Granular knowledge and rational approximation in general rough sets – I

Rough sets are used in numerous knowledge representation contexts and are then empowered with varied ontologies. These may be intrinsically associated with ideas of rationality under certain conditions. In recent papers, specific granular generalisations of graded and variable precision rough sets are investigated by the present author from the perspective of rationality of approximations (and the associated semantics of rationality in approximate reasoning). The studies are extended to ideal-based approximations (sometimes referred to as subsethood-based approximations). It is additionally shown that co-granular or point-wise approximations defined by σ-ideals/filters (for an arbitrary relation σ) fit easily into the entire scheme. Concepts of the rationality of objects (vague or crisp) and their types are introduced and are shown to be applicable to most general rough sets by the present author. Surprising results on these are proved on these by her in this part of the research paper. The present paper is the first of a three part study on the topic.


Introduction
A number of granular and nongranular semantics of rough sets are known in the literature.Concepts of knowledge can be associated with these from multiple perspectives such as those based on classical rough ideas (Orłowska & Pawlak, 1984;Pawlak, 1991;Pawlak & Skowron, 2007), mereological axiomatic granular perspectives (Mani, 2018a(Mani, , 2018b(Mani, , 2020a)), classical granular computing (Yao, 2001;Zadeh, 1997), interpretations of modal logic (Mani et al., 2018;Pagliani & Chakraborty, 2008;Yao & Lin, 1996), constructive logic (Järvinen et al., 2012;Pagliani, 2018), concept analysis (Ciucci et al., 2012;Mani, 2018b;Yao, 2016), evidence theory and machine learning.Importantly, ideas of knowledge are not well developed in every generalisation of rough sets.Researchers also argue about the borders of rough sets from features of knowledge.This has, for example, motivated recent work on ideas of rational knowledge (with applications to validity in clustering) in the context of generalisations (and hybridisation) of variable precision rough sets (VPRS) and granular graded rough sets by the present author (Mani, 2022a;Mani & Mitra, 2022).It may be noted that the literature on VPRS and generalisations has been mostly focused on practical computations and not on semantics towards addressing the problems.Therefore, the issues at stake are about possible algebraic semantics, general frameworks and the very idea of rationality.The last-mentioned concept depends crucially on that of substantial parthood (Burkhardt et al., 2017;Mani, 2018b;Vieu & Aurnague, 2007) as opposed to parthood alone (specifically, VPRS and generalisations are all about a certain specific sense of being a substantial part of ).Mani (2018aMani ( , 2020a) ) are essential for following finer aspects of the granular and non-granular methods discussed in this research.
The concept of rational discourse in a context is typically determined by subjective, normative and rule-based constraints on associated concepts.Formal approaches to rationality are known in non-monotonic logic, evidence theory, general rough sets and epistemology.An overview of older theories of rationality and rational inference across can be found in the article (Kolodny & Brunero, 2020).A specific version of rationality called bounded rationality in which possible inferences are bounded or limited has found much application in applied logic.Bounded rationality is additionally relevant in the context of rough set applications.However, the boundedness aspect need not happen always.While deterministic rationality may be realised through well-constructed or formally-verified computer programs, it is possible to write totally unreliable random code (and compile it).People are likely to think rationally provided they have learnt reliable meta-level models or rules, and their application.Somewhat analogously, general rough approximation operators require additional layers of being well-constructed to qualify as rational ones (though not necessarily in a deterministic point of view).Importantly, the operators may be capable of handling uncertainty and vagueness at multiple stages of their construction.
From a purely theoretical perspective, this multipart study is about features of the defining conditions of approximations that potentially lead to new classifications of general rough sets and connections with related areas.In relation to these, four broad classes of rough sets (graded, VPRS, ideal based and classical) are investigated, and two more remain to be done.From an application point of view, reasonable (or rational) approximations in the context of general rough approximations are essential for applications to contexts that require high quality approximations or predictions (especially when robustness cannot be expected).This is very relevant in applications to human learning, and automated evaluation frameworks in education as shown by the present author in Mani (2020b).In the second part of this multipart research, the general rough approach to cluster validation introduced in Mani ( 2021) is actually shown to be related to rationality and learning frameworks.A hard clustering is said to valid if Apart from these a number of potential application areas like intelligent robust image segmentation with unlearning, epidemiology (where feature selection fails badly), unbalanced class problem and medical diagnostics can be indicatedthese are considered separately.
In Mani (2022a) and Mani and Mitra (2022), distinct concepts of rational approximations and rationally constructed objects are investigated in a mereological perspective.Substantial parthoods of various types are common in application contexts, however, the trend towards simplistic solutions ignoring model complexity has led to solutions of poor quality and reliance on excessive computation.It is therefore essential to build the foundations with such predicates and additionally explore related possibilities.Towards the same, the framework of pre-general pre-substantial granular space Section 2.4 is constructed in the mentioned papers.
In this research, new and unexpected connections of graded rough sets with idealbased (or subsethood) approximations are introduced, and the nature of substantial parthood is investigated in ideal-based rough sets in relation to the general framework for rational approximations.The basic properties of seven such relations are characterised.Of these five involve generalised ideals in their definition, while the remaining two are defined by cardinality constraints.
Suppose in a context, a single object has all the definite features of too many other objects, and not too many objects are the result of having all the definite features of other objects.It can be argued that the context does not involve enough diversity of features or that the process of feature evaluation is faulty among others.How does one formalise this scenario?A partial solution to this problem is additionally proposed through rationality types in general rough sets in this research.These types correspond to the nature of existence of objects in relation to their representation in terms of granules.It is shown to have classificatory value that applies to all types of rough sets.Needless to say, the formalisation of the ontological part depends on the assumptions of the agent.
The next part of this multipart research concerns a number of new results on granular VPRS, related distance functions, connections with ideas of rationality in evidence theory and belief functions, and applications to soft cluster validation are investigated.Most other types of rough sets that can coerced within the theoretical framework are covered in the third part.
This research paper is organised as follows: a guide to the necessary background is provided in the next section.In the third section, extensions of earlier results on graded rough sets (Mani, 2022a) to a type based scenario are introduced.Ideal-based co-granular rough sets and the nature of rational approximations are investigated in the following section.Rationality types are introduced and explored in detail in the fifth section.Further, directions are additionally provided in the sixth.

