Table Of ContentFundamentaInformaticae97(2009)177–198 177
IOSPress
Algebraic Semantics of Similarity-Based Bitten Rough Set Theory
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0 A.Mani∗
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Member,CalcuttaMathematicalSociety
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a 9/1B,JatinBagchiRoad
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Kolkata(Calcutta)-700029,India
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Email: [email protected]
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O Homepage:http://www.logicamani.co.cc
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t
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Abstract. We develop two algebraic semantics for bitten rough set theory ([19]) over similarity
0
spaces and their abstract granular versions. Connectionswith choice based generalized rough se-
.
1
manticsdevelopedin[15]bythepresentauthorandgeneralcoverbasedroughsettheoriesarealso
0
considered.
3
1
:
v
Keywords: Rough Set Theory over Tolerance Approximation Spaces, Bitten Rough Semantics,
i
X GranularRoughSemantics,λ-RoughAlgebras,DiscernibilityinSimilaritySpaces
r
a
1. Introduction
In classical rough set theory, information systems with equivalence relations (or approximation spaces)
arestudied. Thedata tables used have crisp symbolic values (see[16]forexample). Moregeneral vari-
antswithothertypesofrelationsandvalueshavealsobeenextensivelystudiedintheliterature(see[11],
[9],[17],[13],[18]forexample). Themethodsandconcepts, includingthoseofdefinability,approxima-
tion, granular representability of approximations, neighbourhoods and others, need many modifications
for application in these situations. We focus on similarity spaces or tolerance approximation spaces in
thisresearch paper. Tolerance approximation spaces arealso ofinterest inrepresentation ofinexact and
incompleteknowledge (see[17],[21]).
In classical rough set theory, the negative region of a set is the lower approximation of the comple-
mentoftheset. Thisregionisdisjointfromtheupperapproximationofthesetinquestion. Ananalogous
∗Iwouldliketothethanktherefereeforusefulremarksandthereferenceto[21]
178 A.Mani/AlgebraicSemanticsofBittenRoughSets
propertyfailstoholdintoleranceapproximation spaces(TAS).Todealwiththisadifferentsemanticap-
proach (to TAS) involving modified upper approximations is proposed in [19]. These modified upper
approximations are formed from upper approximations by ’biting off’ a part of it to form ’bitten upper
approximations’. Thenew approximations turn out tobedisjoint from the negative region of thesubset
andalsopossess betterproperties.
More specifically, the upper approximation of a subset of a TAS formed by a set of granules (with
respect tothoseonthesubset) isreducedbythedeletion ofthenegativeregionofthesubset toformthe
’bitten upper approximation’ of the subset. This ensures that the new upper approximation is disjoint
from the negative region of the subset. In the formation of approximations explicit constraints are not
imposedonthepossible typeofgranules. Theauthors alsoposetheproblem ofdeveloping analgebraic
semanticsfortheapproach.
In the present paper, we develop two different algebraic semantics for the same. The first of these
actually captures reasoning withinthepowersetofthesetofpossible order-compatible partitions ofthe
set of roughly equivalent objects. We also prove suitable representation theorems in the third section.
Themainresultof[2]isrequired intheproofofthesametheorem.
Between formalizing the interaction of roughly equivalent objects and formalizing the approxima-
tions in a classicalist setting, we prefer the former or a dialectical version thereof ([14]). But whether
this ’rough equivalence’ is to be an equivalence relation or a congruence or something else, is an open
question. We do not abandon the viewpoint that it should at least be an equivalence. Interestingly the
knownalgebraic approaches (like[4])aimedatjusttheapproximations andderivedoperationsinaclas-
sicalist setting (asopposed to thedomain ofroughly equal objects) cannot be directly adapted (at least)
tothepresent context.
