Algebraic Approach to Directed Rough Sets
aa r X i v : . [ c s . L O ] A p r Algebraic Approach to Directed Rough Sets
A Mani ∗ HBCSE, Tata Institute of Fundamental Research9/1B, Jatin Bagchi RoadKolkata (Calcutta)-700029, IndiaHomepage:
Orchid: https: // orcid. org/ 0000-0002-0880-1035
S´andor Radeleczki
Institute of Mathematics, University of Miskolc3515 Miskolc-Ergyetemv´arosMiskolc, HungaryHomepage: ∗ Corresponding author
Email addresses: [email protected] (A Mani), [email protected] (S´andorRadeleczki)
Preprint submitted to Elsevier lgebraic Approach to Directed Rough Sets
A Mani ∗ HBCSE, Tata Institute of Fundamental Research9/1B, Jatin Bagchi RoadKolkata (Calcutta)-700029, IndiaHomepage:
Orchid: https: // orcid. org/ 0000-0002-0880-1035
S´andor Radeleczki
Institute of Mathematics, University of Miskolc3515 Miskolc-Ergyetemv´arosMiskolc, HungaryHomepage:
Abstract
In relational approach to general rough sets, ideas of directed relations are sup-plemented with additional conditions for multiple algebraic approaches in thisresearch paper. The relations are also specialized to representations of generalparthood that are upper-directed, reflexive and antisymmetric for a better be-haved groupoidal semantics over the set of roughly equivalent objects by thefirst author. Another distinct algebraic semantics over the set of approxima-tions, and a new knowledge interpretation are also invented in this researchby her. Because of minimal conditions imposed on the relations, neighborhoodgranulations are used in the construction of all approximations (granular andpointwise). Necessary and sufficient conditions for the lattice of local upperapproximations to be completely distributive are proved by the second author.These results are related to formal concept analysis. Applications to studentcentered learning and decision making are also outlined.
Keywords:
General Approximation Spaces, Up-Directed Relations, Nontransitive Parthoods, Granular Rough Semantics, Groupoidal AlgebraicSemantics, Malcev Varieties, Directed Rough Sets.
1. Introduction
In relational approach to general rough sets various granular, pointwise orabstract approximations are defined, and rough objects of various kinds are ∗ Corresponding author
Email addresses: [email protected] (A Mani), [email protected] (S´andorRadeleczki)
Preprint submitted to Elsevier tudied [48, 38, 56, 4, 55, 6]. These approximations may be derived from infor-mation tables or may be abstracted from data relating to human (or machine)reasoning. A general approximation space is a pair of the form S = h S, R i with S being a set and R being a binary relation ( S and S will be used inter-changeably throughout this paper ). Often, approximations of subsets of S aregenerated from these and studied at different levels of abstraction in theoreticalapproaches to rough sets. It is also of interest to understand ideas of closenessof other relations to the relation R – this includes the problem of computingreducts of a type.Parthood (part of) relations [2, 22, 33, 64, 59] of different kinds play a ma-jor role in human reasoning over multiple perspectives. They may be betweenobjects and properties, or collections of objects or properties, or between con-cepts. For example, one can assert that red is part of maroon or that red is asubstantial part of pink or that redness is part of pinkness – a key feature ofsuch relations is the connection with ontology [2, 40].Rough Y-systems and granular operator spaces, introduced and studied ex-tensively by the first author [48, 49, 43, 38], are essentially higher order ab-stract approaches in general rough sets in which the primitives are ideas ofapproximations, parthood, and granularity. In the literature on mereology[66, 67, 64, 22, 40, 61], it is argued that most ideas of binary part of relationsin human reasoning are at least antisymmetric and reflexive. A major reasonfor not requiring transitivity of the parthood relation is because of the functionalreasons that lead to its failure (see [61]), and to accommodate apparent part-hood [49]. In the context of approximate reasoning interjected with subjectiveor pseudo-quantitative degrees, transitivity is again not common. The role ofsuch parthoods in higher order approaches are distinctly different from theirs inlower order approaches – specifically, general approximation spaces of the form S mentioned above with R being a parthood relation are also of interest. Giventwo concepts ( A and B say), it often happens that there are concepts like E of which A and B are part of. This is, loosely speaking, the idea of the part-hood relation being up-directed. In approximate reasoning with vague objectsor concepts, this property is more common than the existence of supremums (ina general sense).From a purely mathematical perspective, the property of up-directedness(also referred to as directedness) of partial orders and semilattice orders is widelyused in literature, it has also been used in studying concepts of ideals of binaryrelations (see [15, 46]. But the groupoidal approach of [10, 9] is not known inearlier work.In this research general approximation spaces, in which the relation R is anup-directed parthood relation, are studied in detail by the authors. It is alsoshown that the algebraic semantics of such spaces is very distinct from thosein which R is a directed or partial or quasi-order. More specifically, two of thealgebraic models are groupoids with additional operations (these correspond togranular approximations), while the third is based on completely distributivelattices (this corresponds to mixed local approximations).In the following section, some of the essential background is mentioned. In3he third section, directed rough sets are introduced and basic results are proved.Illustrative examples are invented in the following section. Algebraic semanticson the power set and subsets thereof are explored in depth by the first author inthe fifth section. In the sixth section, groupoidal semantics over quotients areinvestigated by the first author. Algebraic semantics of local approximations,connections with formal concept analysis and induced groupoids on subsetsof the power set are explored in the following section by the second author.Subsequently knowledge interpretation over the three semantic approaches isdiscussed and an application to student-centred learning is invented in the nextsection. Further directions are provided in the ninth section.
2. Some Background
The concept of information can also be defined in many different and non-equivalent ways. In the first author’s view anything that alters or has the po-tential to alter a given context in a significant positive way is information . Inthe contexts of general rough sets, the concept of information must have thefollowing properties: • information must have the potential to alter supervenience relations in thecontexts (A set of properties Q supervene on another set of properties T if there exists no two objects that differ on Q without differing on T ), • information must be formalizable and • information must generate concepts of roughly similar collections of prop-erties or objects.The above can be read as a minimal set of desirable properties. In practice,additional assumptions are common in all approaches and the above is abouta minimalism. This has been indicated to suggest that comparisons may workwell when ontologies are justified.The concept of an information system or table is not essential for obtaininga granular operator space or higher order variants thereof. As explained in[48, 51, 49], in human reasoning contexts it often happens that they arise fromsuch tables.Information tables (also referred to as descriptive systems or knowledge rep-resentation system in the literature) are basically representations of structureddata tables. Often these are referred to as information systems in the roughset literature, while it refers to an integrated heterogeneous system that hascomponents for collecting, storing and processing data in AI, computer scienceand ML. From a mathematical point of view, the latter can be described usingheterogeneous partial algebraic systems. In rough set contexts, this generalityhas not been exploited as of this writing. It is therefore suggested in [13] toavoid plural meanings for the same term.4n information table I , is a relational system of the form I = h O , A , { V a : a ∈ A } , { f a : a ∈ A }i with O , A and V a being respectively sets of Objects , Attributes and
Values re-spectively. f a : O ℘ ( V a ) being the valuation map associated with attribute a ∈ A . Values may also be denoted by the binary function ν : A × O ℘ ( V ) defined by for any a ∈ A and x ∈ O , ν ( a, x ) = f a ( x ) .An information table is deterministic (or complete) if ( ∀ a ∈ At )( ∀ x ∈ O ) f a ( x ) is a singleton . It is said to be indeterministic (or incomplete) if it is not deterministic that is ( ∃ a ∈ At )( ∃ x ∈ O ) f a ( x ) is not a singleton . Relations may be derived from information tables by way of conditions ofthe following form: For x, w ∈ O and B ⊆ A , σxw if and only if ( Q a, b ∈ B ) Φ( ν ( a, x ) , ν ( b, w ) , ) for some quantifier Q and formula Φ . The relationalsystem S = h S, σ i (with S = A ) is said to be a general approximation space .This universal feature of the definition of relations in general approximationspaces do not hold always in human reasoning contexts.In particular if σ is defined by the condition Equation 1, then σ is an equiv-alence relation and S is referred to as an approximation space . σxw if and only if ( ∀ a ∈ B ) ν ( a, x ) = ν ( a, w ) (1) In this research, prefix or Polish notation is uniformly preferred for relationsand functions defined on a set. So instances of a relation σ are denoted by σab instead of aσb or ( a, b ) ∈ σ . If-then relations (or logical implications) in a modelare written in infix form with −→ . In Equation 1, if and only if is used becausethe definition is not done in an obvious model.
This research is relevant to all theoretical approaches to rough sets includingthe contamination avoidance based axiomatic granular approach due to the firstauthor [38, 48, 44, 49, 46, 47, 45, 51], modal approaches (for the pointwise ap-proximations) [55, 56], and other abstract approaches [25, 5, 16]. For additionalclarifications on the context, readers may refer to the references suggested.In fact, the specific approximation spaces studied in this paper can be usedto generate a number of High granular operator spaces and variants thereofstudied by the first author [47, 49, 45, 48, 51]. These will be taken up in aseparate paper.