Background
Quantifiers are uniformly enclosed in braces for easier reading.So, ∀a∃b (a, b) is the same as (∀a∃b) (a, b).
Conditional implications of the form for every x whenever (x) holds, then (x) holds as well, are replaced (whenever possible) by Definition 2.1: A collection of subsets L = {L j : j ∈ J} is a τ k -covering of a set S if and only if all the following hold: • L is a set of pairwise incomparable subsets relative to the usual inclusion order, • L is a cover for S (that is, its union is S), and • if A is a subset of S which is not included in any L j , then there exist k elements {a 1 , a 2 , . . .a k } of A which are not included in any L j .
A τ 2 covering is also referred to as a normal cover (Chajda et al., 1976).
A lattice ideal K of a lattice L = (L, ∨, ∧) is a subset (possibly empty) of L that satisfies the following (≤ being the definable lattice order on L): Spec(L) shall denote the set of all prime ideals.A lattice filter is the dual of an ideal.Maximal lattice filters are the same as ultrafilters.In Boolean algebras, any filter F that satisfies (∀a) a ∈ F or a c ∈ F is an ultrafilter.Chains are subsets of a poset in which any two elements are comparable, while antichains are subsets of a poset in which no two distinct elements are comparable.Singletons are both chains and antichains.
For basics of partial algebras, the reader is referred to Burmeister (1986) and Ljapin (1996).
A partial algebra P is denoted by a tuple of the form with P being a set, f i 's being partial function symbols of arity r i .The interpretation of f i on the set P should be denoted by f P i ; however, the superscript will be dropped in this paper as the application contexts are simple enough.If predicate symbols enter into the signature, then P is termed a partial algebraic system.
For two terms s, t, s ω = t shall mean that, whenever both the terms are defined (after interpretation) then they are equal.That is (∀a, b)(s P = a &t P = b −→ a = b).ω = is the same as the existence equality (sometimes written as e =) in the present paper.

Rationality and semantic domains
Semantic domains are important in any logical or mathematical approach that focuses on meaning and models.It suffices to specify these without completely formalising them within specific domains of knowledge.One way of identifying semantic domains in contexts involving approximate reasoning or rough sets is through the type of objects involved in the discourse.This can be specified in a number of ways such as through direct observation of all attributes, observation of parts of an object's attributes, approximations of objects, operations permitted (or admissible predicates) on objects.
For example, a teacher's understanding of a lesson is very different from that of a student, and each may be approximating a certain knowledge structure (see Ball et al., 2008;Mani, 2022b for a more detailed discussion).In the context, more than the described six semantic domains may be constructed for the purpose of modelling from different perspectives simply because multiple agents and knowledge hierarchies are involved.
Consider the sentences',

S-A
The golden mountain is golden and a mountain.

S-B
The tallest mountain is tall and a mountain.
It is common knowledge that S-A is an unverifiable assertion, while S-B is a verifiable fact.However, relative to a knowledge base in which the concept of being tall is missing, both S-A and S-B would be unverifiable.Related semantic domains can be respectively formalised in different ways.
Informally, an information table is a usual table with each row corresponding to a crisp entity and columns corresponding to attributes.Cell entries are the attribute values for the entity.However, the main concerns are about objects defined by subsets of attributes and related valuations (for details, see Mani, 2018aMani, , 2020a)).In more abstract settings, one directly specifies conditions on approximation operators on sets or partial algebraic systems and with no reference to information tables.
Formally, an information table I is a tuple of the form with O, A and V a being respectively sets of Objects, Attributes and Values respectively.
being the valuation map associated with attribute a ∈ A. Values may also be denoted by the binary function ν : Relations may be derived from information tables by way of conditions of the following form: for x, w ∈ O and B ⊆ A, (x, w) ∈ σ if and only if (Qa, b ∈ B) (ν(a, x), ν(b, w), ) for some quantifier Q and formula .The relational system S = S, σ (with S = A) is said to be a general approximation space.In particular if σ is an equivalence relation, then S is referred to as an approximation space.
In the general rough context, the concepts of definite and rough (or vague) objects (these objects are in the rough semantic domain as opposed to the crisp entities of an information table in the classical semantic domain) are specified through one or more approximation operators and additional constraints.Concepts of crisp objects are additionally specified in the same way or may be designated as such.In the case of Pawlakian (or classical) rough sets, if the lower approximation of an object X coincides with itself, then X is said to be crisp or definite.Objects that are not definite are rough.There are several ways of representing rough objects in terms of crisp objects such as a pair of definite objects (A, B) with A ⊂ B. It may additionally be reasonable to think of sets of objects that properly contain common maximal crisp objects as a rough object or take orthopairs (Cattaneo & Ciucci, 2018) as the primary objects of interest.In the penultimate section of Mani (2016), knowledge representation in proto-transitive rough sets is shown to lead to more than three types of semantic domains.For a longer list, see Mani (2018a).Semantic domains restrict the the nature of admissible language, and therefore the nature of rationality, and rational approximations.If only lower approximations of objects are represented in a discourse, then the parthood relation would be restricted to these, and thus affect possible concepts of rational approximations and objects.Naturally, these lead to different rough semantic domains.
Classical rough sets (see the book Pawlak, 1991) starts from approximation spaces (derived from information tables) of the form S, R , where R is an equivalence on the set S. The Boolean algebra on the power set ℘ (S) with lower and upper approximation operations forms a model that does not describe the rough objects alone.This extends to arbitrary general approximation spaces where R is permitted to be any relation or even to those based on covers.The classical semantic domain associated with such classes of models may be understood in terms of the collection of restrictions on possible objects, predicates, constants, functions and low-level operations.
The problem of defining rough objects that permit reasoning about both intensional and extensional aspects posed in Chakraborty (2016) corresponds to identification of suitable semantic domains.Explicit perspectives, as for example in Cattaneo and Ciucci (2018) and Yao (2015), correspond similarly.Other semantic domains, including hybrid semantic domains, can be built from more complicated objects such as maximal antichains of mutually discernible objects (Mani, 2015(Mani, , 2017b) (mentioned earlier) or even over the power set of the set of possible order-compatible partitions of the set of roughly equivalent elements (Mani, 2009(Mani, , 2018a)).In fact, any general rough or soft reasoning context may be associated with a number of semantic domains (Mani, 2012(Mani, , 2014(Mani, , 2016(Mani, , 2017b(Mani, , 2018a(Mani, , 2020a)).These are sometimes vaguely referred to as meta levels in the artificial intelligence and machine learning (AIML) literature.For example, type-I and type-II fuzzy sets are read in terms of descriptions of functions from different meta levels.Thus the concept of semantic domains in abstract model theory (Mundici, 1984) is analogous to the usage here (though the formalisation of domains can be objected to because the additional work involved may not justify the coverage of issues afforded).Contamination (Mani, 2012(Mani, , 2018a(Mani, , 2020a) ) is the unjustified use of concepts from one semantic domain in another.For example, in a domain intended for reasoning about rough objects alone, presuming absolute knowledge of all objects would amount to contamination.Data intrusion, as in introducing poorly justified stochastic assumptions into a rough domain, is additionally a form of contamination.The problem of consistently avoiding such assumptions is the contamination problem.
Even when rough sets are formalised as well-formed formulas in a fixed language they do not refer to the same domain of discourse.For example, Banerjee and Khan (2007), Banerjee and Chakraborty (2004), Düntsch andOrłowska (2011), andMani (2005) refer to semantics of classical rough sets from different perspectives with that of Mani (2005) being a higher order semantics of the ability of objects to approximate -this is not expressible in the others.To see this, note that the algebraic model of Mani (2005) is built over blocks of a tolerance relation on the set of roughly equivalent objects, while the model of Banerjee and Chakraborty (2004) (for example) is essentially about the set of roughly equivalent objects.
For semantic domains in the context of fuzzy sets, the reader is referred to Turksen (2005).On page 46, the author essentially points out that possible solutions of the following problem (of subjective probability arguably) depend on the semantic domain used: A box contains ten balls of various sizes.Several of these are large, and a few are small.What is the probability that a ball drawn at random is neither large nor small?Implicit in the problem is that subjective perceptions determine the being of objects.From a rough perspective, information about attributes of the objects would be necessary.
From the above scenario, it can be expected that possible concepts of rationality of approximations are additional impositions on the domain of discourse.Current practices in rough sets do not explicitly require approximations to be rational (in the sense that the approximations are well-constructed from its constituents) in most cases; however, many conditions on possible decisions can be imposed.This lack of universality across domains further makes it natural to explore concepts of rationality at the object level.Apparently, the most convenient way is to define reasonable concepts through ideas of being a substantial part of that are related to rough approximations and rationality.
Rationality is additionally approached through formalisations of principles in the context of modal or nonmonotonic logics with epistemological concerns.Ideal-based rough sets can potentially be related to such theories, and this is an open problem.The relatively weaker assumptions on approximation operators can be a cause for concern.
Human reasoning is often not about truth of statements or grades of truth of statements.The valuation of statements with grades of truth in {F, T} or hypercubes generated by [0, 1] are typically meaningless and never intended.They are, however, used in soft decision-making with no proper validation or opaque expert approved mechanisms as in the LSP (Logic Scoring Preference) method (Dujmovic, 2018).Related assumptions do not provide a good basis for rationality.