Given that it may not be possible to have suitable abstract representation theorems from partial al-
gebras over the set of roughly equivalent objects in the context, we use choice functions in a crucial
(though implicit) way to introduce simplified algebra of the bitten granular semantics SGBA. These
are developed up to the level of some solvable/open problems. Here the motivation is not to integrate
choice functions with rough set theory as in [15], but to build an easier semantics. At the same time
thesemantics isalsomotivatedbypossible connections withchoiceinclusive similarity basedrough set
theory.
In the following section we mention some of the essential notions and outline the bitten approach.
Thefirstofthealgebraicsemanticsforthebittenapproachisdevelopedinthethirdsection. Somerelated
problemsarealsoposedinthesame. Inthefourthsection,apartialalgebraic semanticsoverarelatively
easier semantic domain (with respect to the previous approach) is developed. These semantics are also
applicable toparticular casesofother general rough settheories including thecoverbased ones. Thisis
indicated inthefifthsection.
2. Background
ByaToleranceApproximation Space(TAS),wemeanapairoftheformS = hS, Ti,withSbeingaset
andT atolerancerelationoverit-thesearealsoknownassimilarityortolerancespaces. Somereferences
for extension of classical rough set theory to TAS are [4], [10], [21], [5], [15] and [11]. These theories
differ inthetypes ofgranules, process ofdefinition ofapproximations, semantics andoncomputational
aspects. We mention some important aspects of rough set theory over TAS before proceeding with the
A.Mani/AlgebraicSemanticsofBittenRoughSets 179
bittenproposal in[19].
An approach ([11]) has been to define a new equivalence θ on S via (x, y) ∈ θ if and only if
0 0
dom (x) = dom (y) with dom (z) = ∩{[x] : z ∈ [x] }. This is essentially an unduly cautious
T T T T T
’clearperspective’ approach.
In [10],atheoryofgeneralized roughsetsbasedoncoversofsubsetsofagivensetSisconsidered.
Let S be a set and K = {K }n : n < ∞ be a collection of subsets of it. We will abbreviate subsets
i 1
of natural numbers of the form {1,2,...,n} by N(n). If X ⊆ S, then consider the sets (with K = ∅,
0
K = Sforconvenience):
n+1
(i) Xl1 = {K : K ⊆ X, i ∈ {0, 1, ...,n}}
i i
S
(ii) Xl2 = {∩i∈I(S \Ki) : ∩i∈I(S \ Ki) ⊆ X, I ⊆ N(n+1)}
S
(iii) Xu1 = {∪i∈IKi : X,⊆∪i∈IKi, I ⊆ N(n+1)}
T
(iv) Xu2 = {S \ K : X ⊆ S \ K , i ∈ {0, ...,n}}
i i
T
The pair (Xl1, Xu1) is called a AU-rough set by union, while (Xl2, Xu2) a AI-rough set by inter-
section (in the present author’s notation [13]). In the notation of [10] these are (F∪(X), F∗(X)) and
∗ ∪
(F∩(X), F∗(X))respectively. WewillalsorefertothepairhS, KiasaAUAI-approximation system.
∗ ∩
In the TAS context, let the granules be [x] = {y; (x, y) ∈ T} for each element x, and K be the
T
collection ofsuch sets. Thenl1andu1approximations ofasetAshould begivenby(i)and(iii)above
respectively. FurtherAl = {[x] ; [x] ⊆ A} = Al1 andAu = {[x] ; [x] ∩ A 6= ∅, x ∈ A} 6=
T T T T
S S
Au1.Notethatthegeneralized coverapproach doesnotprescribe anyspecifictypeofgranules.