For basics of partial algebras, the reader is referred to [3, 34].5 efinition 1. A partial algebra P is a tuple of the form h P , f , f , . . . , f n , ( r , . . . , r n ) i with P being a set, f i ’s being partial function symbols of arity r i . The interpre-tation of f i on the set P should be denoted by f Pi , but the superscript will bedropped in this paper as the application contexts are simple enough. If predicatesymbols enter into the signature, then P is termed a partial algebraic system . In this paragraph the terms are not interpreted. For two terms s, t , s ω = t shall mean, if both sides are defined then the two terms are equal (the quan-tification is implicit). ω = is the same as the existence equality (also written as e = ) in the present paper. s ω ∗ = t shall mean if either side is defined, then theother is and the two sides are equal (the quantification is implicit). Note thatthe latter equality can be defined in terms of the former as ( s ω = s −→ s ω = t ) & ( t ω = t −→ s ω = t ) Various kinds of morphisms can be defined between two partial algebras orpartial algebraic systems of the same or even different types. If X = h X, f , f , . . . , f n i and W = h W , g , g , . . . , g n i are two partial algebras of the same type, then a map ϕ : X W is said tobe a • morphism if for each i , ( ∀ ( x , . . . x k ) ∈ dom ( f i )) ϕ ( f i ( x , . . . , x k )) = g i ( ϕ ( x ) , . . . , ϕ ( x k )) • closed morphism , if it is a morphism and the existence of g i ( ϕ ( x ) , . . . , ϕ ( x k )) implies the existence of f i ( x , . . . , x k ) .Usually it is more convenient to work with closed morphisms. The reader may refer to [20, 21, 24, 14] for lattice theoretical concepts. Someare stated below for convenience.In a complete lattice L , an element x = 0 is said to be completely join-irreducible if and only if ( ∀ K ⊆ L )( _ K = x −→ ( ∃ z ∈ K ) z = x ) The set of join-irreducible elements of L will be denoted by CJ ( L ) . The lattice L is said to be CJ-generated or spatial if and only if every element of L isrepresented as a join of some elements of CJ ( L ) .A lattice in which every descending chain is finite is said to satisfy thedescending chain condition (DCC). In particular, if a complete lattice satisfiesDCC, then it is necessarily spatial. 6 .4. Groupoids and Binary Relations Under certain conditions, groupoidal operations can correspond to binaryrelations on a set. More generally, all binary relations can be read as partialgroupoidal operations in a perspective ([10]) and therefore all general approxi-mation spaces can be transformed into partial groupoids. The connections willbe explored by the first author in a forthcoming paper. In this subsection knownresults for groupoids are stated for convenience.Let S = h S, R i be a relational system, define U R ( a, b ) = { x : Rax & Rbx } S is said to be up-directed if and only if U R ( a, b ) is never empty. That is, ( ∀ a, b ) ¬ U R ( a, b ) = ∅ (up-directed) Definition 2.
If a relational system is up-directed, then it corresponds to anumber of groupoids defined by ( ∀ a, b ) ab = ( b if Rabc c ∈ U R ( a, b ) & ¬ Rab (updg)These are studied in [8]. The collection of groupoids satisfying the above con-dition will be denoted by B ( S ) and an arbitrary element of it will be denotedby B ( S ) . It may be noted that up-directed sets (partially ordered sets that areup-directed) and related constructions are well-known in topology and algebra,but the specific association of up-directedness mentioned is new. Join directoids [31] are groupoids of the form S that admit of a partialorder relation ≤ that satisfies ( ∀ a, b ) a, b ≤ ab and if max { a, b } exists then ab =max { a, b } . Clearly the results of [8] may also be read as a severe generalizationof known results for join directoids. It may also be noted that lambda lattices(that are commutative join and meet directoids) are related special cases (see[63, 36]). Theorem 1 ([8]).
For a groupoid A , the following are equivalent • A up-directed reflexive relational system S corresponds to A • A satisfies the equations aa = a & a ( ab ) = b ( ab ) = ab Definition 3. If A is a groupoid, then two relational systems corresponding toit are ℜ ( A ) = h A, R A i and ℜ ∗ ( A ) = h A, R ∗ A i with R A = { ( a, b ) : ab = b } R ∗ A = [ { ( a, ab ) , ( b, ab ) } heorem 2 ([8]). • If A is a groupoid then ℜ ∗ ( A ) is up-directed. • If a groupoid A | = a ( ab ) = b ( ab ) = ab then ℜ ( A ) = ℜ ∗ ( A ) . • If S is an up-directed relational system then ℜ (( B )( S )) = S . Theorem 3 ([8]). If S = h S, R i is a up-directed relational system, then all ofthe following hold: • R is reflexive if and only if B ( S ) | = aa = a . • R is symmetric if and only if B ( S ) | = ( ab ) a = a . • R is transitive if and only if B ( S ) | = a (( ab ) c ) = ( ab ) c . • If B ( S ) | = ab = ba then R is antisymmetric. • If B ( S ) | = ( ab ) a = ab then R is antisymmetric. • If B ( S ) | = ( ab ) c = a ( bc ) then R is transitive. Morphisms between up-directed relational systems are preserved by corre-sponding groupoids. A relational morphism (as in [35]) from a relational system S = h S, R i to another K = h K, Q i is a map f : S K that satisfies ( ∀ a, b ) ( Rab −→ Qf ( a ) f ( b )) .f is said to be strong if it satisfies ( ∀ c, e ∈ Q )( ∃ a, b ∈ S ) Qf ( a ) f ( b ) & f ( a ) = c, & f ( b ) = e It should be noted that up-directedness is not essential for a relation to berepresented by groupoidal operations. The following construction that differsin part from the above strategy can be used for partially ordered sets as well,and has been used by the first author in [42, 39] in the context of knowledgegenerated by approximation spaces. The method relates to earlier algebraicresults including [30, 29, 32, 17]. The groupoidal perspective can be extendedfor quasi ordered sets.If S = h S, R i is an approximation space, then define (for any a, b ∈ S ) a · b = ( a, if Rabb, if ¬ Rab (2)Relative to this operation, the following theorem (see [30]) holds:8 heorem 4. h S, ·i is a groupoid that satisfies the following axioms (braces areomitted under the assumption that the binding is to the left, e.g. ’ abc ’ is thesame as ’ ( ab ) c ’): xx = x (E1) x ( az ) = ( xa )( xz ) (E2) xax = x (E3) azxauz = auz (E4) u ( azxa ) z = uaz (E5) Theorem 5.
The following are consequences of the defining equations of E (from E1,E2,E3 ): x ( ax ) = x ; x ( xa ) = xa ; ( xa ) a = xax ( xaz ) = x ( az ); ( xz )( az ) = xz ; ( xa )( zx ) = xazxxazxa = xa ; xazaz = xaz ; xcazaxa = xaza ( xazx )( za ) = x ( za ); x ( az ) a = xaza ; ( xaz )( ax ) = ( xza )( zx ); xazxz = xzaz. ( ∀ x )( ex = ea −→ x = a ) ≡ ( ∀ x ) xe = e This purpose of this list is to help with the terminology relating to generalrough sets (and also high granular operator spaces [48, 51]). • Crisp Object : That which has been designated as crisp or is an approxi-mation of some other object. • Vague Object : That whose approximations do not coincide with the objector that which has been designated as a vague object. • Discernible Object : That which is available for computations in a roughsemantic domain (in a contamination avoidance perspective). • Rough Object : Many definitions and representations are possible relativeto the context. From the representation point of view these are usuallyfunctions of definite or crisp objects. • Definite Object : An object that is invariant relative to an approximationprocess. In actual semantics a number of concepts of definiteness is pos-sible. In some approaches, as in [68, 52], these are taken as granules.Related theory has a direct connection with closure algebras and opera-tors as indicated in [48]. 9 . Up-Directed Rough sets: Basic ResultsDefinition 4.
In a general approximation space S = h S, R i consider the fol-lowing conditions: ( ∀ a, b )( ∃ c ) Rac & Rbc (up-dir) ( ∀ a ) Raa (reflexivity) ( ∀ a, b )( Rab & Rba −→ a = b ) (anti-sym) If S satisfies up-dir, then it will be said to be a upper directed approximationspace . If it satisfies all three conditions then it will be said to be a up-directedparthood space . In general, partial/quasi orders, and equivalences need not satisfy up-dir .When they do satisfy the condition, then the corresponding general approxima-tion spaces will be referred to as up-directed general approximation spaces .The neighborhood granulations used for defining approximations are speci-fied next.
Definition 5.
For any element a ∈ S , the following neighborhoods are associ-ated with it [ a ] = { x : Rxa } (neighborhood) [ a ] i = { x : Rax } (inverse-neighborhood) [ a ] o = { x : Rax & Rxa } (symmetric neighborhood)(3) A subset A ⊆ S will be said to be nbd-closed if and only if ( ∀ x ∈ A ) [ x ] ⊆ A Let the set of all nbd-closed subsets of S be E ( S )[ a ] is the set of things that relate to a and [ a ] i is the set of things that a relates to. [ a ] , [ a ] i and [ a ] o are respectively denoted by R − ( a ) , R ( a ) and ( R ∩ R − )( a ) Definition 6.
For any subset A ⊆ S , the following approximations can be de-fined: A l = [ { [ a ] : [ a ] ⊆ A } (lower) A l i = [ { [ a ] i : [ a ] i ⊆ A } (i-lower) A u = [ { [ a ] : ∃ z ∈ [ a ] ∩ A } (upper) A u i = [ { [ a ] i : ∃ z ∈ [ a ] i ∩ A } (i-upper) A l s = [ { [ a ] o : [ a ] o ⊆ A } (s-lower) A u s = [ { [ a ] o : ∃ z ∈ [ a ] o ∩ A } (s-upper)10 efinition 7. In the context of the previous definition, the pointwise and localapproximations are defined as follows: A u + = { x : [ x ] ∩ A = ∅} . (Point-wise Upper) A l + = { x : [ x ] ⊆ A } (Point-wise Lower) A ui + = { x : [ x ] i ∩ A = ∅} . (Point-wise i-Upper) A li + = { x : [ x ] i ⊆ A } (Point-wise i-Lower) A △ = { x : Rax & a ∈ A } (u-Triangle) A ▽ = { x : [ x ] i ⊆ A & x ∈ A } (l-Triangle) A N = [ { [ x ] : x ∈ A } . (ub-Triangle) A H = { x : [ x ] ⊆ A & x ∈ A } (lb-Triangle) Remark 1.
The *-triangle approximations are local in the sense that they aredefined relative to points (as opposed to subsets) in the set being approximated.It is shown below that while ▽ and H are pointwise approximation operators, △ and N are granular approximations because they can be represented as terms in-volving neighborhood granules and set operations alone. Because, neighborhoodsand inverse-neighborhoods are used, the granular and pointwise approximationsare inter related in a complex way. Proposition 1.
