{function $\varphi$ is(fg)‐isomorphism if}it is_bijectiveand VA\subseteq S: $\varphi \gamma$(A)=g $\varphi$(A).

Wewillwnte_{isomorphism, if}no_confusionis_{likely. If}S=T,wewillcall $\varphi$an[fg)‐automorphism, or

smply automorphism.

Proposition3.2 The conditions_forafunction _$\varphi$to bean isomorphism, asdefinedabove, are

equivalenttoeitheronebelow:

(1) $\varphi$ hasaninverse

$\varphi$^{J}-

,and both arecontinuous,

(2)Thereexistsa_{function $\psi$from} TtoSsuch that _{$\varphi \psi$=id_{T}} and _{$\psi \varphi$=id_{S} and both $\varphi$ and $\psi$} are

continuous.

Proposition3.3_Letfbeaclosure_operatoron aset S,gaclosure_operatoronsetT_{s}and $\varphi$ bea

function from StoT Then,every(fg)‐isomorphism $\varphi$generatesalattice isomorphism$\varphi$^{*}betweenthe

completelattices_ofciosed subsets_{L_{f}}and_{L_{g} defined byVA\in L_{f}. $\varphi$^{*}(A) = $\varphi$(A)\in L_{g}}.Also, ifafunction

$\varphi$.\cdot S\rightarrow Tis_bijectiveandis_generatingalatticeisomorphism $\varphi$^{*}between lattices_{L_{f}}and_{L_{g}.}, then $\varphi$ is

an_{Ư,g)‐isomorphism.}

Proposition3.4_{Everyfauthomorphism $\varphi$ of<SJ>generates}a_uniquelatticeautomorphism_{ofL_{P}}

However, more than one _{f‐authomorphism $\varphi$ of<Sf>} can correspond to the same lattice

automorphismofL_{f}

Proposition 3.5 The set_ofall_{fautomorphisms of<S,f> forms} a _group Aut<Sf> under the

functioncomposition.Thisgroup isisomorphicto_{Aut(L)oflattice automorphismsofL_{f^{\mathrm{i}}}} Wewill refertotheconceptofan(antisotone)Galois connectionbetweentwo_posets.

Definition 3.5Let<P,-<>and_<Qg>beposets and $\varphi$ and $\psi$ be anti‐isotone(order inverting)

fitnctions $\varphi$:P\rightarrow Qand _{$\psi$ Q\rightarrow P}. Then thefunctions defmeaGalois connection between theposets

if: Vx\in P:x\underline{<} $\psi \varphi$(x)and _{Vy\in Q:y $\Xi \varphi$ Wy).}

Galoisconnectioncanbedefinedinanequivalent_wayas a_pairoffinctions $\varphi$.\cdot P\rightarrow Qand $\psi$ Q\rightarrow

Psuch that _{Vx\in P\emptyset/\in Q:y\underline{<} $\varphi$(x) iff x\underline{<} $\psi$(y)}.

Proposition3.ó_Ifapairoffunctions $\varphi$.\cdot P\rightarrow Qand _{$\psi$\cdot Q\rightarrow P defines}aGaloisconnection, then the

functions $\psi \varphi$.\cdot P\rightarrow P and _{$\varphi \psi$ Q\rightarrow Q} are closure_operators, i.e. _{they satisfy} the conditions 1₎-3_{) of}

Definition3.1_{generalizedfrom}theinclusion\subseteq tothe_partialorder‐〈.Moreover, the_{functions $\varphi$.\cdot P\rightarrow}

Qand _{$\psi$ Q\rightarrow Pdefme}order_{anti‐isomorphism (orderreversingfunctionspreserving}all_infima and

suprema)betweenthe_completelattices_ofclosedelementsintheposetsPand_Q.

Proposition3.7 Givenananti‐isotonefunction $\varphi$.\cdot P\rightarrow Q. Ifthefunction $\varphi$.\cdot P\rightarrow Q defimes together

with _{$\psi$ Q\rightarrow P}aGaloisconnection, then the_function $\psi$ isunique. However, thereare anti‐isotone

Proposition3.8_Ifposets<P, \underline{<}>and_{<Qg>are complete}lattices,then_forevery anti‐isotone

function $\varphi$.\cdot P\rightarrow Q, there exists (by Prop. 3.6unique)function $\psi$ Q\rightarrow P, such thatthey forma Galois

connection. The_{function $\psi$ Q\rightarrow P}is_{defined by: Vy\in Q}: $\psi$(y)=\displaystyle \vee\oint x\in P:y\leq $\varphi$(x)J, whereisthelowest

upper boundoftheset, whichm\mathrm{t}4Stexist inacompletelattice.

