Lattice-Valued Topological Systems as a Framework for Lattice-Valued Formal Concept Analysis

Recently, Denniston, Melton, and Rodabaugh presented a new categorical outlook on a certain lattice-valued extension of Formal Concept Analysis (FCA) of Ganter and Wille; their outlook was based on the notion of lattice-valued interchange system and a category of Galois connections. This paper extends the approach of Denniston et al. clarifying the relationships between Chu spaces of Pratt, many-valued formal contexts of FCA, lattice-valued interchange systems, and Galois connections.


Introduction
This paper considers a particular application of the theory of variety-based topological systems, introduced in [1] as a generalization of topological systems of Vickers [2], which in turn provide a common framework for both topological spaces and their underlying algebraic structures frames (also called locales), thereby allowing researchers to switch freely between the spatial and localic viewpoints.Vickers' concept, which was motivated by the theory of crisp topology, has recently drawn the attention of fuzzy topologists, who have incorporated the notion in their topic of study.The first attempt in this respect was done by Denniston and Rodabaugh [3], who considered various functorial relationships between topological systems of Vickers and latticevalued topological spaces of Rodabaugh [4].Soon afterwards, the study of Guido [5,6] followed; he used topological systems in order to construct a functor from the category of -topological spaces to the category of crisp topological spaces, thereby providing a common framework for various approaches to the hypergraph functor of the fuzzy community (see, e.g., [7]).
It soon appeared though that the translation of the most important tools of the theory of topological systems (e.g., system spatialization and localification procedures) into the language of lattice-valued topology required a suitable fuzzification of the former concept, which was accomplished successfully by Denniston et al. in [8].The authors brought their theory into maturity in [9], where they considered its applications to both lattice-valued variable-basis topology of Rodabaugh [4] and (, )-fuzzy topology of Kubiak and Šostak [10].Later on, they introduced a particular instance of their concept called interchange system [11], which was motivated by certain aspects of program semantics (the socalled predicate transformers) initiated by Dijkstra [12].It should be noted, however, that an interchange system is a particular instance of the well-known concept of Chu space in the sense of Pratt [13] (the original notion goes back to Barr [14]).Moreover, the interchange system morphisms of [11] are precisely the Chu space morphisms (called Chu transforms) of [13].(In [15], Denniston et al. use the terms transformer system and transformer morphism.)The underlying motivations of interchange systems and Chu spaces are quite different, which is reflected well enough in both their names and their respective theories.
Parallel to the previously mentioned developments, we introduced the concept of variety-based topological system [1], which not only extended the setting of Denniston et al., but also included the case of state property systems of Aerts [16][17][18] (introduced as the basic mathematical structure in the Geneva-Brussels approach to foundations of physics), considered in [19] in full detail, and we brought the functor of Guido to a new level [20] (e.g., made it variable-basis in the sense of Rodabaugh [4], as well as extended the machinery of Höhle [7], to construct its right adjoint functor).Moreover, in [21], we provided a thorough categorical extension of the system spatialization procedure to the variety-based setting, the respective localification procedure being elaborated in [22].Additionally, certain aspects of the new theory bearing links to bitopological spaces of Kelly [23] as well as to noncommutative topology of Mulvey and Pelletier [24,25] were treated extensively in [26,27], respectively.
Seeing the fruitfulness of their newly introduced notions, both Denniston et al. and the author of this paper independently turned their attention to possible applications of the arising system framework to the areas beyond topology.In particular, in [28], we started the theory of lattice-valued soft universal algebra, whereas Denniston et al. presented in [29,30] a challenging categorical outlook on a certain lattice-valued extension of Formal Concept Analysis (FCA) of Ganter and Wille [31].Similar to the case of lattice-valued topological systems, their research motivated us to take a deeper look into the proposed topic of lattice-valued FCA, thereby streamlining the motivating approach of Denniston et al. and also extending some of their results.
Modifying slightly the definition of the category of Galois connections of McDill et al. [32], Denniston et al. [29] consider its relationships to the already mentioned category of interchange systems, the objects of both categories being essentially formal contexts of FCA.The main result is a categorical embedding of each category into its counterpart, the ultimate conclusion though amounting to the categories in question being rather different; that is, the two viewpoints on FCA are not categorically isomorphic.The rest of the paper provides a lattice-valued, fixed-basis extension of the crisp achievements, based on the lattice-valued FCA of Bêlohlávek [33].Moreover, towards the end of the paper, with the help of the constructed embeddings, the authors compare the categories of (lattice-valued) formal contexts and (lattice-valued) topological systems inside the category of (lattice-valued) interchange systems, making a significant (and rather striking) metamathematical conclusion on their being disjoint.