In the above context, for any subset A , A △ = [ { [ x ] i : x ∈ A } ⊆ A u + ⊆ A u i (4) A N = [ { [ x ] : x ∈ A } ⊆ A u (5) Proposition 2.
In the above context, when R is reflexive and c is the comple-mentation operation A l = A ▽ △ and A l i = A HN (6) A u = A △ N and A u i = A N △ (7) A △ = [ { [ x ] i : x ∈ A } ⊆ A u + ⊆ A u i (8) A ⊆ A N = [ { [ x ] : x ∈ A } ⊆ A u (9) A ▽ ⊆ A ⊆ A △ & A H ⊆ A ⊆ A N (10) A △ c = A c ▽ & A N c = A c H (11) Theorem 6.
In a reflexive up-directed approximation space S , the following roperties hold for elements of ℘ ( S ) : ( ∀ a ) a ll = a l ⊆ a (l-id) ( ∀ a ) a ⊆ a u ⊆ a uu (u-wid) ( ∀ a ) a l ⊆ a lu ⊆ a u (lu-inc) ( ∀ a, b )( a ⊆ b −→ a l ⊆ b l ) (l-mo) ( ∀ a, b )( a ⊆ b −→ a u ⊆ b u ) (u-mo) S u = S = S l & ∅ l = ∅ = ∅ u (bnd) ( ∀ a, b )( a ∪ b ) u = a u ∪ b u (u-union) ( ∀ a, b ) a l ∪ b l ⊆ ( a ∪ b ) l (l-union) ( ∀ a, b )( a ∩ b = ∅ −→ a l ∪ b l = ( a ∪ b ) l ) (l-union0) ( ∀ a, b )( a ∩ b ) l ⊆ a l ∩ b l (l-cap) ( ∀ a, b )( a ∩ b ) u ⊆ a u ∩ b u (u-cap) Proof. l-id If x ∈ a ll then there exists a b ∈ S such that x ∈ [ b ] ⊆ a l . So a ll ⊆ a l . This proves l-id . u-wid If x ∈ a then because of reflexivity of R , x ∈ [ x ] ⊆ a u . So u-wid holds. lu-inc The proof of u-wid carries over to that of lu-inc because a l ⊆ a u implies a lu ⊆ a u . l-mo If a ⊆ b then ( ∀ x ∈ S )([ x ] ⊆ a −→ [ x ] ⊆ b ) . This ensures that a l ⊆ a ⊆ b and b l ⊆ b and a l ⊆ b l . u-mo The proof is similar to that of l-mo. bnd
Follows from ( ∀ x ∈ S ) x ∈ [ x ] . u-union • If x ∈ ( a ∪ b ) u , then ( ∃ z ∈ S ) x ∈ [ z ] & [ z ] ∩ ( a ∪ b ) = ∅• the latter condition is ([ z ] ∩ a ) ∪ ([ z ] ∩ b ) = ∅• or ([ z ] ∩ a ) = ∅ or ([ z ] ∩ b ) = ∅ . So x ∈ a u ∪ b u . • Conversely, if h ∈ a u ∪ b u then ( ∃ z ∈ S ) h ∈ [ z ] & ([ z ] ∩ a ) = ∅ ∨ ([ z ] ∩ b ) = ∅• the latter condition is ([ z ] ∩ a ) ∪ ([ z ] ∩ b ) = ∅• So h ∈ ( a ∪ b ) u l-union • x ∈ a l ∪ b l ⇔ x ∈ a l or x ∈ b l • ⇔ ( ∃ z, h ∈ S ) x ∈ [ z ] ⊆ a ∨ x ∈ [ h ] ⊆ b • ⇔ ( ∃ z, h ∈ S ) x ∈ [ z ] ⊆ a ∪ b and so x ∈ ( a ∪ b ) l .Examples for the failure of the converse inclusion are easy to construct. l-union0 • x ∈ ( a ∪ b ) l then ( ∃ z ∈ S ) x ∈ [ z ] ⊆ a ∪ b So ( ∃ z ∈ S ) x ∈ [ z ] ⊆ a Xor x ∈ [ z ] ⊆ b • So x ∈ a l xor x ∈ b l , which implies x ∈ a l ∪ b l . l-cap x ∈ ( a ∩ b ) l • if and only if ( ∃ z ∈ S ) x ∈ [ z ] ⊆ a ∩ b • if and only if ( ∃ z ∈ S ) x ∈ [ z ] ⊆ a & [ z ] ⊆ a • implies x ∈ a l and x ∈ b l • To see possible reasons for the failure of the converse, let x ∈ a l and x ∈ b l • then ( ∃ z ∈ S ) x ∈ [ z ] ⊆ a and ( ∃ z ∈ S ) x ∈ [ z ] ⊆ b • so x ∈ [ z ] ∩ [ z ] ⊆ a ∩ b , but it can happen that [ z ] ∩ [ z ] is not ofthe form z for some z ∈ a ∩ b . Remark 2.
The nature of failure of a l ∩ b l ⊆ ( a ∩ b ) l shown in the proof suggeststhat it can be fixed at a semantic level in many ways. Theorem 7.
In a up-directed approximation space S , the following propertieshold for elements of ℘ ( S ) : ( ∀ a ) a ll = a l ⊆ a (l-id0) ( ∀ a ) a u ⊆ a uu (u-wid0) ( ∀ a ) a l ⊆ a lu ⊆ a u (lu-inc) ( ∀ a, b )( a ⊆ b −→ a l ⊆ b l ) (l-mo) ( ∀ a, b )( a ⊆ b −→ a u ⊆ b u ) (u-mo) S l = S u ⊆ S & ∅ l = ∅ = ∅ u (bnd0) ( ∀ a, b )( a ∪ b ) u = a u ∪ b u (u-union) ( ∀ a, b ) a l ∪ b l ⊆ ( a ∪ b ) l (l-union) ( ∀ a, b )( a ∩ b ) l ⊆ a l ∩ b l (l-cap) ( ∀ a, b )( a ∩ b ) u ⊆ a u ∩ b u (u-cap) Proof.
Most of the proof of Theorem 6 carries over. Because of the absenceof reflexivity, the weaker properties u-wid0, and bnd0 hold.
Remark 3.
It may be noted that the upper cone of a subset A (that is the set { b : ( ∃ a, c ∈ A ) Rab & Rcb } ) is contained in A u .
4. Illustrative Examples
Abstract and practical examples are constructed in this section for illustrat-ing various aspects of up-directed approximation spaces.13 .1. Abstract Example
Let S be the set S = { a, b, c, e, f } and let R = { ac, ae, af, bc, bf, ca, cb, cf, ea, ef, f a, f b } be a binary relation on it ( ac means the ordered pair ( a, c ) and so on for otherelements). In Figure 1, the general approximation space S = h S, R i is depicted.An arrow from e to f is drawn because Ref holds. feca b
Figure 1: Up-Directed Relation R The up-directed approximation space S = h S, R i is irreflexive and R isnot antisymmetric. The antisymmetric completion R + of R coincides with itsreflexive completion and is defined by R + = R ∪ { aa, bb, cc, ee, f f } The groupoid corresponding to S is given by Table 1a b c e fa e c c e fb e c c e fc a b f f fe a b f f ff a b a a a Table 1: A Groupoid of S The neighborhood granules determined by the elements of S are as in Table2 14 [ x ] [ x ] i [ x ] o a { c, e, f } { c, e, f } { c, e, f } b { c, e, f } { c, f } { c, f } c { a, b } { a, b, f } { a, b } e { a } { a, b, f } { a } f { a, b, c, e } { a, b } { a, b } Table 2: Neighborhood Granules
Since ℘ ( S ) has elements, approximations of specific subsets are aloneconsidered next.Let A = { e, c } , then its approximations are as below: • A l = ∅ and A u = S • A ▽ = ∅ and A △ = { a, b, f }• A H = ∅ and A N = { a, b }• A l i = ∅ = A l o and A u i = { c, e, f } = A u o • A l + = ∅ and A u + = a, b, f • A l i + = ∅ and A u i + = { a, b } Suppose a set S of concepts relating to a classroom lesson are given, andthat some of these are vague. For any two concepts a and b , assume that aconcept c that apparently contains the two exists – this type of search for a c amounts to taking decisions. Let this concept of apparent parthood be denotedby R . Depending on the context, the relation R may be a up-directed, reflexiveand antisymmetric relation. Thus S = h S, R i may be a up-directed parthoodspace or definitely an up-directed space. Apparent parthood relation has been considered by the first author in [49] –in general it is not antisymmetric.For two concepts a and b , ab = b may mean that b fulfils the functions of a in some sense (for example). If, on the other hand, ab ∈ U R ( a, b ) then there isa implicit reference to a choice function in the search for a concept that fulfilsthe role of both a and b .For a concept a , the neighborhood [ a ] is the set of concepts that are ap-parently part of it, while [ a ] i is the set of concepts that it is apparently partof, and [ a ] o is the set of concepts that it is apparently part of and conversely.Obviously, when antisymmetry holds, the set [ a ] o will be a singleton. Note thatthese concepts have a directional character – because of up-directedness of R .Each granule of the form [ a ] may be associated with at least one element of S .Is [ a ] determined by a ? The actual interpretation depends on the applicationcontext. In this case, it can be said the investigation of a leads to the set [ a ] .15or a subset of concepts A , the lower approximation is an aggregation ofdirected granules that are included in A . It may also be read as the collectionof relatively definite concepts that are attainable from A (using common sensemethods or through common knowledge).
5. Algebraic Semantics-1
In this section, possible semantics of the approximations l and u on theirimage set are investigated. From Theorem 6 and Theorem 7, it follows thata semantics over ℘ ( S ) without additional constructions is not justified becausethey do not distinguish between closely related general approximation spaces. Definition 8.