Mereology
Mereology (Burkhardt et al., 2017) consists of a number of theoretical and philosophical approaches to relations of parthood (or is a part of predicates) and relatable ones such as those of being connected to, being apart from and being disconnected from.Such relations can be found everywhere, and they relate to ontological features of anybody of soft or hard knowledge (and their representation).In the second part of this paper, the relation between other mereological predicates and the core assumptions about the part-of relations used in this paper are discussed in more detail.It builds on earlier work of the present author (Mani, 2012(Mani, , 2018a(Mani, , 2018b(Mani, , 2020a) ) and others.
Many types of mereologies (Burkhardt et al., 2017;Cotnoir & Varzi, 2021;Janicki & Le, 2007;Lewis, 1991;Mani, 2012) are known in the literature.The differences can be about axioms (when a common formal language is possible) or domains of discourse.Cotnoir and Varzi (2021) imposes a common formal language on a number of approaches.Such reductionist reasoning is additionally evident in rough mereology (Polkowski, 2004;Polkowski & Polkowska, 2008), inspired by the ontology due to Lesniewski, where even the perspective of Grzegorczyk (1955) (that theorems of ontology are those that are true in every model for atomic Boolean algebras without a null element) is accepted.This makes the resulting model unsuitable for modelling human reasoning, though it appears to work for simpler robotic tasks, and such.Note that the relation of being roughly included to a degree r is not transitive.Reasoning about vagueness in mereological perspectives requires one to add additional predicates almost always, and therefore associated axiomatics is involved (Mani, 2012(Mani, , 2018b;;Polkowski, 2011) and may need to be enhanced with ontologies.Another mereology (Maffezioli & Varzi, 2021) that assumes many properties of the mereological predicates is, however, formalised in an intuitionist perspective.The extensionality of parthood and overlap considered in the work do not hold in rough reasoning.In Mani (2018b), predicates such as is-apparently-part-of, is-certainly-part-of, is-certainly-not-a-part-of and figures of opposition generated by these are studied.Of these, the predicate iscertainly-a-part-of is most closely related to the idea of being a-substantial-part-of that plays a central role in this research.
Apart from the motivations and reasons explained in Mani (2012Mani ( , 2018aMani ( , 2018b) by the present author, the need for formalising rationality and flexible languages for teaching and educational research provides additional motivations.The latter is explained in Mani (2022b).Basically, in classroom teaching of a subject like mathematics (and especially from a student-centric perspective; Jacobs et al., 2016;Mani, 2020b), it is useful to use a language that is easy to understand and at the same time be formalisable (possibly in a number of ways).In Mani (2022b), the use of mereology is suggested as a universal solution to the problem.It may be noted that partial formalisations of negative valuations in multisets (Felisiak et al., 2020) are related.

Granules and granulations
A granule may be vaguely defined as some concrete or abstract realisation of relatively simpler (or crisper) objects through the use of which more complex concerns may be solved.They exist relative to the issues being solved in question and can be specified in different non-equivalent ways.For example, they can be specified by the internal attributes of objects, precision levels of possible solutions or precision levels attained by objects.An axiomatic approach to granular computing is proposed by the present author in Mani (2012).It is based on a comparative study of the properties of various concrete approximations studied in earlier work.The basic assumptions about the underlying structure is weakened in her later works (Mani, 2016(Mani, , 2018a(Mani, , 2020a)).Some form of generalised transitivity of the part-of relation is still assumed in Mani (2018a).However, it is not required in later generalisations (Mani, 2020a) primarily because the transitivity property can be practically generalised in too many ways.Further improvements are in her later works (Mani, 2013(Mani, , 2018a(Mani, , 2020a)).Differences between primitive granular computing and classical granular computing (typically involving numeric precision values) (Lin, 2009;Lin & Liu, 1994;Liu, 2006;Yao, 1996Yao, , 2001Yao, , 2007;;Zadeh, 1979) are additionally explained.In Mani (2018aMani ( , 2020a)), it is actually argued that the latter can be traced to algorithms in ancient mathematics.
The axiomatic frameworks (AGCP) (Mani, 2012(Mani, , 2018a) ) do not refer to numeric precision for defining granules, and the problem of defining or rather extracting concepts that qualify require much work in the specification of semantic domains and process abstraction.A granulation is a collection of all granules and is denoted by G.For a granulation to be admissible, it is required that every approximation is term-representable by granules, that every granule in G coincides with its lower approximation (granules are lower definite), and that all pairs of distinct granules are part of definite objects (those that coincide with their own lower and upper approximations).
By a neighbourhood granulation G on a set W will be meant a subset of the power set ℘ (W) for which there exists a map λ : W −→ G such that λ is referred to as a neighbourhood map.Given a collection of granules, it may often be possible to generate newer types of interesting granules and granulations using settheoretical constraints.Atef et al. (2020), Al-shami and Ciucci (2022), Allam et al. (2006) and Mani (2017a) use special cases of neighbourhood granules to investigate subset neighbourhoods.
Let L, U : A)} for some formula (possibly involving λ).Pointwise upper approximations are defined analogously.An example of a pointwise lower approximation is the one defined by The definition is necessarily tied to the language.
In general rough contexts involving any number of approximation operators, it is possible to speak of definite and rough objects in multiple senses that arise from preferred concepts of approximate equality and approximations.Even situations in which the goal is to look for possible explanations that fit approximations specified by agents without explanation can be handled by the general frameworks of granular operator spaces/partial algebras introduced by the present author.For detailed explanations of the assumptions underlying the frameworks of this section, the reader is referred to the papers (Mani, 2020a;Mani & Mitra, 2022) and related references.The high adjective is used as an abbreviation for higher order.Key definitions are mentioned below for convenience.
Definition 2.2: An high mereological approximation Space (mash) S is a partial algebraic system of the form S = S, l, u, P, ≤, ∨, ∧, ⊥, where S is a set, and l, u are operators : S −→ S that satisfy the following (S is replaced with S if clear from the context.∨ and ∧ are idempotent partial operations and P is a binary predicate): In a high general granular operator space (GGS), defined below, aggregation and co-aggregation operations (∨, ∧) are conceptually separated from the binary parthood (P) and a basic partial order relation (≤).Parthood is assumed to be reflexive and antisymmetric.It may satisfy additional generalised transitivity conditions in many contexts.Real-life information processing often involves many non-evaluated instances of aggregations (fusions), commonalities (conjunctions) and implications because of laziness or supporting metadata or for other reasons -this justifies the use of partial operations.Specific versions of a GGS and granular operator spaces have been studied in the research paper (Mani, 2018a).Partial operations in GGS permit easier handling of adaptive granules (Skowron et al., 2016) through morphisms.The universe S may be a set of collections of attributes, labelled or unlabelled objects among other things.