The just mentioned approach of Denniston et al. can be extended, taking into consideration the following facts.Firstly, it never gives much attention to the links between interchange systems and Chu spaces, thereby making no use of the well-developed machinery of the latter.Secondly, while pursuing their approach to many-valued FCA, the authors never mention the already existing many-valued formal contexts of Ganter and Wille [31], which (being similar to Chu spaces) essentially extend their own.Thirdly, being categorically oriented, the paper introduces two types of formal context morphisms, in terms of Chu transforms and Galois connections, respectively, but misses the wellestablished context morphisms of FCA, which are different from those of the paper.Lastly (and most important), turning to fuzzification of their notions, the authors concentrate on the fixed-basis lattice-valued approach, notwithstanding that the more promising variable-basis setting (initiated by one of them) has already gained its popularity in the fuzzy community, currently being a kind of de facto standard for fuzzy structures.
The main purposes of this paper are to extend the results of Denniston et al. and to clarify the relationships between Chu spaces, many-valued formal contexts of FCA, latticevalued interchange systems, and Galois connections.In particular, we introduce three categories (each of them having a twofold modification, arising from its respective variablebasis), whose objects are lattice-valued versions of formal contexts of FCA, but whose morphisms are quite different, each instance reflecting the approach of Pratt, Ganter and Wille, and Denniston et al., respectively.Following the results of [29], we embed (nonfully) each of the categories into its two counterparts, showing, however, that the constructed embeddings fail to provide a categorical isomorphism (even an adjoint situation) between any two of the presented three approaches.As a consequence, we obtain three different categorical outlooks on FCA, which are done in the variable-basis lattice-valued framework of [4].Moreover, the respective underlying lattices of the fuzzified formal contexts are quantales [34,35] instead of complete residuated lattices of Denniston et al.; that is, the heavy assumption on commutativity of the quantale multiplication is dropped.In particular, we show lattice-valued generalizations of formal concept, preconcept, and protoconcept of [36], which are based on an arbitrary (namely, non-commutative) quantale (actually, an algebra, having a quantale as its reduct).
Similar to Denniston et al., we develop the required machinery of quantale-valued Galois connections, which extends that of Erné et al. [37] and which is motivated by the commutative approach of fuzzy Galois connections of Bêlohlávek [38].For example, we provide yet another generalization to the quantale setting of the well-known fact that every order-reversing Galois connection between powersets arises as a pair of Birkhoff operators with respect to a binary relation [39].The generalization though appears to be somewhat truncated, in the sense that while latticevalued binary relations still give rise to order-reversing Galois connections, the converse way is not always available ( [38,40] do restore the full result even in the lattice-valued case, but rely on the more demanding notion of fuzzy Galois connection).Moreover, we define a category of order-preserving Galois connections as a certain subcategory of the category of Chu spaces, in which (unlike McDill et al. and Denniston et al.) Galois connections play the role of morphisms rather than objects, thereby providing particular instances of Chu transforms.We fail to provide a similar machinery for the order-reversing case.
At the end of the paper, we will notice that the approach to FCA through order-reversing Galois connections taken up by Denniston et al. does not suit exactly the underlying ideas of FCA of Ganter and Wille, which are formulated in terms of the incidence relations of formal contexts.The main sticking point here is the previously mentioned fact that the important crisp case one-to-one correspondence between order-reversing Galois connections on powersets and pairs of Birkhoff operators breaks down in the latticevalued case (quantales).As a consequence, the incidence relation viewpoint provides one with a Galois connection, whereas the latter alone lacks the strength of bringing a respective incidence relation in play (cf.Problems 1 and 2).
The paper is based on both category theory and universal algebra, relying more on the former.The necessary categorical background can be found in [41][42][43].For algebraic notions, the reader is referred to [34,35,44,45].Although we tried to make the paper as much self-contained as possible, it is expected from the reader to be acquainted with basic concepts of category theory, that is, those of category and functor.