On the set ( ℘ ( S )) u = { x u : x ∈ ℘ ( S ) } = S u , the followingoperations can be defined (apart from the induced ∪ operation): a ∧ b = ( a ∩ b ) u (iu1) a ∨ b = ( a ∪ b ) (iu2) ⊥ = ∅ (iu3) ⊤ = S u (iu4) and the resulting algebra S u = (cid:10) S u , ∨ , ∧ , ∪ , l, u, ⊥ , ⊤ (cid:11) will be called the alge-bra of upper approximations in a up-directed space (UUA algebra). If R isa up-directed parthood relation or a reflexive up-directed relation respectively,then it will be said to be a up-directed parthood algebra of upper approximations(UAP algebra) or a reflexive algebra of upper approximations (UAR algebra)respectively. Theorem 8.
The UUA, UAP and UAR algebras are well-defined, and an al-gebra of upper approximations satisfies all of the following: ( ∀ a ) a ∨ a = a = a ∨ ⊥ (idemp1) ( ∀ a, b ) a ∧ b = b ∧ a (comm2) ( ∀ a, b ) a ∨ b = b ∨ a (comm1) ( ∀ a, b, c ) a ∨ ( b ∨ c ) = ( a ∨ b ) ∨ c (assoc1) ( ∀ a ) ( a ∧ a ) ∨ a = a ∧ a = a u (absfail) ( ∀ a, b, c ) ( a ∨ b = b −→ ( a ∧ c ) ∨ ( b ∧ c ) = b ∧ c ) (mo1) Proof.
The lower approximation operation is redundant and so the algebrasare well-defined. idemp1 a ∨ a = a ∪ a = a . comm2 a ∧ b = ( a ∩ b ) u = ( b ∩ a ) u = b ∧ a .16 omm1 Follows from definition. assoc1
Follows from associativity of set union. absfail a ∧ a = a u . So absorptivity fails in general.Absorptivity can be improved by defining the operations differently.Let S lu = { x : x = a l or x = a u & a ∈ S } Definition 9. On S lu , the following operations can be defined (apart from l and u by restriction): a ⋓ b = ( a ∩ b ) l (Cap) a ⋒ b = ( a ∪ b ) u (Cup) ⊥ = ∅ (iu3) ⊤ = S u (iu4) The resulting algebra S lu = (cid:10) S lu , ⋓ , ⋒ , ∪ , l, u, ⊥ , ⊤ (cid:11) will be called the algebraof approximations in a up-directed space (UA algebra). If R is a up-directedparthood relation or a reflexive up-directed relation respectively, then it will besaid to be a up-directed parthood algebra of approximations (AP algebra) or areflexive up-directed algebra of upper approximations (AR algebra) respectively. Theorem 9.
A AP algebra S lu satisfies all of the following: ( ∀ a ) a ⋓ a = a & ( a ⋒ a ) ⋓ a = a (idemp3) ( ∀ a ) a ⋒ a = a u (quasi-idemp4) ( ∀ a, b ) a ⋓ b = b ⋓ a & a ⋒ b = b ⋒ a (comm12) ( ∀ a, b ) a ⋓ ( b ⋒ a ) = a (half-absorption) ( ∀ a, b, c ) a ⋒ ( b ⋒ c ) = ( a ⋒ b u ) ⋒ c u (quasi-assoc1) ( ∀ a, b, c )( a ⋒ ( b ⋒ c )) ⋒ (( a ⋒ b ) ⋒ c ) = (( a ⋒ a ) ⋒ ( b ⋒ b )) ⋒ ( c ⋒ c ⋒ c ) (quasi-assoc0) Proof. idemp3 • a ⋓ a = ( a ∩ a ) l = a l = a • a ⋒ a = a u and a u ∩ a = a quasi-idemp4 a ⋒ a = ( a ∪ a ) u = a u . comm12 This follows from definition. half-absorption • a ⋓ ( b ⋒ a ) = ( a ∩ ( b ∪ a ) u ) l = (( a ∩ a u ) ∪ ( a ∩ b u )) l = ( a ∪ ( a ∩ b u )) l = a l = a quasi-assoc1 • a ⋒ ( b ⋒ c ) = ( a ∪ ( b ∪ c ) u ) u = ( a u ∪ b uu ∪ c uu ) • = ( a ∪ b u )) u ∪ c uu = ( a ⋒ b u ) ⋒ c u quasi-assoc0 This can be proved by writing all terms in terms of ∪ . In fact ( a ⋒ ( b ⋒ c )) ⋒ (( a ⋒ b ) ⋒ c ) = a uuu ∪ b uuu ∪ c uuu . The expression on theright can be rewritten in terms of ⋒ by quasi-idemp4 .The above two theorems in conjunction with the properties of the approxi-mations on the power set, suggest that it would be useful to enhance UA-, AP-,and AR-algebras with partial operations for defining an abstract semantics. Definition 10.
A partial algebra of the form S ∗ lu = (cid:10) S lu , ⋓ , ⋒ , ∪ , ⊓ , κ , l, u, ⊥ , ⊤ (cid:11) will be called the algebra of approximations in a up-directed space (UA partialalgebra) whenever S lu = (cid:10) S lu , ⋓ , ⋒ , ∪ , l, u, ⊥ , ⊤ (cid:11) is a UA algebra and ⊓ and kappa are defined as follows ( ∩ and c being the intersection and complementationoperations on ℘ ( S ) ): ( ∀ a, b ∈ S lu ) a ⊓ b = ( a ∩ b if a ∩ b ∈ S lu undefined otherwise (12) ( ∀ a ∈ S lu ) a κ = ( a c if a c ∈ S lu undefined otherwise (13) If R is an up-directed parthood relation or a reflexive up-directed relationrespectively, then it will be said to be a up-directed parthood partial algebra ofapproximations (AP partial algebra) or a reflexive algebra of upper approxima-tions (AR partial algebra) respectively. Theorem 10. If S is a up-directed approximation space, then its associatedenhanced up-directed parthood partial algebra S ∗ lu = (cid:10) S lu , ⋓ , ⋒ , ∪ , ⊓ , κ , l, u, ⊥ , ⊤ (cid:11) satisfies all of the following: (cid:10) S lu , ⋓ , ⋒ , ∪ , l, u, ⊥ , ⊤ (cid:11) is a AP algebra (app1) ( ∀ a ) a ⊓ a = a & a ⊓ ⊥ = ⊥ & a ⊓ ⊤ = a (app2) ( ∀ a, b, c ) a ⊓ b ω = b ⊓ a & a ⊓ ( b ⊓ c ) ω = ( a ⊓ b ) ⊓ c (app3) a ⊓ a u = a = a ⊓ a l & a κκ ω = a (app4) a ⊓ ( b ∪ c ) ω = ( a ⊓ b ) ∪ ( a ⊓ c ) & a ∪ ( b ⊓ c ) ω = ( a ∪ b ) ⊓ ( a ∪ c ) (app5) ( ∀ a, b ) ( a ⊓ b ) κ ω = a κ ∪ b κ & ( a ∪ b ) κ ω = a κ ⊓ b κ (app6) Proof.
The theorem follows from the previous theorems in this section.18 . Groupoidal SemanticsDefinition 11.
In the powerset ℘ ( S ) generated by a upper directed approxi-mation space S , the following operation can be defined (apart from the roughapproximations and induced Boolean operations) ( ∀ A, B ∈ ℘ ( S )) A · B = { ab : a ∈ A & b ∈ B } (g0) The resulting algebra, S b = D ℘ ( S ) , · , ∪ , ∩ , l, u, c , ⊥ , ⊤ E of type (2 , , , , , , , will be called a basic power up-directed algebra ( BP -algebra). If l and u arereplaced by l s and u s , then the resulting algebra will be called a basic symmetricpower up-directed algebra ( BPS -algebra) a ⊆ b will be used as an abbreviation for a ∪ b = b in what follows. Theorem 11.
The algebra D ℘ ( S ) , ∪ , ∩ , c , ∅ , ℘ ( S ) E is a Boolean algebra. Fur-ther, the following properties are satisfied by a BP-algebra S b : ( ∀ a, b, c )( a ∪ b = b −→ ac ∪ bc = bc ) (order-comp) ( ∀ a ) ∅ a = a ∅ = ∅ & aS ⊆ S & Sa ⊆ S (bnd2) ( ∀ a, b, h )( a ∪ b ) h = ( ah ) ∪ ( bh ) & ( a ∩ b ) h = ( ah ) ∩ ( bh ) (comp2) Conditions mentioned in eqn.6. (lu-properties)
Proof. order-comp If x ∈ ac , then it is of the form ef with e ∈ a and f ∈ c .By the premise, e ∈ b , so the conclusion follows. comp2 x ∈ ( a ∪ b ) h if and only if x ∈ ah or x ∈ bh . Similarly for the secondpart. Remark 4.