In[4],thefollowingimprovedapproximations aredefined:
Al∗ = {x; (∃y)(x, y) ∈ T, [y] ⊆ A}
T
Au∗ = {x; (∀y)((x, y) ∈ T −→ [y] ∩ A 6= ∅)}
T
Proposition2.1. ForanysubsetA,Al ⊆ Al∗ ⊆ A ⊆ Au∗ ⊆ Au
Using this approximation, we can define a Brouwerian orthocomplementation on ℘(S), via A# =
{x ∈ S; (∀y ∈ A)(x, y) ∈/ T}. These approximations and possible variants may be seen as a way of
distinguishing betweenobjectsonaheuristic. TheBZandQuasi-BZ-algebraicapproach([5])makeuse
of a preclusivity relation P, which is defined via (a,b) ∈ P if and only if (a, b) ∈/ T. Semantically
thederivedroughoperatorsoflowerandupperapproximation aregeneratedbythepreclusivityoperator
andthecomplementation relation onthepowersetoftheapproximation space oronacollection ofsets
undersuitableconstraintsinamoreabstractsetting. ForanyH ⊆ S,theBrouwerianorthocomplementis
H♯ = {x : (∀y ∈ H)(x, y) ∈ P}. Thenthe operators defined byL (H) = Hc♯♯c and M (H) = H♯♯
♯ ♯
behave like lower and upper approximation operators on ℘(S) and are proper generalizations of the
corresponding notions intheapproximation spacecontext.
SemanticallytheBZ-algebraandvariantsdonotcaptureallthepossiblewaysofarrivingatconcepts
ofdiscernibility oversimilarity spaces. Whilethequasi-BZ lattice doesnotencompass aparadigm shift
relativetheBZ-algebra,theBZMVvariantsaredesignedtocapturefuzzyaspects. Amajorproblemwith
thisapproach isthattheintended semanticdomainisuniformly classicalist andattheobjectleveltoo.
180 A.Mani/AlgebraicSemanticsofBittenRoughSets
In subjective terms reducts are minimal sets of attributes that preserve the quality of classification.
An important problem is in getting good scalable algorithms for the computation of the different types
of reducts (or supersets that are close to them) (see [12, 1]). These depend on the concept of granules
used.
Someothernotions thatwewillusearestatedbelow:
Definition2.1. AdoubleHeytingalgebra L = {L, ∧, ∨, →, ⊖, 0, 1}isanalgebrasatisfying:
• L = {L, ∧, ∨, →, 0, 1}isacompleteatomicHeytingalgebra
• x ⊖ x = 0;x ∨ (x ⊖ y) = x;(x ⊖ y)∨,y = x ∨ y
• (x ∨ y) ⊖ z = (x ⊖ z) ∨ (y ⊖ z)
• z ⊖ (x ∧ y) = (z ⊖ x) ∨ (z ⊖ y)
Definition2.2. By a choice function χ on a set S, we shall mean a function χ : ℘(S) 7−→ S, which
satisfiesallofthefollowing:
• (∀x ∈ S)χ({x}) = x
• (∀A ∈ ℘(S))χ(A) ∈ A
Definition2.3. LetP = hP, <ibeapartially ordered set and ifAisany subset ofP, letits lowerand
upper cone be L(A) = {x; (∀a ∈ A)x ≤ a} and U(A) = {x; (∀a ∈ A)a ≤ x} respectively. A
function λ : ℘(P) 7→ P will be said to be lattice-coherent with < if and only if the condition a ≤ b
thenλ(L(a, b) = a)andλ(U(a, b)) = b.
λ-lattices wereconsidered asageneralization oflattices in[20]. Inthepartially ordered setP above
Ifλisalatticecoherent operation, leta · b = λ(L(a, b))anda + b = λ(U(a, b)),thenthealgebra
Q = hP, ·, +iissaidtobeaλ-lattice. Itcanbeshownthatifapartiallyorderedsethasalattice-coherent
operation thenitcanbeendowedwithaλ-lattice structureandconversely.