Remark 3.9 Wewereusingonlythefactthat the_{poset <P,\underline{<}>is}acompletelattice.

In the_{abstract fornulation of geometry}ontheplanein thetermsofclosurespaces theonlyclosed subsets are entire plane, _{empty subset,} points and straight lines. Geometric configurations are

collections of_points orlines. However,the concepts ofclosure spaces donot _giveus _{any tools for}

analysisofsuch_{configurations beyond}theintersections of lines_{producing points}and_pairsof_points

defininglines. Our_goal isto _providethe tools forthe_analysisof such _{configurations}not _onlyfor abstract_geometries,but for_{arbitraryclosure spaces. Theapproach presented}belowwasinformedby

the_analogywith_{geometric symmetriesin the choice of grouptheory}as afoundation.

Thus,it is a _studyof symmetry of_{configurations} of closed subsets ina selected,butarbitrary

closurespace <S,f> with the group \mathrm{G}=\mathrm{A}\mathrm{u}\mathrm{t}<S,f> of its \mathrm{f}‐automorphisms.Aconfigurationinthis space will beanarbitrary,butnot_emptyset s\leftarrowof \mathrm{f}‐closedsubsets of S. It isanaturalquestionhow the

complete lattice of_subgroups of the _{group \mathrm{G} is related} to _symmetries of_{configurations,} i.e. to

symmetriesof subsets of the_completelattice_{\mathrm{L}_{\mathrm{f}}}of closed subsets in <S,f>. Themain result of the

earlier research _presented below was that for_aỉuitraĩy closure space there is a Galois connection

betweenthelattice of_subgroupsof the group of all itsautomorphisms and the_partiallyorderedsetof its_{configurations}ofclosed subsets.

To avoidconfusion it is _importantto notice thatwe are not interested in stabilizers ofsets of elements of the closure space <S, f>, Uut ofthe families of closed subsets. The asterisk in the formulation of the_followinglemmareferstothelattice_{isomorphism $\varphi$^{*}}between the_completelattices of closed subsets \mathrm{L}_{\mathrm{f}}and _{\mathrm{L}_{\mathrm{g}} generated by (f,g)‐isomorphism} $\varphi$ between closure spaces <Sf> and

<T,g>,whichalwaysexistsby Proposition3.3.

Lemma 3.10 Let Hbeasubgroup ofthe group_G=Aut(L).Definethefamilyd_{H}ofsubsetsofL_{f}

by W\subseteq L_{f}\cdot K\in d_{H}iff VA\in K V $\varphi$\in H: $\varphi$^{*}(A)\in K. Then d_{H}is acomplete lattice with respectto the

order_ofinclusion_ofsets.

Lemma 3.11 Function $\Phi$:H\rightarrow d_{H}defimedinLemm 3.10 is _{anti‐isotonefunction} betweentwo posets, oneofthem (thelatticeofsubgroups ofa_{group G})isacompletelattice.

Nowwecandefine aGaloisconnection._UyProposition3.8andRemark 3.9weknow that there

exists a Galois connection between the poset of_complete lattices ớH and the _complete lattice of

subgroupsof_{\mathrm{G}=\mathrm{A}\mathrm{u}\mathrm{t}(\mathrm{L}_{\mathrm{f}})\cong}Aut_{<S,\triangleright.}

Theorem3.12 The_{following twofunctionsform}aGalois connection: $\Phi$:H\rightarrow \mathrm{c}\neq_{H}defined byVK\subseteq L_{f}K\in d_{H}iff VA\in J\{ V $\varphi$\in H:$\varphi$^{*}(A)\in Kand $\Psi$:ớ\rightarrow Hdefined by \vee fK subgroup ofG: ớ \subseteq\'{o}_{K}\`{i} = í

$\varphi$\inG: $\varphi$(ớ)\subseteqớJ.The lastequality is a

consequenceofthefactthat\ovalbox{\tt\small REJECT} $\varphi$\in G: _$\varphi$(ớ)_\subseteqờJisasubgroup of G.