Algebraic Preliminaries
For convenience of the reader, we begin with those algebraic preliminaries, which are crucial for the understanding of the results of the paper.Experienced researchers may skip this section, consulting it for the notations of the author only.

Varieties of Algebras.
An important foundational aspect of the paper is the abstract algebraic structure (called algebra, for short) which is a set, equipped with a family of operations, satisfying certain identities.The theory of universal algebra calls a class of finitary algebras (induced by a set of finitary operations), which is closed under the formation of homomorphic images, subalgebras, and direct products, a variety.In this paper, we extend the notion of varieties to include infinitary algebraic theories.Our motivation includes ideas of [45][46][47].

Definition 1.
Let Ω = (  ) ∈Λ be a (possibly, proper or empty) class of cardinal numbers.An Ω-algebra is a pair (, (   ) ∈Λ ), which comprises a set  and a family of maps    Theorem 6.The residuations ⋅ →  ⋅ and ⋅ →  ⋅ of a quantale  have the following properties.
Proof.To give the flavor of the employed machinery, we show the proofs of some of the claims of the items.
Definition 10.Frm is the variety of frames, that is, strictly two-sided quantales, in which multiplication is the meet operation.
Definition 11.CBAlg is the variety of complete Boolean algebras, that is, frames , in which ⋀ are considered as primitive operations (cf.Definition 1) and which additionally are equipped with the complement operation (−)  , assigning to every  ∈  an element   ∈  such that  ∧   = ⊥  and  ∨   = ⊤  .
The varieties CBAlg, Frm, STSQuant, UQuant, and CSLat(⋁) provide a sequence of variety reducts in the sense of Definition 3. Additionally, Frm is an extension of CQuant.
The notations related to varieties in this paper follow the category-theoretic pattern (which is different from the respective one of the fuzzy community).From now on, varieties are denoted by A or B, whereas those which extend the variety CSLat(Ξ) of Ξ-semilattices with Ξ ∈ {⋁, ⋀} will be distinguished with the notation L. S will stand for either subcategories of varieties or the subcategories of their dual categories.The categorical dual of a variety A is denoted by A  .However, the dual of Frm uses the already accepted notation Loc [48].Given a homomorphism , the corresponding morphism of the dual category is denoted by   and vice versa.Every algebra  of a variety A gives rise to the subcategory S  of either A or A  , whose only morphism is the identity map