Theorem2.1. Inaλ-latticeQ = hP, ·, +iallofthefollowinghold:
• +isanidempotentandcommutativeoperation
• ·isanidempotent andcommutativeoperation
• a · (a + b) = a
• a + (a · b) = a
• a · ((a · b) · c) = (a · b) · c
• a + ((a + b) + c) = (a + b) + c
A.Mani/AlgebraicSemanticsofBittenRoughSets 181
We will use two kinds of equalities in a partial algebra (of a single sort) P = hP, f , ... f ,νi,
1 n
f s being partial or total function symbols and ν being an interpretation of these on the set P. Strictly
i
P
speaking, the interpreted partial function should be written as f , but we will drop the superscript for
i
simplicity. Thebasictheoryofpartialalgebras canbefoundin[3]forexample.
ω∗
Fortwotermssandt,thestrongweakequality isdefinedasfollows: t(x) = s(x)ifandonlyif
(∀x ∈ dom(t) = dom(s)) t(x) = s(x).
ω
Incontrast, theweakequalityisdefinedviat(x) = s(x)ifandonlyif
(∀x ∈ dom(t)∩ dom(s)) t(x) = s(x).
2.1. Bitten Approach
Inthissectionwerecapitulatetheessentialbittenproposalof[19]withsomemodifications. Actuallythe
authorsintroducebitedupperapproximationsinastudyontolerancespaces. Howeverin[21],’bittenap-
proximations’arealsoconsidered. Wewilluniformlyrefertotheseandapproachbytheadjectivebitten.
Thetheoryapparently laysemphasisondesiredmereological properties atthecostofrepresentation.
Let Gr(S) ⊆ ℘(S) be the collection of granules for a TAS defined by some conditions including
Gr(S) = S. AsubsetXisgranularlydefinable ifandonlyif∃B ⊆ Gr(S)X = B. Thecollection
S S
of granularly definable sets shall be denoted by Def (S). The lower and upper approximations of X
Gr
is defined via Gr∗(X) = {A : A ⊆ X, A ∈ Gr(S)} and Gr∗(X) = {A : A ∩ X 6= ∅, A ∈
S S
Gr(S)}. Thepositiveandnegativeregion,definedbyPOSGr(X) = Gr∗(X)andNEGGr(X) = Gr∗(Xc)
respectively,aregranularlydefinable. ButinthisschemeofthingsGr∗(X)∩NEG (X) 6= ∅ispossible.
Gr
Toavoid this,aconcept ofbittenupperapproximation isdefinedviaGr∗(X) = Gr∗(X) \ NEG (X).
b Gr
Relativethistheboundary isgivenby
BNGr(X) = Gr∗b(X) \ Gr∗(X) = S \ (POSGr(X) ∪ NEGGr(X)).
Gr(S)maybetakentobethesetofT-relatedsorthesetofblocksofT orsomethingelse. Forexam-
ple, if Gr(S) is the set of all sets of the form T (T = {y; (x, y) ∈ T, y ∈ S}), then S = Gr(S)
x x
S
andofcourse Gr(S) ⊆ ℘(S). Giventhelatter twoproperties, the upper and lowerapproximations ofa
subsetXarethengivenby:
∗
Gr∗(X) = {Y ∈ Gr(S); Y ⊆ X}, Gr (S) = {Y ∈ Gr(S); Y∩,X 6= ∅}.
[ [
Thebittenupperapproximation issimplyGr∗b(X) = Gr∗(X) \ Gr∗(X). Iftheyarerelatedtoaspecific
tolerance, thenwewillcallthetuplehS, Gr(S), T, Gr∗, Gr∗biabittenapproximation system(BAS).