Remark3,13Summaryof the_Conceptof_Symmetryin Closure_Spaces:

G=Aut(L_{j}jisthegroupofautomorphisms ofthelogic L_{f} ofaclosurespace<S,f> H isasubgroup ofthe groupG=Aut(L)

K\subseteq L_{f}isaconfigurationof closed subsets(e.g.inthe geometryon aplane ofpointsorlines)

We_getamutualcorrespondencebetweensubgroups\mathrm{H}oftransformationsof_<Sf>andinvariant

families of_{configurationsK}defined_bythe Galois connectionbetween the latticeofsubsetsofG= Aut(L_{f}) and the latticeoffamilies of closed subsets ofthe closurespace<Sf>defined_bytwo

ThisGalois connectiondefines_{anti‐isomorphism}of the lattice of_subgroupsof\mathrm{G} andthe lattice of invariantfamilies of closed subsets of_<Sf>

Theexistence ofthis Galois connection and its definition_giveustools to_{analyzesymmetry}of

configurationsofclosed subsetsinanarbitraryclosure space. Theselection of transformations for the

symmetrysubgroupsis determined_bythe condition of_continuitywith_respecttothe closure_operator. Of _{course, in the} case of a closure space describing Euclidean _geometry, the continuity of

transformationsmeans _isometry,i.e._preservation ofEuclidean distance. But there is_nothingin this formalism which_requires anyparticularform of coordinatization. Allweneed is the restriction to

transformationsfor which action of closureoperatorisinvariant.

Now, we can observe that the restriction of the _{symmetric group of all permutations} to the

specific subgroup correspondingto_symmetriesofsome_{type (in}thecaseof Euclidean geometry, the

restrictiontothegroupofisometries)canbedetermined_bythe_completelattice of_{\mathrm{L}_{\mathrm{f}}}closedsubsets. Sincewe areconcerned hereonlywiththerestrictionof the group oftransformationsto_{the group of}

\mathrm{f}_{‐authomorphisms,}we canthink in_purelylattice theoreticterms. Wecancallthislattice a_“logicfor

symmetry”.Thus,establishingofthe Galoisconmection_requiresthechoice of the_logicfor_symmetry. After Galois connection is _established, we can _proceed to the _study of_symmetries of_particular

configurations.

4. Meta‐Closure

_Space,

its

_{Logic &Meta‐Symmetry}

The most _striking feature ofthe _theory of closure _{spaces is that it is “autological”.} The

descriptionof all_{possibleclosure operators}on agivenset\mathrm{S} canbe achievedbyonespecificclosure

operatoronthepowersetofS.Thisclosureoperatorextends anyfamily fflof subsets of\mathrm{S}tothe least Moore_family \mathscr{M}_including \mathscr{R}, but ofcoursemanydifferent families canbe extended tothe same

\mathrm{f}u\mathrm{n}\mathrm{i}\mathrm{l}\mathrm{y}_{ $\epsilon$ \mathrm{f}}\mathrm{a}.

Definition 4.1 Wecandefinethisclosure_operatorfon

f=\wp(S)

by:

V\ovalbox{\tt\small REJECT}\subseteq F:f(\ovalbox{\tt\small REJECT})=

íB\subseteqS:_{\Re\subseteq ae:B=\cap \mathfrak{C}J}. The powersetequippedwith this closure

operatorcanbe calledameta‐closure space.

The factthat oneclosure_operatoronthe_powerset2^{\mathrm{s}}of\mathrm{S}describes all closureoperators on \mathrm{S}

wasknown from the_beginningof the studies ofclosure_{spaces, but} the_propertiesof this operator

started to be _explored in more _{systematic way relatively recently} and _mainly in the context of combinatorics _(finiteclosure_spaces)which in this case_{significantly}limits the _generalityof results. This has tobe_carefullyconsidered asthereare someresults in literaturepresentedwithoutexplicit

assumptionof the finitness of thesetsarefalse in the infinitecase.[8]

Oneoftheresults of_Caspard&_{Monjardet [9,10]}which_actuallycan_easilybe extendedtoinfinite

caseisthat the lattice of closedsubsets_{L_{f}} is atomistic_(i.e.everynon‐zeroelement_{\mathrm{o}\mathrm{f}L_{f}}is_ajoinof

theatomsbelow_it). Atoms_(i.e.minimalnon‐zeroelements ofthe_lattice)in_{L_{f}}aredefined_byvery

simpleMoorefamilies _{\{\mathrm{A}, \mathrm{S}\}} foreach of proper subsets A of\mathrm{S} _(if\mathrm{A}=\mathrm{S},then theMoorefamily defines the least element of_{L_{f}}orf(\emptyset)). Itis _surprising that the _{following property of}the closure

operatorfwiththe well‐known _{importantconsequences for the latticeL_{f}}of closed subsetsof_fwas

apparentlynever_recognized.