Definition 2.3:
A High General Granular Operator Space (GGS) S is a partial algebraic system of the form S = S, γ , l, u, P, ≤, ∨, ∧, ⊥, where S = S, l, u, P, ≤, ∨, ∧, ⊥, is a mash and γ is a unary predicate that determines G (by the condition γ x if and only if x ∈ G) an admissible granulation(defined below) for S. Further, γ x will be replaced by x ∈ G for convenience.Let P stands for proper parthood, defined via Pab if and only if Pab & ¬Pba.A granulation is said to be admissible if there exists a term operation t formed from the weak lattice operations such that the following three conditions hold: If a granulation is admissible, then the approximations l and u are granular.
Definition 2.4: • In the above definition, if the anti-symmetry condition PT2 is dropped, then the resulting system will be referred to as a Pre-GGS.If the restriction Pa l a is removed from UL1 of a pre-GGS, then it will be referred to as a Pre*-GGS.
• In a GGS (resp Pre*-GGS), if the parthood is defined by Pab if and only if a ≤ b, then the GGS is said to be a high granular operator space GS (resp.Pre*-GS).• A higher granular operator space (HGOS) (resp Pre*-HGOS) S is a GS (resp Pre*-GS) in which the lattice operations are total.• In a higher granular operator space, if the lattice operations are set theoretic union and intersection, then the HGOS (resp.Pre*-HGOS) will be said to be a set HGOS (resp.set Pre*-HGOS).In this case, S is a subset of a power set, and the partial algebraic system reduces to S = S, γ , l, u, ⊆, ∪, ∩, ⊥, with S being a set, γ being a unary predicate that determines G (by the condition γ x if and only if x ∈ G).Closure under complementation is not guaranteed in it.
In general rough sets, approximations may be granular (in the axiomatic sense), or pointwise, or abstract or co-granular.Abstract approximations are those operators that are merely required to satisfy a few universal conditions.Classical rough approximations can be formalised in all four perspectives.

Frameworks for rationality
In earlier papers (Mani, 2022a;Mani & Mitra, 2022), the concept of a pre-general presubstantial granular space (pGpsGS) and stronger forms of partial algebraic systems were proposed to describe a framework for comparing multiple types of rough sets.This assumes that approximations are granular in the sense of the present author (Mani, 2012(Mani, , 2020a(Mani, , 2022a) because all pGpsGS are assumed to be pre*-GGS.To accommodate non-granular approximations, it may be necessary to weaken or remove the granularity axioms of weak representability (WRA), lower stability (LS) and full underlap (FU).A full discussion of the connections are done in the next part of this paper, as the approximations are not always granular in this paper.For convenience, the main definition is The reader is referred to the mentioned papers for relevant explanations and missing concepts.
(∀a) P s aa (sub1) If a pGsGS satisfies the condition PT1, then it will be referred to as a General Substantial Granular Space.

Representation of concept maturation
In Mani (2020b), a model for the purpose of evaluation of descriptive responses/expla nations to structured questions based on directed rough sets is proposed.The questions are part of concept inventories designed to test the understanding of concepts (and possibly the reasons for failure to understand) of students (see Mani, 2020b;Sands et al., 2018).
A practical example concerning the evaluation contexts is constructed to demonstrate aspects of the framework in this section.Suppose that labelled or partially labelled data about the understanding of concepts in a mixed class (consisting of students aged 5-11 years, say) is available in the form (Student_Id, Concept).This can be used to generate a pGpsGS S under the following interpretation: S is a set of sets of crisp and vague concepts, ∨: union, ∧: intersection, : a universal set of collections of concepts, ⊥ is the set containing the empty set of concepts or whatever that should be regarded as empty, l: the greatest set of crisp concepts that are part of the given element, u: the smallest set of crisp concepts of which the given element is part of, and the granulation predicate γ is determined by a suitable subset of sets of concepts.A parthood relation P can be defined on the set of concepts in a perspective as follows: P ab ↔ b is an accessible super concept from a Transitivity of P can fail possibly because the agency of accessibility may require more.In many cases, it may be transitive.Antisymmetry cannot be expected to be true without an additional layer of equivalencing because different concepts that mutually explain each other are always possible.For example, straight lines can be modelled by different types of algebraic equations.

Proposition 2.5: P is a reflexive relation that is not antisymmetric on the set of concepts, that in turn induces a reflexive non-antisymmetric relation on the granulation G.
Irrespective of whether P is a quasi-order or not, a filter of P is any subset F ⊆ S that satisfies Filters, in this sense, may be read as a weak generalisation of sets closed under the consequence afforded by P, or as sets that consist of elements that are not part of elements beyond those in the set.
Such a filter can be used to define a substantial part of relation as follows: P s ab if and only if a ∈ F and Pab.Concrete definitions of rational lower approximation relative to such a substantial parthood can then be formulated as The lower approximation l s is rational because it consists of concepts that are not unbounded in the sense afforded by P. Rational upper approximations on the other hand would need to be computed by its definition.