Theproperties oftheapproximations areasfollows:
182 A.Mani/AlgebraicSemanticsofBittenRoughSets
l1-Property u2-Property
1a.) Gr∗(X) ⊆ X 1b.) X ⊆ Gr∗(X)
b
2a.) (X ⊆ Y −→ Gr∗(X) ⊆ Gr∗(Y)) 2b.) (X ⊆ Y −→ Gr∗(X) ⊆ Gr∗(Y))
b b
3a.) Gr∗(∅) = ∅ 3b.) Gr∗(∅) = ∅
b
4a.) Gr∗(S) = S 4b.) Gr∗b(S) = S
5a.) Gr∗(Gr∗(X)) = Gr∗(X) 5b.) Gr∗b(Gr∗b(X)) = Gr∗b(X)
6a.) Gr∗(X ∩ Y) ⊆ Gr∗(X) ∩ Gr∗(Y) 6b.) Gr∗(X ∩ Y) ⊆ Gr∗(X) ∩ Gr∗(Y)
b b b
7a.) Gr∗(X) ∪ Gr∗(Y) ⊆ Gr∗(X ∪ Y) 7b.) Gr∗(X) ∪ Gr∗(Y) ⊆ Gr∗(X ∪ Y)
b b b
8a.) Gr∗(X) ⊆ Gr∗(Gr∗(X)) 8b.) Gr∗(Gr∗(X)) ⊆) Gr∗(X)
b b b
9a.) (Gr∗(X))c = Gr∗b(Xc) 9b.) (Gr∗b(X))c = Gr∗(Xc)
10A.) X∈ DefGr(S) ←→ X = Gr∗(X)
10B.) X ∈ CrGr(S) ←→ Gr∗(X) = Gr∗b(X)
11A.) X, Y ∈ Def (S) −→ X ∪ Y ∈ (Impliesequalityin7a)
Gr
Def (S)
Gr
11B.) X, Y ∈ Cr (S) −→ X ∩ Y, X ∪ Y ∈ (Impliesequalityin6a, 6b, 7a, 7b, 8a, 8b)
Gr
Cr (S)
Gr
In the samepaper ([19]), the authors pose the problem ofdeveloping analgebraic semantics forthe
approach.
InaTASS,ablock B ⊆ SisamaximalsetthatsatisfiesB2 ⊆ T. In[21],thesearetermedclasses,
while the former is the standard terminology in universal algebra. If H is the collection of all blocks
T
of T, then let A = {∩F : F ⊆ H }. Taking this large collection as the set of granules, the authors
T T
definethethelowerandbittenupperapproximationofaX ⊆ SasXl = {A: A ⊆ X, A ∈ A }and
T
S
Xu = {A: A ∩ X 6= ∅, A ∈ A } \(Xc)l. Onthesetofdefinableobjects∆(S),let
b S T
• X → Y = {A ∈ A ; X ∩ A ⊆ Y}
T
S
• X ⊖Y = {B ∈ A ; X ⊆ Y ∪ A}.
T
T
Thenthefollowingtheorem providesatopological algebraic semantics ([21]):
Theorem2.2. h∆(S), ∩, ∪, →, ⊖, ∅, SiisacompleteatomicdoubleHeytingalgebra.
Howeverextensions ofthe theorem to other types of granules are not known. Noabstract represen-
tationtheoremisalsoprovedintheparticular situation.
3. Semantics for Bitten Rough Set Theory
The concept of granules to be used in the theory is essentially kept open. Many types of granules may
notpermitnicerepresentiontheory. Despitethis,ourfirstsemanticsoverroughlyequivalentobjectsdoes
well. Thisisduetothehigherorderapproach used.
A.Mani/AlgebraicSemanticsofBittenRoughSets 183
Definition3.1. IfSisaTAS,thenover℘(S)let
A ∼ B iff Gr∗(A) = Gr∗(B)and Gr∗b(A) = Gr∗b(B)
The following proposition and theorem basically say the quotient structure (or the set of roughly
equivalentobjects)hasverylittlestructurewithrespecttodesirableproperties ofapartialalgebra. They
are clearly deficient from the rough perspective as we do not have proper conjunction and disjunction
operations. But’biting’ mayactually makethepartialoperations totalinmanycontexts.