Note that x ∈ a c h then x is of the form ef with e ∈ a c and f ∈ h ,but ef may be in ah or ( ah ) c . So, in general, a c h = ( ah ) c .6.1. Meaning of the Groupoidal Operation In the first author’s opinion, the groupoid operation can be read in at leasttwo ways. The operation obviously adds information to the general approxima-tion space – this addition can be read as a decision because it involves choiceamong alternatives . In fact, the collection of all possible groupoidal operationscan be used to generate a decision space. As such this aspect can be investi-gated in the given form or by taking the exact region to which the result of theoperation belongs relatively. For the latter perspective, the groupoidal opera-tion over ℘ ( S ) can be read as a combination of operations that are relativelybetter behaved relative to the approximations, aggregation and commonalityoperations. This permits easier interpretation, and semantics.19 efinition 12. For any
A, B ∈ ℘ ( S ) , the following operations can be defined: n ( A, B ) = { b : ( ∃ a ∈ A ∃ b ∈ B ) ab = b } (normal) o ( A, B ) = { c : ( ∃ a ∈ A ∃ b ∈ B ) ab = c ∈ U R ( a, b ) \ A } (outer-1) o ( A, B ) = { c : ( ∃ a ∈ A ∃ b ∈ B ) ab = c ∈ U R ( a, b ) \ B } (outer-2) i ( A, B ) = { c : ( ∃ a ∈ A ∃ b ∈ B ) ab = c ∈ U R ( a, b ) ∩ A } (inner-1) i ( A, B ) = { c : ( ∃ a ∈ A ∃ b ∈ B ) ab = c ∈ U R ( a, b ) ∩ B } (inner-2) o ( A, B ) = o ( A, B ) ∩ o ( A, B ) (outer)In the above definition, the global groupoid operation has been split intomultiple operations based on the relative values assumed. For any two sets A, B ∈ ℘ ( S ) , • n ( A, B ) is the set of things in B that have some part or approximate partin A , • o ( A, B is the set of things in the outer core determined by elements of A × B that are not in A , • o ( A, B is the set of things in the outer core determined by elements of A × B that are not in B , • i ( A, B is the set of things determined by elements of A × B that are in A , • i ( A, B is the set of things determined by elements of A × B that are in B , and • o ( A, B is the set of things determined by elements of A × B that are notin A or B .These can also be read as generalizations of natural concepts of g-ideals inthe context that can be defined as follows: Definition 13.
A subset A of the groupoid S = h S, ·i is an g-ideal if and onlyif A is a subgroupoid and ab = b & b ∈ A −→ a ∈ A (14) A subset B of the groupoid S is a g-filter if and only if B is a subgroupoid and ab = b & a ∈ A −→ b ∈ B (15)The g-ideal generated by a subset A will be the smallest g-ideal I ( A ) con-taining the subgroupoid Sg ( A ) generated by A . If A is a singleton, then theg-ideal will be said to be principal . The set of all g-ideals (resp principal, finitelygenerated) on S will be denoted by I ( S ) (resp. I p ( S ) , I f ( S ) ).20 efinition 14. If S is an up-directed parthood space, then the algebra S ♯ = D ℘ ( S ) , n, i , i , o , o , o, ∪ , ∩ , l, u, c , ∅ , ℘ ( S ) E defined above will be referred to as the expanded up-directed parthoodgroupoidal Boolean algebra (EUPGB) Theorem 12.
In the context of a EUPGB S ♯ , all of the following hold (for any A, B ∈ S ♯ ): n ( A, B ) ⊆ B (n) o ( A, B ) ⊆ A c (o1) o ( A, B ) ⊆ B c (o2) i ( A, B ) ⊆ A (i1) i ( A, B ) ⊆ B (i2) o ( A, B ) ⊆ ( A ∪ B ) c (o) Proof. • For any a ∈ A and b ∈ B , ab = b yields ab ∈ B . • For any a ∈ A and b ∈ B , ab = c ∈ U R ( a, b ) \ A yields ab ∈ A c . So o1follows. • Note that if a ∈ A , b ∈ B , and ab ∈ U R ( a, b ) then it is possible that ab ∈ B . o ensures that this does not happen. • Other parts can be verified from definition.
Corollary 1. If B ⊆ A in the context of the previous theorem then n ( A, B ) = B (1) i ( A, B ) ⊆ i ( A, B ) ⊆ A (2) o ( A, B ) ⊆ o ( A, B ) (3) AB = B ∪ i ( A, B ) ∪ o ( A, B ) (summary) Remark 5.
Clearly the operations n, i , i , o , and o are better behaved thanthe groupoid operation · . From the above considerations, it can also be deduced that
Proposition 3.
In a EUPGB algebra S ♯ , for any a, b ∈ S • n ([ a ] , [ b ]) ⊆ [ b ] • i ([ a ] , [ b ]) ⊆ [ a ] • i ([ a ] , [ b ]) ⊆ [ b ] • o ([ a ] , [ b ]) ⊆ [ a ] c and • o ([ a ] , [ b ]) ⊆ [ b ] c .2. Rough Equalities and Inequalities Particular rough equalities of natural interest are defined next.
Definition 15.
For any a, b ∈ S ∗ , let a ≈ b if and only if a l = b l & a u = b u (standard req) a ≈ l b if and only if a l = b l (l-standard req) a ≈ u b if and only if a u = b u (u-standard req) ≈ , ≈ s , ≈ l , ≈ u and τ will respectively be referred to as the standard, l-standard,and u-standard rough equalities respectively. Obviously, the relations ≈ , ≈ s , ≈ l , and ≈ u are equivalences on ℘ ( S ) . Themeaning of the above relations is closely connected with the following roughinequalities on ℘ ( S ) : Definition 16.
In a EUGB H , the following relations are definable: a ⊑ l b if and only if a l ⊆ b l (l-rough inequality) a ⊑ u b if and only if a u ⊆ b u (u-rough inequality) a ⊑ b if and only if a ⊑ l b & a ⊑ u b (rough inequality) Proposition 4.
The relations ⊑ l , ⊑ u and ⊑ are quasi orders on the EUGB H . Moreover, they are partly compatible with the operations ∪ and ∩ in thefollowing sense: ( ∀ a, b, c )( a ⊑ l b −→ a ∩ c ⊑ l b ∩ c ) (cs1) ( ∀ a, b, c )( a ⊑ u b −→ a ∪ c ⊑ u b ∪ c ) (cs2) Proof.
It is obvious that ⊑ l is reflexive. It is transitive because for any a, b and c , a l ⊆ c l follows from a l ⊆ b l and b l ⊆ c l .In general, antisymmetry does not hold because a l ⊆ b l and b l ⊆ a l need notimply a = b .The property cs1 can be verified by considering the neighborhoods that maybe included in a , b and c , and observing that the neighborhoods included in a ∩ c must also be included in b ∩ c .The proof for ⊑ u is analogous. For cs2 , note that neighborhoods havingnonempty intersection with a ∪ c must also have nonempty intersection with b ∪ c .In the context of a EUPGB, on the quotient ℘ ( S ) | ≈ , the following operationscan be defined. Definition 17.
In the quotient ℘ ( S ) | ≈ generated on a EUPGB ℘ ( S ) = D ℘ ( S ) , n, i , i , o , o , o, ∪ , ∩ , l, u, c , ∅ , ℘ ( S ) E , he following operations can be defined ( ∀ A, B ∈ ℘ ( S )) ˘ α ([ A ] ≈ , [ B ] ≈ ) = [ [ { α ( F, H ) : F ∈ [ A ] ≈ & H ∈ [ B ] ≈ } ] ≈ where α is any of · , n, i , i , o , o , o, ∪ and ∩ . Further, ( ∀ A, B ∈ ℘ ( S )) [ A ] ≈ ⊛ [ A ] ≈ = h\ { F ∩ H : F ∈ [ A ] ≈ & H ∈ [ B ] ≈ } i ≈ ( ∀ A ∈ ℘ ( S )) ¬ ([ A ] ≈ ) = h[ { F c ∈ [ A ] ≈ } i ≈ ( ∀ A ∈ ℘ ( S )) L ([ A ] ≈ ) = h[ { F l : F ∈ [ A ] ≈ } i ≈ ( ∀ A ∈ ℘ ( S )) U ([ A ] ≈ ) = h[ { F u : F ∈ [ A ] ≈ } i ≈ In addition, the -ary operations ⊥ and ⊤ can be defined as [ ∅ ] ≈ and [ ℘ ( S )] ≈ respectively.The algebra Z = D ℘ ( S ) | ≈ , ˘ n, ˘ i , ˘ i , ˘ o , ˘ o , ˘ o, ˘ ∪ , ˘ ∩ , L, U, ¬ , ⊥ , ⊤ E will be referred to as a up-directed rough parthood algebra (RPA). Remark 6.
In the above definition, the L and U operations are not likely tobehave as modal operators, and this is consistent with the semantic intent. Definition 18. If a is an element of ℘ ( S ) | ≈ then it can also be interpreted asa subset of ℘ ( S ) , and its representative approximations a l and a u are a l = x l for any x ∈ a and a u = x u for any x ∈ a Theorem 13.
All operations in Def.17 are well defined.
Proof.
The definition of each operation over ℘ ( S ) | ≈ is based on forming aset of equivalent sets from a union and so is well defined.In the next theorem, key relations between representatives and operationson a RPA are established. Proposition 5. If x ∈ a ∈ ℘ ( S ) | ≈ , then x can be represented in the form a l ∪ K subject to the condition a l ∪ K u = a u and K l = ∅ . Proof.
Let a = { A , . . . , A n } for some integer n ≤ ∞ with A i ∈ ℘ ( S ) , then A i = a l ∪ K i for some set K i Because a ll ∪ K li ⊆ A li = a l , it can be assumed that K li = ∅ and that K i ∩ a l = ∅ .In this situation, A ui = ( a l ∪ K i ) u = a ul ∪ K ui = a u .23 heorem 14. In the context of Prop. 5, all of the following hold: ( ∀ a ) a u ⊆ ( U a ) u (Uu) ( ∀ a ) a l = ( La ) l ⊆ ( U a ) l (Ll) ( ∀ a, b )( a ˘ ∪ b ) u = a u ∪ b u (ujoins) ( ∀ a )( ¬ a ) u ⊆ a cul (uc) Proof. Uu Using the representation of a in Prop. 5, it follows that a u =( a l ∪ K i ) u = a ul ∪ K ui for any i , while U a = [ [ { a l ∪ b : b ∩ K i = ∅ or b ∩ a = ∅ & b ∈ G} ] So U a = ( a l ) u ∪ S K ui . So a u ⊆ ( U a ) u . Ll In the same representation, La = [( S A i ) l ] = [ a l ∪ ( S K i ) l ] . So a l = ( La ) l . ujoins Suppose a ˘ ∪ b = h[ { X i ; X i ∈ a or X i ∈ b } i The X i can be written as A i ∪ B i = a l ∪ K i ∪ b l ∪ J i . Further for each i ( a l ∪ K i ) u = a u = a ul ∪ K ui ans similarly b u = b ul ∪ J ui . Using these forsubstituting X i results in ( a ˘ ∪ b ) u = a u ∪ b u . uc Using the same strategy as in the proof of the previous properties, • ¬ a = (cid:2) { S A ci : A i ∈ a } (cid:3) = • = [ { S ( a l ∪ K i ) c } ] = [ a cl ∩ ( S K ci )] . • So ( ¬ a ) u ⊆ a cul Remark 7.