Theorem4.1

_{Let<2^{s},f>is}

_definedforanarbitrarysetS_by:

\nabla \mathscr{R}\subseteq 2^{s}:f(\mathscr{R})=

ỉB\subseteqS. $\Xi$ \mathfrak{C}\underline{c=}\mathscr{X}:B=\cap \mathfrak{C}J. Thenfsatisfiestheanti‐exchangeproperty:

Proof:Let..ttbeany Moorefamilyon\mathrm{S} and\mathrm{A}, \mathrm{B}\subseteq \mathrm{S}, \mathrm{A}\neq \mathrm{B}, \mathrm{A},\mathrm{B}\not\in \mathscr{M}and \exists ã\subseteq\ovalbox{\tt\small REJECT}:\mathrm{A}=\cap \mathscr{X}\cap \mathrm{B} and\exists \mathfrak{C}\subseteq\ovalbox{\tt\small REJECT}:\mathrm{B}=\cap \mathfrak{C}\cap \mathrm{A}.Then A\subseteqB&B\subseteqA,i.e.\mathrm{A}=\mathrm{B}

,contradiction.

Corollary4.2. Meta‐closure closure space<2^{s},f>isa convex_geometry,i.e.the latticeofclosed

subsets is meet‐distributive.

Definition 4.2_Let<S,f>beaclosure_{space. Subsets AofS satisfying}the condition:

Vx\in A: x\not\in_f(A\backslash _íxJ)arecalledf‐independent. Thefamily ofall independentsubsets is_{represented by}

the_{symbolf‐Ind.}

Subsets A_{ofS satisfying}the condition:_f(A)=Sare_{calledfgenerating}S.The_{family ofall}

fgeneratingsubsetsis_{represented byf}Gen.

AsubsetA_ofSiscalled_{anf‐base (orjust base) iff\in f}‐Ind_\mathrm{A}f‐Gen._{Obviously everyf‐base}isthe

same asminimalgeneratingsubset. Notall_{closure spaces}havebases!

Remark 4.3 Inthe fmitecase(i.e.forafiniteset _S),convex_geometry

<2^{\mathrm{s}}p

alwayshasabasis

andmoreoverthis basis is_unique.This makes the_studyof the group ofauthomorphismsof

_{<2^{\mathrm{s}},f>}

relatively simple, as itcan becarried outinterms of the groupofpernutationsofitsuniquebase. Howeverthis isnot truewhen \mathrm{S} is infinite.

Theorem 4.4_lf<Sf>isa convex_geometryandthesetS isinfinite, then_{foreveryinfinite}andco‐

fmitesubset_BofSthereexistsanf‐closedfamily ofsubsets \mathscr{B}_{B}ofS,such that thereisnominimal

subfamily31 ofãBsatisjỳingf(\mathscr{B})=f(\mathscr{R}_{B}).

Proof: Consider \mathscr{B}_{\mathrm{B}}the_principal filter of B. Itcan be shown that it does nothaveaminimal

generatingsubset.

Corollary4.5_IfSis_infinite,

_{then<2^{s},\ovalbox{\tt\small REJECT}>does}

nothaveabase.

This_Corollaryis consistentwith the fact that for theinfiniteset

\mathrm{S}<2^{\mathrm{s}}p

is_{“essentially”}infinite. Definition 4.3 Wecallaclosurespace<S,f> offinitecharacterif

[fC) VA\subseteq SVX _{$\epsilon$ S.\cdot x\in f(A)\supset $\Xi$ B\in Fin(A):x\in f(B)}.

Ofcourseall closure_{spaces with finite \mathrm{S}} are of finite character. Infiniteclosurespaces of finite character retain manycharacteristics of finiteclosure spaces. In absence of finitecharacter,typically

most ofthe results for finite closure spaces cannotberecovered. This is_{unfortunately} thecaseof meta‐closurespace.

Proposition4.ó_IfSis_infinite,

then<2^{s},f>is

not_offinitecharacter_ƯC).