Summary and extensions: granular graded rough sets
In graded rough sets, approximations are constructed relative to integral grades that are related to the cardinality of sets.Neighbourhood granulations of points are used to construct point-wise approximations in Yao and Lin (1996), while equivalent granular approximations generated by equivalences are studied in Zhang et al. (2012).These are generalised to arbitrary granulations and explored by the present author in a recent work (Mani, 2022a).While such granulations can be related to variable precision rough sets, they can be suggestive of the use of mutually inconsistent procedures in their construction.A few incorrect semantic claims in Zhang et al. (2012) are also rectified in the research (Mani, 2022a).Additionally, many examples and applications are part of the paper.Semantics of graded modal logics (see Chen et al., 2021;de Rijke, 2000 and related references) and their limitations are applicable to the non-granular context of the paper (Yao & Lin, 1996).
A grade k is a fixed positive integer that refers to the cardinality of granules or sets used.Let S be a collection of subsets of a universe H, and G a subset of S. If x is a set in the collection S, then the k-lower and k-upper approximations, and related regions are (\ is used for the set difference operation) The lower approximation can result in strange values, and the following regularised version comes naturally: (k-reg.lower) Proposition 3.1: The range of the operations u k and l k on ℘ (H) need not be equal even when G is a partition of H, and S = ℘ (H).
In the context of granular graded rough sets, the following substantial parthood predicates are defined (Mani, 2022a): For any a, b ∈ S, and a fixed positive integer k The compatibility of these with defining conditions in the pGpsGS framework is given in Table 1 (Tr, Sy and Asy respectively abbreviate transitivity, symmetry and antisymmetry respectively.* indicates the property is satisfied and − that it is not necessarily satisfied) .This shows that the introduced framework may be sufficient for representing substantial parthood in the context of granular k-grade rough sets.Additional variants are also motivated.

Extensions by derived neighbourhoods
In a few recent papers (Al-shami & Ciucci, 2022;Allam et al., 2006;Atef et al., 2020), a number of neighbourhoods derived from more basic ones generated in a general approximation space are studied from a mathematical perspective.Some of them approximate better than approximations defined by successor neighbourhoods (in the point-wise perspective).However, the parthoods defining the approximations may fail to be substantial.It is of natural interest to investigate graded variants of the same.The main ideas are abstracted and granular graded variants are proposed below.A more thorough investigation of these cases will appear in the second part of this paper.They can serve as examples of point-wise co-granular graded rough sets -the cogranularity being in the sense of the present author (Mani, 2017a).In the mentioned paper, approximations based on lattice-theoretical concepts of ideals are generalised, characterised, and their spatial mereological features are explored.They directly relate to the style of the approximations used in Al-shami and Ciucci (2022), Allam et al. (2006) and Atef et al. (2020).The substantial parthoods of co-granular rough sets do not depend on arbitrary assumptions about grade or precision degrees, and they carry over to the neighbourhoods of this section too.On a general approximation space S = S, R , the basic neighbourhoods of a point x ∈ S are (the N notation is used in (Al-shami & Ciucci, 2022;Atef et al., 2020)) From these other neighbourhoods such as () These eight neighbourhoods are generally denoted by N ρ (x) (with ρ being a member of {l, r, l , r , i, u, i , u }).Relations between these neighbourhoods are used to generate other classes (E, P, C, S) of neighbourhoods in the same spirit.For example, the last mentioned class of neighbourhoods are The standard point-wise approximations for any given neighbourhood operation n : S −→ ℘ (S) the n-lower and n-upper approximations are defined as follows for any B ⊆ S The logical meaning of these classes of neighbourhoods and associated approximations is not explained in detail in the literature.While it is easy to obtain alternative granulations (finer, under additional conditions on the original relation), the neighbourhoods represent the interpretation of patterned connections in the model.For example, v is in the S l neighbourhood of x provided anything R-related to x is also R-related to v.So the neighbourhood is essentially a higher order representation of a derived relation.
Let N l (S) = {N l (x) : x ∈ S}.This collection is partially ordered by set inclusion, and it can be proved that Proposition 3.2: For each x ∈ S, the principal filter generated by N l (x) in N l (S) coincides with the set of N l -neighbourhoods of the elements of S l (x).
The proof follows by a direct set-theoretic argument for the two inclusions.It should be noted that the connection of filters or ideals generated similarly is a central motivation for defining co-granular approximations as a generalisation of pointwise approximations in the next section.This eventually means that a typetheoretic generalisation as opposed to the granular graded approach to rationality is also possible.The following example is used to illustrate the difference The table of neighbourhoods of R are given in Table 2.
These neighbourhoods can also be regarded as granulations, and relative to the granulation G = {S l (x) : x ∈ S}, the 2-graded approximations of the subset F = {a, b, e} are By contrast, the A S l and A S l approximations of F are respectively {e} and F. In addition, the A S l and A S l approximations of F are both equal to F. The graded lower approximations are necessarily proper.

Ideals and filters for rationality
If R is a binary relation on a set Z, then concepts of R-ideals and R-filters are definable.Under additional assumptions, such concepts may be interpreted as subsets closed under generalised consequence.In Abo-Tabl (2011), Allam et al. (2008), and Kandil et al. (2016), point-wise approximations that involve order-theoretic ideals in their definition are studied.They are generalised set-theoretically to general approximation spaces and R-ideals in Mani (2017a) relative to rough approximations by the present author.However, all considerations are restricted to point-wise approximations alone.
In this section, the essential definitions and results are reinterpreted, granular variations are introduced and potential connections with rationality are investigated.In this regard, it should be noted that a substantial part of the connections is already mentioned in Mani (2017a).The approximation framework embodies a kind of rationality in the following sense: if a property has little to do (in a structured way) with what something is not, then that something has the property in an approximate sense.The idea of having little to do with or to set no value of relative to operations is intended to be captured by concepts of ideals.Thus substantial parthood is potentially ensured by discarding the inessential.
Apart from these, the obvious cases of general approximation spaces of the form (Z, R) in which R is a partial or quasi or preference order (see the research chapter Mani, 2018a for details) should be usable to define filters and ideals, and induce generalised orders on granulations.These would easily lead to a number of applicable concepts of substantial parthoods and rational approximations.However, in the literature such investigations are not known, and these will be considered in a separate paper.