Proposition3.1. ∼ is an equivalence on the power set ℘(S). Moreover the following operations and
relationson℘(S)| ∼arewell-defined:
• L([A]) = [Gr∗(A)]
• ¬[A] = [Ac]ifdefined
• (cid:7)([A]) = [Gr∗(A)]
b
• [A] ≤ [B] if and only if, for any A ∈ [A] and B ∈ [B] Gr∗(A) ⊆ Gr∗(B) and Gr∗b(A) ⊆
Gr∗(B).
b
• [A] ⋓ [B] = [C] if and only if [C] is the infimum of [A] and [B] w.r.t ≤. It shall be taken to be
undefinedinothercases.
• [A] ⋒ [B] = [C]ifandonly if[C]isthesupremum of[A]and[B]w.r.t≤. Itshallbetaken tobe
undefinedinothercases.
Moreover≤isapartialorderon℘(S)| ∼thatispartiallycompatible withLonthecrispelements.
Proof:
• ForanyA ∈ ℘(S)andanyB ∈ [A],Gr∗(Gr∗(A) = Gr∗(A) = Gr∗(B)andGr∗(Gr∗(A)) =
b
Gr∗(Gr∗(B)). ThisprovesthatLiswell-defined.
b
• Forany A ∈ ℘(S) and any B, E ∈ [A], (Gr∗(B))c = Gr∗b(Bc) = (Gr∗(E))c = Gr∗b(Ec) and
(Gr∗(B))c = Gr∗(Bc) = (Gr∗(E))c = Gr∗(Ec).
b b
Wecanverifytherestinasimilarway. ⊔⊓
Proposition3.2. Intheabovecontext,
([A] ≤ [B] −→ ¬[B] ≤ ¬[A])
Theorem3.1. Allofthefollowing holdin℘(S)| ∼(weassume thatunaryoperators bind morestrongly
thanbinaryones):
1. ((cid:7)x ⋒ (cid:7)y = a, (cid:7)(x ⋒ y) = b −→ (cid:7)x ⋒ (cid:7)y ≤ (cid:7)(x ⋒ y))
2. x ⋓ (cid:7)x = x
184 A.Mani/AlgebraicSemanticsofBittenRoughSets
3. (cid:7)(cid:7)x = (cid:7)x
4. ((cid:7)x ⋓ (cid:7)y = a, (cid:7)(x ⋓ y) = b −→ (cid:7)(x ⋓ y) ≤ (cid:7)x ⋓ (cid:7)y)
5. (Lx ⋒ Ly = a, L(x ⋒ y) = b −→ Lx ⋒ Ly ≤ L(x ⋒ y))
6. Lx ⋓ (cid:7)Lx = Lx
7. (cid:7)x ⋒ L(cid:7)x = (cid:7)x
8. ¬(cid:7)x = L¬x
9. ¬Lx = (cid:7)¬x
Proof:
1. If A ∈ x and B ∈ y, then (cid:7)x = [Gr∗(A)], (cid:7)y = [Gr∗(B)]. Given the existence of the terms
b b
in the premise, we can assume that there exists C ∈ [Gr∗(A)] ⋒ [Gr∗(B)]and E ∈ (cid:7)(x ⋒ y).
b b
Gr∗(C) ⊆ Gr∗(E) and Gr∗b(C) ∈ Gr∗b(E). So, given the existence of the terms in the premise,
wehave(cid:7)x ⋒ (cid:7)y ≤ (cid:7)(x ⋒ y).
2. IfA ∈ x,then(cid:7)x = [Gr∗(A)]. AsA ⊆ Gr∗(A),sox ⋓ (cid:7)x = x.
b b
3. IfA ∈ x,then(cid:7)(cid:7)x = (cid:7)[Gr∗(A)] = [Gr∗(Gr∗(A))] = [Gr∗(A)] = (cid:7)x.
b b b b
4. The proof of ((cid:7)x ⋓ (cid:7)y = a, (cid:7)(x ⋓ y) −→ (cid:7)(x ⋓ y) ≤ (cid:7)x ⋓ (cid:7)y) is similar to that of the
firstitem.