This result suggests that a better (but relatively difficult) operationon the quotient ℘ ( S ) | ≈ can be L a = h { [ { A : A ∈ a }} l i ≈ (bL) U a = h { [ { A : A ∈ a }} u i ≈ (bU) It can be checked that while L a is the same as La , but ( U a ) l ⊆ ( U a ) l and ( U a ) u ⊆ ( U a ) u in general. Theorem 15. If Z is an RPA, then ( ∀ a ∈ Z ) a l ⊑ ( La ) l ⊑ ( La ) u (rep1) ( ∀ a ∈ Z ) a u ⊑ ( U a ) u (rep2) ( ∀ a, b ∈ Z ) a l ⊆ ( a ˘ ∪ b ) l (rep3) ( ∀ a, b ∈ Z ) a u ⊆ ( a ˘ ∪ b ) u (rep4) The converse of rep1 holds if Z is reflexive. roof. The proof depends on Thm 6, and Thm. 7. • Suppose a = { b , b , . . . , b n } with b li = b lj = a l for all i, j . By definition, La = [ S b li ] ≈ . So a l ⊑ ( La ) l . The converse holds if Z is reflexive. • Suppose a = { b , b , . . . , b n } with b ui = b uj = a u for all i, j . By definition, U a = [ S b ui ] ≈ . So a u ⊑ ( U a ) u . • Suppose a = { a , . . . a v } and b = { b , b , . . . , b n } , then a ˘ ∪ b is by definitionequal to [ S { a i ∪ b j } ] ≈ . Since a i ⊆ a i ∩ b j for all i , rep3 follows. • The proof of rep4 is similar to that of rep3 . Remark 8.
Note that the following can fail to hold in general: ( ∀ a, b ∈ Z ) ( a ˘ ∩ b ) l ⊆ a l (rep5) ( ∀ a, b ∈ Z ) ( a ˘ ∩ b ) u ⊆ a u (rep6) Theorem 16.
All of the following properties hold in a RPA Z : ˘ n ( a, a ) = a (n-idemp) a ˘ ∪ b = b ˘ ∪ a (join-comm) a ˘ ∩ b = b ˘ ∩ a (meet-comm) a ⊑ a ˘ ∪ a (join-explosion) a ⊑ a ˘ ∩ a (meet-explosion) a ⊛ b = b ⊛ a (star-comm) a ⊑ ( a ˘ ∪ b )˘ ∩ a (abs-fail) Proof.
Let Z = D ℘ ( S ) | ≈ , ˘ n, ˘ i , ˘ i , ˘ o , ˘ o , ˘ o, ˘ ∪ , ˘ ∩ , L, U, ¬ , ⊥ , ⊤ E As mentioned in the introduction, join directoids were introduced in [31]. Abetter equational way of defining these is as follows:
Definition 19.
Directoids (join) are groupoids of the form H = h H, ·i thatsatisfy the following conditions: aa = a (dir1) ( ab ) a = ab (dir2) b ( ab ) = ab (dir3) a (( ab ) c ) = ( ab ) c (dir4) Proposition 6 ([9]).
A groupoid of the form H = h H, ·i is a join directoid ifand only if there exists a partial order ≤ on H that satisfies ( ∀ a, b ) a, b ≤ ab (jd1) ( ∀ a, b )( a ≤ b −→ ab = ba = b ) (jd2)25o it follows that a up-directed partially ordered set can be written as agroupoid and the groupoid in turn determines the partial order uniquely.From the proposition it follows that Theorem 17.
When an up-directed parthood space is also transitive, then ajoin directoid operation is definable on it (as per Equation 2).
7. Algebraic Semantics of Local ApproximationsDefinition 20. If ℘ ( S ) △ = { A △ : A ∈ ℘ ( S ) } , then let ( ∀ A, B ∈ ℘ ( S ) △ ) A g B = A ∪ B (T1) ( ∀ A, B ∈ ℘ ( S ) △ ) A f B = ( A ∩ B ) H △ (T2) ( ∀{ A j } j ∈ J ∈ ℘ ( S ) △ ) g j ∈ J A j = [ A j (ET1) ( ∀{ A j } j ∈ J ∈ ℘ ( S ) △ ) f j ∈ J A j = ( \ j A j ) H △ (T2) Theorem 18.
The algebra D ℘ ( S ) △ , g , f E is a complete bounded lattice. Thecorresponding lattice order on the algebra is ⊆ (induced from set inclusion on ℘ ( S ) ). Proof. If A = X △ for some X , then A = S x ∈ X [ x ] i . For arbitrary collections { A j } J in ℘ ( S ) △ , it is easy to see that ( ∀ B )(& A j ⊆ B −→ [ A j ⊆ B ) This ensures that the union is a complete join semilattice operation and B = Z △ for some Z , then A = S x ∈ Z [ x ] i f j ∈ J A j = [ { Z : Z ∈ ℘ ( S ) △ & Z ⊆ \ j ∈ J A j } = [ {∪ [ x ] i : ∪ [ x ] i ⊆ \ j ∈ J A j } = [ { [ x ] i : [ x ] i ⊆ \ j ∈ J A j } = ( \ j ∈ J A j ) H △ Theorem 19. h ℘ ( S ) ▽ , ⊆i is dually isomorphic to (cid:10) ℘ ( S ) △ , ⊆ (cid:11) as a completelattice. Proof.
Define a map f : ℘ ( S ) △ ℘ ( S ) ▽ according to ( ∀ Z ∈ ℘ ( S ) △ ) f ( Z ) = Z c • Since any Z ∈ ℘ ( S ) △ has the form Z = X △ for some X ⊆ S , so f ( Z ) = X △ c = X c ▽ ∈ ℘ ( S ) ▽ . 26 This ensures that the map f is well defined. • For any Z , Z ∈ ℘ ( S ) △ , Z ⊆ Z ↔ Z c ⊆ Z c ↔ f ( Z ) ⊇ f ( Z ) . From this it follows that f is a dual order-isomorphism.Hence in view of Theorem 18, h ℘ ( S ) ▽ , ⊆i is also a complete lattice. Definition 21.
On the image ℘ ( S ) N = { X N : X ⊆ S } of N , the induced rela-tion ⊆ can be associated with the following operations: g ∗ i ∈ I A i = [ i ∈ I A i (bt-join) f ∗ i ∈ I A i = ( \ i ∈ I A i ) ▽N (bt-meet)Note that relation of ℘ ( S ) N to R − corresponds to the relation of ℘ ( S ) △ with R . Theorem 20. h ℘ ( S ) N , ⊆i and h ℘ ( S ) H , ⊆i are dually isomorphic complete lat-tices. Proof.
The proof is analogous to that of Theorem 19.Definable sets in rough sets can be described in different ways. From alattice-theoretical perspective, it is of interest to see if the set of lower or upperdefinable or at least the set of lower and upper approximations form distributivelattices. In this section, it is shown that the algebras formed by the set ofapproximations ℘ ( S ) △ , ℘ ( S ) ▽ , ℘ ( S ) N , and ℘ ( S ) H are completely distributivelattices. It may be noted that the second author has studied these sets froma similar perspective in the context of approximations generated by tolerancerelations in [27].In view of Theorem 20, this condition is equivalent to the condition thatthe concept lattice L ( S, S, I ) is (completely) distributive. In [19] several condi-tions equivalent to the complete distributivity of L ( S, S, I ) are formulated. Forinstance, the following was established: Theorem 21 ([19]: Thm.40).
A concept lattice L ( G, M, I ) is completely dis-tributive if and only if for any object attribute pair ( g, m ) / ∈ I there exists anobject h ∈ G and an attribute n ∈ M with ( g, n ) / ∈ I , ( h, m ) / ∈ I and such that h ∈ { k } II , for any k ∈ G (cid:31) { n } I . As an immediate consequence, in case of the concept lattice L ( S, S, I ) andthe lattice ℘ ( S ) △ we can formulate the following:27 heorem 22. The lattice (cid:10) ℘ ( S ) △ , ⊆ (cid:11) is completely distributive if and only iffor any a, b ∈ S satisfying Rab there exist some elements n, h ∈ S satisfying Ran & Rhb and such that for any x ∈ S satisfying ( Rxn we have [ h ] i ⊆ [ x ] i .That is ( ∀ a, b ) Rab −→ ( ∃ n, h )( ∀ x ) Ran & Rhb & [ h ] i ⊆ [ x ] i Proof.
In view of Theorem 18, (cid:10) ℘ ( S ) △ , ⊆ (cid:11) is completely distributive if andonly if the concept lattice L ( S, S, I ) is completely distributive. This is equivalentto the condition formulated in Theorem 19. • Let a, b ∈ S and Rab . • In the context L ( S, S, I ) , Rab ↔ ¬
Iab . So the above theorem applieswith g := a and m := b and there exists n, h ∈ S with Ran & Rhb andsatisfying h ∈ { x } II , for any x ∈ S (cid:31) { n } I . • As S (cid:31) { n } I = { s ∈ S : ¬ Isn } , x ∈ S (cid:31) { n } I means that Rxn . Since h ∈ { x } II is equivalent to [ x ] ci = { x } I ⊆ { h } I = [ h ] ci , we deduce that Rxn implies [ h ] i ⊆ [ x ] i , for any x ∈ S .Therefore the condition in the present theorem is equivalent to the conditionformulated in Theorem 21 and the conclusion follows.By using this theorem, two characterizations of the (complete) distributivityof (cid:10) ℘ ( S ) △ , ⊆ (cid:11) can be deduced. Also note that it is easy to check that anycompletely distributive element of (cid:10) ℘ ( S ) △ , ⊆ (cid:11) has the form [ s ] i (for some s ∈ S )– but the converse statement is not true in general. Theorem 23.