Nowwe can seethat while formeta‐closure_spacesonfinitesetsthereare_{many results}waitingin

the literature of finiteconvexgeometries,little isknown aboutmoregeneralcases.

Finally,wecan observe thatthere is another_exampleofaclosure spaceonthe powersetof \mathrm{S}of

specialinterest.

Definition 4.4 A_binaryrelation Ton asetS isa weak toleranceifit is_symmetricand_satisf $\iota$ es

the condition: _{\mathrm{h}\in S.\cdot[xr_{X}\Rightarrow \emptyset\prime\in S.\cdot x $\Gamma$ y]}.Everyweak tolerancewhichisreflexive (Vx\in S:xTx)is

Proposition 4.7There is abijective correspondencebetween weak tolerancerelationsonsetS which_{generalize equivalence}relations_extendingthemtoa_{generalconceptofsimilarity}and closed

subsets_ofthe closure_operatoronthe powerset_{ofS_{2}} i.e.closure_space

_<2^{s},f>

_{defined by:} vã\subseteqP: f(\ovalbox{\tt\small REJECT})=jB\underline{\mathrm{c}}S: \mathrm{h}y $\epsilon$ B $\Xi$ 4 $\epsilon$ã:_{íx yJ}\subseteqAÌ.

OpenProblem:This article is_{concluded with the openproblem.}We couldseethattwo_particular

closure spacesonthe_powersetof \mathrm{S} define andcharacterizeinone caseall closure spaceson\mathrm{S} in the

othercaseall_binaryrelations_{generalizing equivalence.}Whataretheotherstructuresontheset\mathrm{S}that

aredetermined and characterized_Uyclosureoperatorsdefinedonthe powersetof \mathrm{S} ?

References

[1]Schroeder,M. J._Conceptof_Symmetryin ClosureSpacesasaToolfor Naturalization ofInformation,In

Horiuchi,K._{(Ed.) Algebraic System, Logic, Language}and_ComputerScience.RIMSKokyuroku, Kyoto:

Research_{InstinuteforMathematical}Sciences, KyotoUniversity,2016,No._2008,pp.29‐36. Availableat

http://\mathrm{w}\mathrm{w}\mathrm{w}.kurims.kyoto‐u.ac.jp/\sim \mathrm{k}\mathrm{y}\mathrm{o}\mathrm{d}\mathrm{o}/\mathrm{k}\mathrm{o}\mathrm{k}\mathrm{y}\mathrm{u}\mathrm{r}\mathrm{o}\mathrm{k}\mathrm{u}/\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}/2008.html

[2]Klein, $\Gamma$.C. AComparativeReviewofRecentResearchesinGeometry (Vergleichende Betrachtungenüber

neueregeometrische Forschungen, l872).Haskell,M. W._(Transl.),2008. arXiv:0807.3161vl

[3] Weyl,H._Symmetry.Princeton: Princeton Univ.Press.,1952.

[4]deSaussure, $\Gamma$.CourseinGeneralLinguistics.Transl.Wade Baskin. New York: Columbia Univ.Press.,

2011.

[5] Lévi‐Strauss,C._{StructuralAnthropology.}Transl. Claire JacobsonandBrooke Grundfest_SchoepfNew York:_DoubledayAnchor_Books,1967.

[6] Piaget,J. Lestructuralism. Paris: Presses Universitaires deFrance, 1968;Engltr.Structuralism.NewYork:

BasicBooks,_Harper&Row,1970.

[7]Birkhoff,G.Lattice_Theory, 3^{rd}.ed. American MathematicalSociety ColloquiumPublications,VolXXV,

Providence,R. I.:American Mathematical_Society,1967.

[8]Pfaltz,J.L.Closure Lattices.DiscreteMath., 154_(1996),217‐236.

[9] Caspard,N.&Monjardet,B.The lattices ofclosure_systems,closure_operators,and_{implicationalsystems}on

afiniteset:a_{survey. Discrete}Applied_Mathematics,127_(2003),241‐269.

[10] Caspard,N.&_Monjardet,B.Some lattices of closure_systems,on afiniteset.DiscreteMathematicsand Theoretical_{ComputerScience,}6_(2004),163‐190.

[11]Schroeder,M.J&_Wright,M.H.Tolerance and weak tolerance_relations,J. Combin. Math. and Combin.