5. Given the existence of the terms in the premise, if A ∈ x and B ∈ y, then Lx = L[A] =
[Gr∗(A)]andLy = [Gr∗(B)]. IfC ∈ Lx ⋒ LyandE ∈ L(x ⋒ y),thenGr∗(C) ⊆ Gr∗(A) ∪
Gr∗(B),Gr∗(A) ∪ GrB ⊆ Gr∗(E)andGr∗b(C) ⊆ Gr∗b(E). So,Lx ⋒ Ly ≤ L(x ⋒ y).
6. IfA ∈ x,thenLx = L[A] = [Gr∗(A)]and(cid:7)Lx = [Gr∗Gr∗(A)]. ButGr∗(A) ⊆ Gr∗Gr∗(A).
b b
SoLx ⋓ (cid:7)Lx = Lx.
7. If A ∈ x, then (cid:7)x = (cid:7)[A] = [Gr∗(A)] and L(cid:7)x = [Gr∗Gr∗(A)]. But Gr∗Gr∗(A) ⊆
b b b
Gr∗(A). So,(cid:7)x ⋒ L(cid:7)x = (cid:7)x.
b
8. If A ∈ x, then ¬(cid:7)x = ¬(cid:7)[A] = ¬[Gr∗b(A)]. ¬[Gr∗b(A)] = [(Gr∗b(A))c],= [Gr∗(Ac)] =
Lneg[A]. So,¬(cid:7)x = L¬x.
9. IfA ∈ x,then ¬Lx = ¬[Gr∗(A)] = [(Gr∗(A))c]. But[(Gr∗(A))c] = [Gr∗b(Ac)]. Thisyields
¬Lx = (cid:7)¬x.
⊔⊓
Asemanticsusingthepartialalgebraovertheassociatedquotientmaybedifficultbecauseofaxioma-
tisability issues. So weuse a higher order approach, taking care not to introduce extraneous properties.
Eventually the constructed algebra ends up with three partial orders. In the following construction the
useofamodifiedconcept offilterssimplifiestheeventualrepresentation theorem.
A.Mani/AlgebraicSemanticsofBittenRoughSets 185
If ℘(S)| ∼= K, then let K∗ = {f : f : K 7→ I is isotone}, I being the the totally ordered two
elementset{0, 1}under0 < 1. ForanyA ⊂ K∗,asubsetFisanA-idealifandonlyif
F = x−1{0}.
\
x∈A
DuallyFisanA-filter ifandonlyif
F = x−1{1}.
\
x∈A
AllA-idealsareorderideals(w.r.ttheinducedorderonK∗),buttheconverseneednothold. A ⊂ K∗
issaidtobefullif∀p (cid:2) q∃x ∈ Ax(p) = 1, x(q) = 0. Aissaidtobeseparating ifforanydisjoint
idealIandfilterF,thereexistsax ∈ Asuchthatx = 0andx = 1
|I |F
Lemma3.1. IfAisaseparating subset ofK∗ and(∀p, q ∈ K)(q (cid:2) p −→ p ↓ ∩q ↑ = ∅), then
A A
Aisfull.
If p ∈ K, then let UP(p) = {x : x(p) = 1} and LO(p) = {x : x(p) = 0}, then we can define
twoclosureoperators C , C via
1 2
C1 = clos{UP(p)}p∈K
(aC1-closed setisanintersection ofelementsof{UP(p)}p∈K)and
C2 = clos{LO(p)}p∈K
Note that elements of UP(p)}p∈K are in fact C1O2-sets (that is sets that are open w.r.t the second
closure system and closed with respect to the first). The set of such sets on a system (S, C , C ),
1 2
will be denoted by C O (S, C , C ). The associated closure operators will be denoted by cl and cl
1 2 1 2 1 2
respectively.