If the lattice (cid:10) ℘ ( S ) △ , ⊆ (cid:11) is spatial, then the following assertionsare equivalent: i The lattice (cid:10) ℘ ( S ) △ , ⊆ (cid:11) is completely distributive. ii If [ s ] i is an arbitrary completely join-irreducible element of (cid:10) ℘ ( S ) △ , ⊆ (cid:11) , then [ s ] i " [ { [ x ] i : [ x ] i [ s ] i } (ei3) Proof (i). ⇒ [ii] • Let [ s ] i be a completely join-irreducible element of (cid:10) ℘ ( S ) △ , ⊆ (cid:11) . • Assume the contrary [ s ] i ⊆ S { [ x ] i : [ x ] i [ s ] i } . • Since the lattice (cid:10) ℘ ( S ) △ , ∪ , ∧ (cid:11) is completely distributive, we have [ s ] i =[ s ] i ∧ S { [ x ] i : [ x ] i [ s ] i } = S { [ x ] i ∧ [ s ] i : [ x ] i [ s ] i } . • Since [ s ] i is a completely join-irreducible, we obtain [ s ] i = [ s ] i ∧ [ x ] i , i.e. [ s ] i ⊆ [ x ] i , for some x ∈ S with [ x ] i [ s ] i – a contradiction. • This proves the implication. 28ii] ⇒ [i] • Assume that (ii) holds, and let
Rab for some a, b ∈ S . Then b ∈ [ a ] i . • If [ a ] i is completely join-irreducible, then in view of (ii), there exists anelement n ∈ [ a ] i \ S { [ x ] i : [ x ] i [ a ] i } . If we set h := a , then Ran & Rhb . • For any k ∈ S satisfying Rkn , n ∈ [ k ] i excludes the case [ k ] i [ a ] i , hencewe obtain [ k ] i ⊇ [ a ] i = [ h ] i . • Now suppose that [ a ] i is not completely join-irreducible. Then b ∈ [ a ] i = S { [ p ] i : [ p ] i ∈ CJ ( ℘ ( S ) △ ) } , and this yields b ∈ [ p ] i for some completelyjoin-irreducible element [ p ] i ) of ℘ ( S ) △ with [ p ] i ⊆ [ a ] i . Further Rpb and [ p ] i " [ { [ x ] i : [ x ] i [ p ] i }• Therefore there exists an element n ∈ [ p ] i \ S { [ x ] i : [ x ] i R ( p ) } ⊆ [ a ] i and hence we get Ran . Set h := p . This yields Rhb . • For any k ∈ S satisfying Rkn , n ∈ [ k ] i and n / ∈ S { [ x ] i : [ x ] i [ p ] i } exclude [ k ] i [ p ] i . Hence we obtain [ h ] i ⊆ [ k ] i .From this it follows that the lattice (cid:10) ℘ ( S ) △ , ⊆ (cid:11) is completely distributive.Replacing the relation R with R − in the above theorem we obtain: Theorem 24.
If the lattice h ℘ ( S ) N , ⊆i is spatial, then the following assertionsare equivalent: The lattice h ℘ ( S ) N , ⊆i is completely distributive. If R − ( s ) is a completely join-irreducible element of h ℘ ( S ) N , ⊆i , then [ s ] " [ { [ x ] : [ x ] [ s ] } (ei4) Theorem 25.
Let R be a reflexive antisymmetric relation. Then i (cid:10) ℘ ( S ) △ , ⊆ (cid:11) is completely distributive if and only if R is transitive. ii (cid:10) ℘ ( S ) △ , ⊆ (cid:11) is completely distributive if and only if h ℘ ( S ) N , ⊆i is completelydistributive. Proof. • If R is a reflexive and antisymmetric relation, then in view ofTheorem 22, (cid:10) ℘ ( S ) △ , ⊆ (cid:11) and h ℘ ( S ) N , ⊆i are spatial lattices and for any s ∈ S , [ s ] i is a completely join-irreducible element of ℘ ( S ) △ , and [ s ] iscompletely join-irreducible in ℘ ( S ) N . • Therefore, in view of Theorem 23 and Theorem 24, the relations (ei3) and(ei4) are satisfied for all s ∈ S . 29 • Suppose that (cid:10) ℘ ( S ) △ , ⊆ (cid:11) is completely distributive, and let Rus & Rsv . • Then [ s ] i " S { [ x ] i : [ x ] i [ s ] i } , according to (ei3). • Let n ∈ [ s ] i \ S { [ x ] i : [ x ] i [ s ] i } , then Rsn and n ∈ [ n ] i implies that [ s ] i ⊆ [ n ] i . • Since s ∈ [ s ] i , we also get Rns . By the antisymmetry of R we obtain n = s . Hence s ∈ [ s ] i \ S { [ x ] i : [ x ] i [ s ] i } . • Since s ∈ [ u ] i , we have [ s ] i ⊆ [ u ] i . As v ∈ [ s ] i (by assumption), weobtain v ∈ [ u ] i & Ruv . This proves the transitive property of R . • Conversely, if it is assumed that R is transitive, then R is a partialorder, and by the result in [28] ℘ ( S ) △ and ℘ ( S ) N are completelydistributive lattices. ii • Assume that (cid:10) ℘ ( S ) △ , ⊆ (cid:11) is completely distributive. Then, in view of (i) R and R − are partial orders. • Then by [28] h ℘ ( S ) N , ⊆i is also completely distributive. • The proof of the converse implication is completely analogous.By applying the above definitions and Proposition 5, we obtain:
Theorem 26.
Let ( S, R ) be a directed relational system and h B ( S ) , ·i the cor-responding groupoid. Then (cid:10) ℘ ( S ) △ , ⊆ (cid:11) is completely distributive if and onlyif ( ∀ a, b ∈ S )( ab = b −→ ( ∃ n, h ∈ S )( ∀ k, s ∈ S ) an = n & hb = b & kn = n & ( hs = s → ks = s )) (triagrp)
8. Knowledge Perspective
In a general approximation space S , if R is an equivalence, a partial order or aquasi order, then it is also possible to associate other groupoidal operations (see[42, 48, 50, 39]) on S . This is discussed in brief in Sec.2.5. But the associatedoperation is distinct from the one considered in this paper.General and classical rough sets have been associated with concepts of knowl-edge and studied from that perspective in a number of papers by the first author[47, 45, 41, 49, 37] and others[58, 57, 56, 11, 18]. The basic idea in the contextof classical approximation spaces [58] is to associate definite objects with con-cepts and consequently the equivalence relation R is associated with knowledge.In more general situations, granularity has a bigger role to play, and knowl-edge is defined relative to granular axioms used and other desirable properties.Examples of such conditions are GK1
Individual granules are atomic units of knowledge.30 K2 If collections of granules combine subject to a concept of mutual inde-pendence, then the result would be a concept of knowledge. The ’result’may be a single entity or a collection of granules depending on how oneunderstands the concept of fusion in the underlying mereology.
GK3
Maximal collections of granules subject to a concept of mutual indepen-dence are admissible concepts of knowledge.
GK4
Parts common to subcollections of maximal collections may be inter-preted as knowledge.
GK5
All stable concepts of knowledge consistency should reduce to correspon-dences between granular components of knowledges. In particular, tworelations R and R may be said to be consistent if and only if the setof granules associated with the two general approximation spaces havebijective correspondence.In [37] and [54] choice operations over granules are involved. But they do notgenerate groupoid operations on the general approximation space itself. Neitherdo the granular knowledge axioms of the kind mentioned. All this means that thegroupoid operation provides an additional layer of decision making that needsto integrated with existing work. A concrete practical example is considerednext to illustrate key aspects of this. In student-centered learning students are put at the center of the learn-ing process, and are encouraged to learn through active methods. Arguably,students become more responsible for their learning in such environments. Intraditional teacher-centered classrooms, teachers have the role of instructors andare intended to function as the only source of knowledge. By contrast, teach-ers are typically intended to perform the role of facilitators in student-centeredlearning contexts. A number of best practices for teaching in such contexts [23]have evolved over time. Teachers need to constantly improve their methods insuch teaching contexts because that is part of the methodology.Because of the open-ended aspect of the learning process, it is not expectedthat teachers have absolute control over the concepts learned. Students maythemselves arrive at new methods of solution or define new concepts as part ofthe learning process. In this scenario it is of interest to suggest potential higherconcepts that relate to the progress of the work in question. Teachers canpossibly provide some initial suggestions and subsequently these can be workedupon by algorithms relying upon datasets of concepts for improved suggestions.From the perspective of this research this becomes the problem of constructionof the best groupoid operations.In more precise terms, L1 Let A and B be two concepts arrived at by the learner. The open-endednature of the learning process means that a general rough set model ofconcepts must be adaptive or permit supervision.31 Teacher observes that concept C among others contains A and B in somesense, and offers suggestions relating to the scenario. S1 Software aid for the learning context provides better suggestions based on L1 and T1 using a groupoidal decision model instead of the former alone.In general available strategies that can be used to arrive at suggestionsbased on L1 alone are likely to be unintelligent.It may be noted that the impact of AI on enhancing classroom learning andlearning in general has been very limited (see [12] and related references). In factdigital technology in the context of mathematics teaching has been stagnatingbecause most of the effort has been on non-intelligent software that merely aidcommunication. There is no dearth of motivation for such work – Often teachersdo not have sufficient knowledge about the working of their students mind, havean excess of work load at hand and may be suffering from cognitive dissonancesof specific types.In a forthcoming paper by the first author, the rough methodology suggestedin this subsection is applied to specific practices such as opening of exercisesin the context of mathematics teaching [62, 53], use of explicit mathematicallanguage [65], and software for student expression [1, 12].