Definition 4.1:
• Let Z, R be a general approximation space where Z is a set and R a reflexive binary relation on Z • Let Z be an algebra of subsets of Z, I(Z) the lattice of lattice ideals of Z and ∈ I} ∪ A. The approximations will be referred to as Set difference approximations by ideals (IASD approximations).
This general approach proposed in Mani (2017a) is based on concepts of generalised ideals determined by binary relations on a set and those of ideals of partially ordered sets (Duda & Chajda, 1977;Rudeanu, 2015;Venkataranasimhan, 1971).It is additionally possible to use a binary relation on the power set to construct generalised ideals consisting of some subsets of the set.Essential definitions are recalled below.Definition 4.2: Let σ be any binary relation on a set H then • F ⊂ H is a σ -filter if and only if F is L-directed, and • The set of σ -ideals and σ -filters is denoted by I(H) and F(H) respectively.These are all partially ordered by the set inclusion order.If the intersection of all σ -ideals containing a subset B ⊂ H is an σ -ideal, then it will be called the σ -ideal generated by B and denoted by B .The collection of all principal σ -ideals is denoted by I σ (H).If x exists for every x ∈ H, then H is said to be σ -principal (principal for short).
If F is a subset of an U-directed set B, then it is possible that it is not U-directed.For this reason, the intersection of two σ -ideals is not necessarily a σ -ideal.
In general, if a lower or upper rough approximation (⊕) is defined by expressions of the form X ⊕ = {a : λ(a) X * ∈ I} where G ⊂ ℘ (S), λ : S −→ G is a neighbourhood map, * a complementation or identity operation, I a set of σ -ideals, and ∈ {∩, ∪}, then the approximation is said to be co-granular.
Theorem 4.4: The following hold in a GOSI S: The proof can be found in Mani (2017a).
Remark 4.1: Monotonicity of the approximations need not hold in general, the granulation is not admissible, and the approximations l * , u * are not granular.This is because the choice of parthood is not sufficiently coherent with σ in general.A sufficient condition is that σ be at least a quasi order.

Issues in granular generalisation
This generalisation was not considered earlier in Mani (2017a) as the focus was on the point-wise approximations.Let σ be a binary relation on a set H, an arbitrary collection of subsets of H, and [[g]] be the intersection of ideals contained in a g ∈ .The following four pseudo-approximations of a subset A of H may be said to be pseudo granular: While all four are representable in a strong sense relative to the condition WRA of Definition 2.3 (also see Mani, 2012Mani, , 2018a)), the approximations would be granular only if they satisfy the conditions LS and FU.This justifies the terminology.Without additional conditions on σ or , it is easy to see that the four are not even reasonable approximations.So rationality related conditions of the point-wise case cannot be extended, apparently.

Substantial parthood
The defining conditions of the approximations by σ -ideals are nicely relatable to part of relations.However, for the point-wise/co-granular approximations to be rational, the relation σ and the neighbourhood granulation should be well grounded in the context in the sense that σ and λ determine or express relevant causal relations.

Definition 4.5:
In what follows a and b are subsets of the universe, F a set of nonempty lower (l + ) definite subsets, and ξ Fa means there is a b ∈ F such that b ⊆ a.New substantial parthood relations can be defined as follows (the conditions s* and s3 are repeated from the context of Section 3): Of these, P i * s and P i3 s are barely relatable to the approximations or properties of σ or σ -ideals, while the last three are better grounded.For other approximations in the class, similar substantial parthoods are definable.The next few theorems concern representative properties of the other four.Additional properties of σ with weaker forms of the conditions in si1, si2, si5, si6, si5+ can define other interesting parthoods.
Theorem 4.6: If I σ (S) is closed under set intersection then the following hold for all a, b, e ∈ ℘ (S): Proof: (1) I σ (S) is closed under set intersection, therefore a ∩ b c , a ∩ e c and a ∩ b c ∩ e c are σ -ideals.So P i1 s a(b ∪ e) and si1A hold.(2) si1B is an extension of si1A.It is valid because additional constraints are not imposed on the first argument of P i1 s .
(3) The empty set is a trivial σ -ideal.So si1C holds.
(5) si1F follows from the properties of l * and si1E.Theorem 4.7: P i5 s satisfies all the following: ξ Fa −→ P i5 s aa (F-reflexivity) In general, a ⊆ b P i5 s ab (anti set inclusion) F9: F is closed under intersection.
(2) a l + ⊆ b l + ⊆ e l + follows from the premise.In conjunction with ξ Fa, it follows that P i5 s ae.
(3) Because, ξ Fa is not true for all a, the property of antiset inclusion follows.
Counterexamples are easy to construct.(4) P i5 s ab&P i5 s ba imply that ξ Fa and ξ Fb.This does not contradict the consequence a l + = b l + .
Theorem 4.8: P i6 s has the following properties: Proof: (1) If a ⊆ b, then a ∩ b c = ∅, and this is a trivial σ -ideal.ξ Fa in conjunction with this ensures that P i6 s ab.
(2) That P i6 s aa implies ξ Fa follows from the definition.(3) Monotony of approximations with ξ Fa that follows from P i6 s aa, ensures property i63.Theorem 4.9: (1) P i2 s is a symmetric, partially reflexive relation.
(2) If a = ∅, then for any b, ¬P i2 s ab (3) P i9 s is a not necessarily symmetric, partially reflexive relation.(4) If ξ Fa then for some b, ¬P i2 s ab.(5) If ξ Fa and ξ Fb then P i9 s ab implies P i9 s ba.

Proof:
(1) P i2 s ab if and only if a ∩ b / ∈ I σ (S), and because a ∩ b = b ∩ a, it follows that P i2 s ba.In general, for a specific a, P i2 s aa may or may not hold.So P i2 s is symmetric and partially reflexive.
(3) In general, a ∩ b / ∈ I σ (S)&ξ Fa does not imply that ξ Fb.So P i9 s is not necessarily symmetric.(4) The existence of a lower definite element that is a subset of a is guaranteed by ξ Fa.
From the above, it is clear that the last three conditions can serve as nice conditions defining substantial parthood, while s* and s3 are difficult to work with in general.The rest have the potential when the trivial cases are eliminated.This class of approximations is therefore very different from both VPRS and graded rough sets.It is, however, closer to rough sets with generalised orders on the set of attributes or collections of subsets of attributes.A number of parthoods are possible with additional conditions on the collection of σ -ideals.