On any subset A ⊆ K∗, we can define closure operators via C (X) = C ∩ A, with associated
iA i
closuresystemsUP (p) = UP(p) ∩ AandLO (p) = LO(p) ∩ Arespectively. Itcanbeseenthat,
A A
inthesituation, C = clos{UP(p)} ∈ PandC = clos{LO(p)} ∈ P.
1A p 2A p
Theorem3.2. If A ⊆ K∗ and σ : K 7→ C O (A, C , C ) is a map defined by σ(p) = UP(p)
1 2 1A 2A
then
1. σisisotone
2. IfAisfull,thenσisinjective
3. IfAisseparating, thenσissurjective.
Proof:
Thistheorem andthefollowingtheorem areprovedforanarbitrarypartially orderedsetKin[2]. ⊔⊓
Theorem3.3. IfAisafulland separating subset ofK∗, then K =∼ C O (A, C , C ). Inparticular,
1 2 1A 2A
K =∼ C O (K∗, C , C )(asK∗ isafullandseparating set). EvenK∗\{0,1}isafullandseparating set.
1 2 1 2
186 A.Mani/AlgebraicSemanticsofBittenRoughSets
We state the following for clarifying the connection with the more common way of using closure
operators.
Proposition 3.3. A ∈ C O (K∗, C , C )ifandonlyifA ⊆ K∗ and(∃B, E ⊆ K∗)A = cl B, A =
1 2 1 2 1
K∗ \ cl (E).
2
K∗ can be interpreted asthe set of partitions ofthe set of roughly equivalent elements into an upper
and lower region subject to the new order being a coarsening of the original order. Theimportant thing
isthatthisrestrictedglobalobjectiscompatiblewiththe’naturalglobal’versionsoftheotheroperations
andleadstoapropersemantics. Weshowthisinwhatfollows.
Definition3.2. OnK∗,thefollowingglobaloperations (relativethoseonK)canbedefined:
• If A ∈ C O (K∗, C , C ), then L(A) = L(i(A)), i being the canonical identity map from
1 2 1 2
C O (K∗, C , C )onto℘K.
1 2 1 2
• A ∨ B = cl (A ∪ B) (if the RHS is also open with respect to the second closure system), ∪
1
beingtheunionoperation overK∗
• A ∧ B = cl (A ∩ B) (if the RHS is also open with respect to the second closure system), ∩
1
beingtheintersection operation overK∗
• IfA ∈ C O (K∗, C , C )then♦A = (cid:7)i(A)
1 2 1 2
• IfA ∈ C O (K∗, C , C )then∼ A = ¬i(A)
1 2 1 2
• cl , cl canbetakenasunaryoperators onK∗
1 2
• 1, ⊥, ⊤,shallbe0-aryoperations withinterpreted valuescorresponding toK, ∅, K∗ respectively
• IfA, B ∈ C O (K∗, C , C )thenA ⊓ B,= i(A) ⋓ i(B)
1 2 1 2
• IfA, B ∈ C O (K∗, C , C )thenA ⊔ B,= i(A) ⋒ i(B)
1 2 1 2
Proposition 3.4. A partial operation is well defined if it is either uniquely defined or not ambiguously
defined. Inthissensealloftheoperations andpartialoperations arewell-defined.
Proof:
Mostoftheverificationisdirect. ⊔⊓
Definition3.3. Analgebra oftheform
W = ℘(K∗), ∨, ∧, ⊓, ⊔, ∪, ∩, c, ,cl , cl , ∼, L, ♦, ⊥, 1, ⊤
D 1 2 E
of type (2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0) in which the operations are as in the above definition will
becalled aconcrete bitten algebra. ∪, ∩, c arethe union, intersection and complementation operations
respectively on℘(K∗). Weuseξ(x)asanabbreviation forcl x = x, cl xc = xc. Furtherξ(x, y, ...)
1 2
shall mean ξ(x), ξ(y) and so on. If S is the original TAS, then we will denote its associated concrete
bittenalgebrabyBite(S).