9. Further Directions and Remarks
In this research • the concepts of up-directed and up-directed parthood approximationspaces are invented, • their potential role in weak decision making is illustrated, • algebraic semantics of sets of granular, nongranular and local approxima-tions are invented and investigated in depth and shown to be nonequiva-lent, • algebraic semantics of roughly equivalent objects that involve additionalgroupoidal operations of decision making are invented and investigated, • their connection with knowledge and formal concept analysis are explored,and • possible applications to student centred learning is proposed.The results on connection with FCA supplement the work in [26].Parthood and apparent parthood relations have been the focus in higherorder granular approaches to rough sets in a number of papers by the firstauthor [47, 49, 45, 48, 51]. The results of this paper motivate connectionsbetween those and the lower order approach of this paper. Specifically it isof interest to identify the cases that are representable in terms of lower order32emantics. The groupoidal approach of this paper is also extended to the higherorder approaches in a separate paper.A groupoid S is tolerance trivial if every definable compatible tolerance on itis a congruence. Key results can be found in [60, 7]. This concept extends to allalgebras including the AR, AP, EUPGB and algebras of local approximations.In relation to knowledge interpretation, tolerance triviality amounts to a selforganizing aspect of knowledge. In other words, much less computational effortwould be required to impose an interpretation on the semantics. This aspectis also explored in concrete terms by the first author in a forthcoming paper inthe frameworks proposed in this research. References [1] P. Allen, Show And Tell Software To Explore Maori Ways Of Communi-cating Mathematically, in: J. Subramanian, et al. (Eds.), Proceedings ofMES10, Mathematics Education Society, 2019, pp. 207–211.[2] H. Burkhardt, J. Seibt, G. Imaguire, S. Gerogiorgakis (Eds.), Handbook ofMereology, Philosophia Verlag, Germany, 2017.[3] P. Burmeister, A Model-Theoretic Oriented Approach to Partial Algebras,Akademie-Verlag, 1986, 2002.[4] G. Cattaneo, Algebraic Methods for Rough Approximation Spaces by Lat-tice Interior–closure Operations, in: A. Mani, I. Düntsch, G. Cattaneo(Eds.), Algebraic Methods in General Rough Sets, Trends in Mathematics,Springer International, 2018, pp. 13–156.[5] G. Cattaneo, D. Ciucci, Lattices With Interior and Closure Operators andAbstract Approximation Spaces, in: J.F. Peters, et al. (Eds.), Transactionson Rough Sets X, LNCS 5656, Springer, 2009, pp. 67–116.[6] G. Cattaneo, D. Ciucci, Algebraic Methods for Orthopairs and inducedRough Approximation Spaces, in: A. Mani, I. Düntsch, G. Cattaneo (Eds.),Algebraic Methods in General Rough Sets, Birkhauser Basel, 2018, pp.553–640.[7] I. Chajda, Algebraic Theory of Tolerance Relations, Olomouc UniversityPress, 1991.[8] I. Chajda, H. Langer, Groupoids Assigned to Relational Systems, MathBohemica 138 (2013) 15–23.[9] I. Chajda, H. Langer, Weak Lattices, Italian Journal of Pure and AppliedMathematics (2013) 125–140.[10] I. Chajda, H. Langer, P. Sevcik, An Algebraic Approach to Binary Rela-tions, Asian European J. Math 8 (2015) 1–13.3311] M.K. Chakraborty, P. Samanta, On Extension of Dependency and Consis-tency Degrees of Two Knowledges Represented by Covering, in: J.F. Peters,A. Skowron (Eds.), Transactions on Rough Sets IX, LNCS 5390, SpringerVerlag, 2008, pp. 351–364.[12] S. Chorney, Digital Technology in Teaching Mathematical Competency: AParadigm Shift, in: A. Kajander, et al. (Eds.), Advances in MathematicsEducation, Springer International, 2018, pp. 245–256.[13] D. Ciucci, Back To The Beginnings: Pawlak’S Definitions of The TermsInformation System and Rough Set, in: G. Wang, et al. (Eds.), ThrivingRough Sets, Studies in Computational Intelligence 708, Springer Interna-tional, 2017, pp. 225–236.[14] B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, CambridgeUniversity Press, second edition, 2002.[15] J. Duda, I. Chajda, Ideals of Binary Relational Systems, Casopis pro pesto-vani matematiki 102 (1977) 280–291.[16] I. Düntsch, E. Orłowska, An Algebraic Approach To Preference Relations,in: Relational and Algebraic Methods in Computer Science - RAMICS’2011Proceedings of, pp. 141–147.[17] R. Freese, J. Jezek, J. Jipsen, P. Markovic, M. Maroti, R. Mckenzie, TheVariety Generated by Order Algebras, Algebra Universalis 47 (2002) 103–138.[18] B. Ganter, C. Meschke, A Formal Concept Analysis Approach to RoughData Tables, in: J.F. Peters, et al. (Eds.), Transactions on Rough Sets XIVLNCS 6600, volume LNCS 6600, 2011, pp. 37–61.[19] B. Ganter, R. Wille, Formal Concept Analysis: Mathematical Foundations,Springer, Berlin/Heidelberg, 1999.[20] G. Grätzer, General Lattice Theory, Birkhauser, 1998.[21] G. Gratzer, F. Wehrung (Eds.), Lattice Theory: Special Topics and Appli-cations Volume 1, Birkhauser Basel, 2014.[22] R. Gruszczyński, A. Varzi, Mereology Then and Now, Logic and LogicalPhilosophy 24 (2015) 409–427.[23] G.M. Jacobs, W.A. Renandya, A. Power, Simple, Powerful Strategies forStudent Centered Learning, Springer Briefs in Education, Springer Nature,2016.[24] J. Järvinen, Lattice Theory for Rough Sets, in: J.F. Peters, et al. (Eds.),Transactions on Rough Sets VI, volume LNCS 4374, Springer Verlag, 2007,pp. 400–498. 3425] J. Järvinen, P. Pagliani, S. Radeleczki, Information Completeness in NelsonAlgebras of Rough Sets Induced by Quasiorders, Studia Logica (2012) 1–20.[26] J. Järvinen, S. Radeleczki, Rough Sets Determined by Tolerances, Interna-tional Journal of Approximate Reasoning 55 (2014) 1419–1438.[27] J. Järvinen, S. Radeleczki, Representing Regular PseudocomplementedKleene Algebras by Tolerance-Based Rough Sets, Journal of The AustralianMathematical Society (2017) 1–22.[28] J. Jarvinen, S. Radeleczki, L. Veres, Rough Sets Determined by Qua-siorders, Order 26 (2009) 337–355.[29] J. Jezek, T. Kepka, Quasitrivial and Nearly Quasitrivial DistributiveGroupoids and Semigroups, Acta Univ. Carolinae Math et Phys 19 (1978)22–44.[30] J. Jezek, R. Mcenzie, Variety of Equivalence Algebras, Algebra Universalis45 (2001) 211–219.[31] J. Jezek, R. Quakenbush, Directoids: Algebraic Models of Up-DirectedSets, Algebra Universalis 27 (1990) 49–69.[32] T. Kepka, Quasitrivial Groupoids and Balanced Identities, Acta Univ. Car-olinae Math et Phys 22 (1981) 49–64.[33] K. Koslicki, Towards a Neo-Aristotelian Mereology, dialectica 61 (2007)127–159.[34] E.S. Ljapin, Partial Algebras and Their Applications, Academic, Kluwer,1996.[35] A.I. Malcev, The Metamathematics of Algebraic Systems – Collected Pa-pers, North Holland, 1971.[36] A. Mani, Algebraic Semantics of Similarity-Based Bitten Rough Set The-ory, Fundamenta Informaticae 97 (2009) 177–197.[37] A. Mani, Choice Inclusive General Rough Semantics, Information Sciences181 (2011) 1097–1115.[38] A. Mani, Dialectics of Counting and The Mathematics of Vagueness, Trans-actions on Rough Sets XV (2012) 122–180.[39] A. Mani, Towards Logics of Some Rough Perspectives of Knowledge, in:Z. Suraj, A. Skowron (Eds.), Intelligent Systems Reference Library dedi-cated to the memory of Prof. Pawlak ISRL 43 , Springer Verlag, 2013, pp.419–444.[40] A. Mani, Ontology, Rough Y-Systems and Dependence, Internat. J ofComp. Sci. and Appl. 11 (2014) 114–136. Special Issue of IJCSA on Com-putational Intelligence. 3541] A. Mani, Algebraic Semantics of Proto-Transitive Rough Sets, Transactionson Rough Sets XX (2016) 51–108.[42] A. Mani, Granular Foundations of the Mathematics of Vagueness, AlgebraicSemantics and Knowledge Interpretation, University of Calcutta, 2016.[43] A. Mani, Probabilities, Dependence and Rough Membership Functions,International Journal of Computers and Applications 39 (2016) 17–35.[44] A. Mani, Pure Rough Mereology and Counting, in: WIECON,2016, IEEX-Plore, 2016, pp. 1–8.[45] A. Mani, Approximations From Anywhere and General Rough Sets, in:L. Polkowski, et al. (Eds.), Rough Sets-2, IJCRS,2017, LNAI 10314,Springer International, 2017, pp. 3–22.[46] A. Mani, Generalized Ideals and Co-Granular Rough Sets, in: L. Polkowski,et al. (Eds.), Rough Sets, Part 2, IJCRS,2017 , LNAI 10314, SpringerInternational, 2017, pp. 23–42.[47] A. Mani, Knowledge and Consequence in AC Semantics for General RoughSets, in: G. Wang, et al. (Eds.), Thriving Rough Sets, volume 708 of