Discussion and meaning
In the above, conditions of the form g \ A ∈ I for a neighbourhood or granule g essentially refer to the fact that those differences are similar to a generalised zero or an algebraically closed and absorptive subset.In the point-wise perspective, a point that generates such an instance is regarded as a part of the lower approximation.This happens because ideals of an algebra have absorptive properties relative to the algebraic operations and are closed.Further, in universal algebras with 0 (that are ideal determined), ideals are the 0-class of a congruence (Freese et al., 2022;Gumm & Ursini, 1984;Mckenzie et al., 1987).Additionally, the requirement that g \ A is an ideal means the following: • Whatever is a substantial part of g is also a substantial part of A (because the difference is a generalised zero) • if g is substantial, then it is a substantial part of A, and g \ A is not a substantial part of S.
These assertions are consistent.Moreover, if K is an ideal in a GOSI S, A a subset of S, and x ∈ S, then ( 1 ) This extends to the higher order variant of GOSI (GOSIH) (Mani, 2017a) as well.For reference, a higher co-granular operator space by ideals GOSIH is a structure of the form S = S, σ , G, l o , u o where S is a set, σ a binary relation on the powerset ℘ (S), G a neighbourhood granulation over S and l o , u o o-lower and o-upper approximation operators : ℘ (S) −→ ℘ (S) defined as follows (for any X ∈ ℘ (S), and a fixed I ∈ I σ (℘ (S))): By contrast, note that the condition g ∩ A / ∈ I means that g ∩ A may be a part of some σ -filters and σ -ideals, but is not a σ -ideal.Therefore, in a nice enough situation (e.g. when g is a neighbourhood of a point), g ∩ A is not an σ -ideal, and possibly closed under a generalised consequence afforded by the model.g may additionally be said to share a common substantial part with A. This is the point that there exists a c such that P s cg and P s cA.A paraconsistent interpretation of the scenario is also justified.
However, as noted in Mani (2017a), a subset of an ideal need not behave like a generalised zero in general.This statement is dialectically opposed (Mani, 2018b) to (or alternatively, runs counter to) the idea that the subset is part of a generalised zero.A higher order approach helps to simplify the interpretation.Partial interpretation of operations may be appropriate because of the nature of objects in a context (e.g.attribute values of objects may need to be integers).Such a contamination avoidance procedure can affect the nature of possible models, and in fact, serves as a motivation for introducing the notion of a GOSIH.In the relational approximation contexts of the papers (Abo-Tabl, 2011;Allam et al., 2008;Kandil et al., 2016), subsets of generalised zeros are generalised zeros.This results in wide differences with properties of their generalisations.Therefore, if a property has little to do (in a structured way) with what something is not, then that something has the property in an approximate sense.The idea of little to do with or set no value of relative operations is intended to be captured by concepts of ideals.
If σ -ideals are seen as essentially empty sets, then they have a hierarchy of their own and function like definite entities.The σ -ideals under some weak conditions permit the following association.If A is a subset, then it is included in the smallest σ -ideal containing it and a set of maximal σ -ideals contained in it.These may be seen as a representation of rough objects of a parallel universe.

Rationality of objects
The idea of rationality of objects can be read from the perspective of ontology (as in an object's existence is justified by its constructive definition) or from a distributive cognition perspective (as in the object's defining process, and therefore its environment, is a part of the agent's existence) -the latter involves a broader understanding of objects (see Mani, 2022b;Werner, 2020).In common language discourses, across languages, people often use phrases such as PCIE-5 slots on computer main-boards are insane (because their bandwidth cannot be saturated by anything that can be inserted there), this smartphone is not rational, or it has a mind of its own (to mean that it behaves erratically to input instructions), and the apples taste sweet (in reference to a collection of apples).The ability to ontologically represent such information is important for making robust AIML algorithms.Additionally, such representation can help in the construction of algorithms that avoid learning toxic behaviours (from text or speech data that objectifies women, for example Godoy & Tommasel, 2021) and understand jargon better when it matters among others.
Rough objects are defined in relation to approximations.However, such objects need not be functionally defined at the object level as can be seen from possible definitions.Their rationality can be expected to be definable in terms of the language of the rough domain.This is because the concept of rationality can only be relative to available reasoning machinery and associated facts.The concepts of substantial parthood used in the earlier papers in the context of granular graded/variable-precision rough sets, and the general frameworks may not be easily relatable as the language is expected to have built-in predicates.In the light of these observations, two new concepts of rational and co-rational objects are proposed next.
A rationally existing object will be an exact object that approximates as few objects as is possible.This would ensure that it remains distinct from most others from the rough perspective.A dual of this concept that will be referred to as a co-rationally existing object will be an exact object that approximates as many objects as is possible.The two formulations can be easily formalised in a number of ways in a rough language that has at least a parthood predicate, and unary lower and upper approximation operator symbols.Scope for expressing granularity is not essential.The following definitions over models may be rewritten to express similar ideas in a language.

Derivations for classical rough sets
For classical rough sets, sets of reasonable integers can represent the quality, and type of rational existence of objects in relation to its rough semantic domain.This is a surprising new result.
Let X = X, R be an approximation space with R being an equivalence relation on the finite set X. Further, let S = ℘ (X), the equivalence classes of R be g 1 , g 2 , . . .g k , and Card(X) = n, Card(g i ) = n i (for each i).The set of equivalence classes of R is the granulation G = {g 1 , g 2 , . . .g k }.An element x ∈ S (or equivalently, a subset of X) is said to lower (upper) definite if and only if x l = x (x u = x).As mentioned earlier, it is definite, if it is both lower and upper definite.Denoting the set of all lower (upper) definite elements by δ l (X) (δ u (X)), it is obvious that δ l (X) = δ u (X) = δ(X) -the set of definite elements.
Definition 5.1: For any element x ∈ δ(X), let l (x) = {z : z l = x} (lower approximates) Proposition 5.2: The following hold: In other words, μ l is an antitone valuation, while μ u is monotone.
Proof: l and u are many-one functions on S. l and u are respectively their inverse images.Therefore, if a = b, then it is necessary that both l (a) ∩ l (b) and u (a) ∩ Elements of l (a) will have the form a ∪ x, where x ⊂ ∪{G \ {g 1 , g 2 , . . .g r }} and that x does not include any granule.Elements of l (b) will also have the form b ∪ z, where z is a union of proper subsets of granules.Therefore, it follows that μ l (b) < μ l (a).The actual formula is derived in the following theorem.
Elements of u (a) will have the form x ⊆ a subject to the condition that at least one element of each of the granules included in a are part of x.Since a is included in b, elements of u (b) would be constructible in more number of ways.This ensures the monotonicity of μ u .

Theorem 5.3: For each granule g
For a definite object a = r i=1 g i , Proof: The essence of the matter is the statement in the last proof that elements of l (a) will have the form a ∪ x, where x is a subset of ∪{G \ {g 1 , g 2 , . . .g r }} that does not include any granule.This means that x can include 0, 1, . . .n r+1 − 1 elements from g r+1 and so on.Each of these number of elements can be selected in n j f ways with the variable f ranging over 0 ≤ f < n j ) and so the possible combinations is the sum.The product is the result of combining the selections over the allowed granules.