Powerset Operators.
One of the main tools in the subsequent developments of the paper will be generalized versions of the so-called forward and backward powerset operators.
More precisely, given a map  1   →  2 , there exists the forward (resp., backward) powerset operator P( 1 ) The two operators admit functorial extensions to varieties of algebras.(For a thorough treatment of the powerset operator theory, the reader is referred to the articles of Rodabaugh [49,[51][52][53], dealing with the lattice-valued case, or to the papers of the author himself [21,54], dealing with a more general varietybased approach.) To begin with, notice that given an algebra  of a variety A and a set , there exists the -powerset of , namely, the set   of all maps    → .Equipped with the pointwise algebraic structure involving that of ,   provides an algebra of A. In the paper, we will often use specific elements of this powerset algebra.More precisely, every  ∈  gives rise to the constant map    →  with the value .Moreover, if A extends the variety CSLat(⋁) of ⋁-semilattices, then every  ⊆  and every  ∈  give rise to the map Of particular importance will be the characteristic maps of singletons, that is, {} , the latter case assuming additionally that A extends the variety UQuant of unital quantales.
Coming back to powerset operators, the following extension of the forward one has already appeared in a more general way in [28], being different from the respective lattice-valued approaches of Rodabaugh.
Theorem 12.Given a variety L, which extends CSLat(⋁), every subcategory S of L provides the functor Set × S (−) →   → CSLat(⋁), which is defined by (( 1 ,  1 ) To show that the functor preserves composition, notice that given Set × S-morphisms ( and every The case S = S  is denoted by (−) →  and is called a (variety-based) fixed-basis approach, whereas all other cases are subsumed under (variety-based) variable-basis approach.Similar notations and naming conventions are applicable to all other variety-based extensions of powerset operators introduced in this subsection and, for the sake of brevity, will not be mentioned explicitly.
There is another extension of the forward powerset operator, which uses the idea of Rodabaugh [52] (the reader is advised to recall the notation for lower and upper adjoint maps from the previous subsection).Before the actual result, however, some words are due to our employed powerset operator notations.In all cases, which are direct analogues of the classical powerset operators of the fuzzy community (e.g., those of Rodabaugh [51] and his followers), we use the solid arrow notation, that is, "(−) ← " (resp., "(−) → ") for backward (resp., forward) powerset operator.In the cases, which are not well known to the researchers, we use dashed arrow notation, that is, "(−) \ " (resp., "(−) [ ") for backward (resp., forward) powerset operator, possibly, adding some new symbols (e.g., "⊣" or "⊢") to underline the involvement of upper or lower adjoint maps in the definition of the operator in question (e.g., (−) ⊣[ or (−) ⊢[ ) as in the following theorem.

Theorem 13.
(1) Given a variety L, which extends CSLat(⋀), every subcategory S of L  gives rise to the functor Set × S (2) Let L be a variety, which extends CSLat(⋁), and let S be a subcategory of L  such that for every Proof.We rely on the fact that every ⋀-(resp., ⋁-) semilattice is actually a complete lattice and, therefore, is a ⋁-(resp., ⋀-) semilattice.The proof then follows the steps of those of Theorem 12, employing the fact that given CSLat(⋀)-(resp., The respective generalization of the backward powerset operator is even stronger (see, e.g., [21]), in the sense that the employed varieties do not need to be related to any kind of lattice-theoretic structure.Theorem 14.Given a variety A, every subcategory S of A  induces the functor Set × S (−) ←  → A  , which is defined by To show that the functor preserves composition, notice that given Set × S-morphisms ( 1 ,  1 ) → ( 2 ,  2 ) and → ( 3 ,  3 ), for every  ∈ Similar to the case of variety-based forward powerset operators, we have the following modifications of the functor introduced in Theorem 14.
In the paper, we will use the following dualized versions of the functors of Theorems 12 and 13.

Theorem 16.
(1) Given a variety L, which extends CSLat(⋁), every subcategory S of L  gives rise to the functor Set  × S (−) →   → (CSLat(⋁))  defined by (( 1 ,  1 ) (2) Given a variety L, which extends CSLat(⋀), every subcategory S of L gives rise to the functor Set  × (3) Let L be a variety, which extends CSLat(⋁), and let S be a subcategory of L such that for every S- Additionally, the functors of Theorems 14 and 15 can be dualized as follows.

Theorem 17.
(1) Given a variety A, every subcategory S of A induces the functor Set  × S (−) ←   → A, which is defined by The reader should notice that the functors of Theorem 17 are not standard in the fuzzy community.
Notice that the second item of Theorem 19 is responsible for the term "order-preserving" (resp., "order-reversing") in Definition 18.Moreover, in view of the results of Theorem 19, there exists the following equivalent definition of Galois connections.
To conclude, we would like to notice that this paper studies the properties of the classical Galois connections on lattice-valued powersets.In contrast [38,40], employ the notion of fuzzy Galois connection, which is particularly designed to fit the lattice-valued powerset framework.Such a modification is needed to preserve the main results valid on crisp powersets.The preservation in question comes, however, at the expense of a more complicated definition of fuzzy Galois connections themselves, which will be off our topic of study.