Theorem 5.4: For each granule g
For a definite object a = r i=1 g i , Proof: The essence of the matter is that elements of u (a) will have the form ∪ i=1 rx i , where x i is a nonempty subset of g i .This means that x can include 1, . . .n i elements from g i for i = 1, . . ., r.Each of these number of elements can be selected in n j f ways with the variable f ranging over 1 ≤ f ≤ n j and so the possible combinations are the sum.The product is the result of combining the selections over the r granules.
Remark 5.1: This approach offers an easier higher order approach to classify classical rough set models and instances.Additionally, it takes the ability of definite objects to approximate others into account.
Definition 5.5: The l-rationality type of an approximation space X will be the totally ordered multi-set lrt(X) = {μ l (x) : x ∈ δ l (X)} (in decreasing order).Similarly, the u-rationality type of an approximation space X will be the totally ordered multi-set urt(X) = {μ u (x) : x ∈ δ u (X)} (in increasing order).
In practice, the identification of the values of μ l and μ u corresponding to granules would be essential for using the l-rationality and u-rationality types effectively for comparing different situations from a rough perspective.Definition 5.6: Let B l : δ l (X) −→ Z + , and B u : δ u (X) −→ Z + be two functions defined by the local l-rationality and local u-rationality types will be the totally ordered multi-set llrt(X) = { μ l (x) * 2 n B l (x) : x ∈ δ l (X)} (in decreasing order) and the totally ordered multi-set nurt(X) = { μ u (x) * 2 n B u (x) : x ∈ δ u (X)} (in increasing order).
The local types are intended to account for the number of elements in the lower and upper boundaries, and therefore can be related to the quality of approximation in the context.
Problem 5.7: Problems on bounds on the size of μ l and μ u can possibly be solved from a purely combinatorial perspective.
The number of equivalences with exactly k classes on a set X of cardinality n is given by While the set of all equivalence relations on the set is given by Bell's equation: The first few Bell numbers in order are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, . . . .If an equivalence relation has k classes, then the cardinality of δ(X) = Card(℘ (G)) would be 2 k .However, the rationality types are intended to work across different values of n.
The approximations of the sets are as in Table 4.In the table, sets of the form {1, 2} have been abbreviated as 12.The neighbourhood granules generated by the elements are given in Table 3.Note that G = {[x] : x ∈ S}, Card(G) = 5, the empty set is a definite element and not a granule.

More general rough contexts
Initially, tolerance spaces are explored, and the concepts are partly extended to more general contexts.These carry over to the general frameworks for rational approximation and granularity.Rough approximations over tolerance spaces (or similarity approximation spaces) can be defined in a number of nonequivalent ways, and these may or may not involve granulations.This helps in illustrating the difficulty of characterisations based on cardinalities of granules alone in the approaches.
Let X = X, T be a general approximation space (tolerance space) with T being a tolerance relation on the finite set X. Possible semantics depend on choice of granulation (see Mani, 2018a for details).Some choices of granulations in the context are the following: (1) The collection B = {g 1 , g 2 , . . .g k } of blocks (maximal subsets B of S that satisfy B 2 ⊆ T) (2) The collection of successor N and predecessor N i neighbourhoods generated by T and (3) The collection T = {∩( ) : ⊆ B}.These will be called the collection of squeezed blocks.
B may appear to be the most natural choice because it is a proper generalisation of the concept of a partition associated with an equivalence relation.In fact, it forms a normal cover.Related representation theorems can be found in the chapter on duality in Mani (2018c).For this case, multiple semantic approaches have been developed by the present author (Mani, 2009(Mani, , 2011(Mani, , 2017b)).Choice functions over blocks are involved in the construction of approximations in Mani (2011), while partial algebraic models for bitten approximations are proposed in Mani (2009).The approach in It can be seen that the concepts extend from the classical case to both granular, and point-wise approximations because the entire definition is about the ability to approximate.For any subset x of X, approximate bounds on μ l * (x) and μ u * (x) can be computed given additional assumptions about the cardinality of g i ∩ g j for g i , g j ∈ G.The bounds are ensured by the representation results for tolerances by normal covers (see Chajda et al., 1976).Rationality types can however be defined by analogy.
Definition 5.11: The l*-rationality type of a tolerance space X will be the totally ordered multi-set l * rt(X) = {μ l * (x) : x ∈ δ l * (X)} (in decreasing order).Similarly, the u*-rationality type of X will be the totally ordered multi-set u * rt(X) = {μ u * (x) : x ∈ δ u * (X)} (in increasing order).
In practice, the identification of the values of μ l and μ u corresponding to granules would be essential for using the l-rationality and u-rationality types effectively for comparing different situations from a rough perspective.Further, the types can be used to classify general rough sets.
In Table 6, U(x, x) = [x] i and L(x, x) = [x].Given the above information, it can be deduced that the nontrivial σ -ideals are For distinct lattice ideals many approximations of A by l i and u i can be computed.GOSIH related computations of approximations are bound to be cumbersome even for four element sets and so have been omitted.

Further directions and conclusions
From the present research, it is clear that only some generalised rough contexts have a mechanism for controlling the rationality of approximations or objects: graded rough sets, VPRS, probabilistic rough sets, dominance based rough sets, ideal based rough sets, and generalised order based rough sets (including their generalisation and hybridisation).Of these, ideal-based versions and the rationality of objects are investigated in this first part of a three-part research paper.Generalised order based rough sets will be considered separately as it is a huge topic.
The suitability of the introduced framework for handling substantial parthood is fairly clear from the properties satisfied.Some definitions of substantial parthoods in granular graded and VPRS contexts involve generalised metric condition.This is not the case for ideal-based rough sets (and those defined by subset neighbourhoods) that require specifications based on ideals (or other types of sets).The universal properties of the substantial parthood relations form a basis for deeper explorations.
Rationality types introduced are shown to be useful for encoding patterns of approximation and would be useful for comparing different rough contexts of the same theoretical type (like probabilistic, VPRS, and ideal-based), and are by no means restricted by the necessity of substantial parthoods.Once again, they work the best for classical rough sets.These results are new and a bit surprising.In the anti-chain based semantics of Mani (2015Mani ( , 2017b)), many operations are studied over antichains constructed in relation to discernibility.The connections of the rationality types, and related results are of natural interest in the context.The learning theory proposal is work in progress.

Definition 4. 3 :
By a Co-Granular Operator Space By Ideals, GOSI, is meant a structure of the form S = S, σ , G, l * , u * where S is a set, σ a binary relation on S, G a neighbourhood granulation over S determined by the neighbourhood map λ, and l * , u * are operators : ℘ (S) −→ ℘ (S) satisfying the following (S is replaced with S if clear from the context): (4) P i6 s ab&P i6 s ae imply that a ∩ b c and a ∩ b e are σ -ideals and by F9 that a ∩ b c ∩ e c = a ∩ (b ∪ e) c .Combining this with ξ Fa yields the conclusion P i6 s a(b ∪ e).

u
(b) are empty.If a ⊂ b, then they must differ by the union of some granules.So let

Table 1 .
Summary k-grade rough sets.

Table 2 .
A few neighbourhoods.

Table 5 .
Upper and lower bounds.