Variety-Based Topological Systems and Their Modifications
In [1], we introduced the concept of variety-based topological systems as a generalization of Vickers' topological systems [2] mentioned in the Introduction section.Moreover, in [22], we presented a modified category of variety-based topological systems, to accommodate Vickers' system localification procedure.For convenience of the reader, we recall both definitions, restating them, however, in a slightly more general way, to suit the framework of the current paper.
Definition 22.Given a variety A, a subcategory S of A  , and a reduct B of A, S B -TopSys is the category, concrete over the product category Set × B  × S, which comprises the following data.
The following notational remark applies to all the categories of systems introduced in this paper.
Remark 23.The case S = S  is called (variety-based) fixedbasis approach, whereas all other cases are subsumed under (variety-based) variable-basis approach.Moreover, if A = B, then the notation S B -TopSys is truncated to S-TopSys, whereas if S = S  , then the notation is shortened to  B -TopSys.In the latter case, the objects (resp., morphisms) are truncated to  = (pt , Ω, ⊨) (resp.,  = (pt , Ω)).
The next example gives the reader the intuition for the new category, showing that the latter incorporates many of the already existing concepts in the literature.
(2) If A = B = Frm, then the category Loc-TopSys is the category of lattice-valued topological systems of Denniston et al. [8,9].
(3) If A = CQuant, then the category  Set -TopSys is the category -IntSys of lattice-valued interchange systems of Denniston et al. [11,29,30], which is also the category -TransSys of lattice-valued transformer systems of Denniston et al. [15] (notice that both categories are fixed-basis).
(4) If A = B = Set, then the category -TopSys is the category Chu  of Chu spaces over the set  of Pratt [13].In particular, the objects of Chu 2 are called formal contexts in Formal Concept Analysis of Ganter and Wille [31], whereas the objects of the category Chu  itself are called many-valued formal contexts [31,Section 1.3].Additionally, the respective context morphisms proposed by Ganter and Wille [31,Chapter 7] are different from the morphisms of Chu spaces, which will be dealt with in the subsequent sections of the paper.
Moreover, the following definition (extended and adapted for the current setting from [15]) illustrates the concept of a topological system morphism.
The reader can easily see (or verify) that the objects of the category GalCon are posets, and its morphisms are precisely the order-preserving Galois connections, which is reflected in its name.This category is essentially different from the category GAL of Galois connections of [32], in the sense that while the latter category uses Galois connections as its objects, the category GalCon uses them as morphisms.It will be the topic of our future research, to consider the properties of the category GalCon (at least) up to the extent of the respective ones of the category GAL.Notice, however, that the topic of the category GAL will find its proper place in the subsequent developments of the paper.
The reader should also be aware of the important fact that the case of order-reversing Galois connections to the knowledge of the author) does not allow the previously mentioned system interpretation.Indeed, the already mentioned adjunction condition now reads as  1 ⩽ ( 2 ) iff  2 ⩽ ( 1 ) for every  1 ∈  1 and every  2 ∈  2 .With the help of the dual partial order ⩽  on  2 , we do get an order-preserving Galois connection It is unclear though how to incorporate in the system setting another order-reversing Galois connection ( 2 , ⩽) , that is, how to define the morphism composition law of the possible category (notice that ( 2 , ⩽) and ( 2 , ⩽  ) are different as posets; i.e., the codomain of the first connection is not the domain of the second one).
In the next step, we introduce the modified category of topological systems inspired by [22].
Definition 26.Given a variety A, a subcategory S of A, and a reduct B of A, S B -TopSys  (notice that "" stands for "modified") is the category, concrete over the product category Set × B  × S, which comprises the following data.
Objects.Tuples  = (pt , Ω, Σ, ⊨), where (pt , Ω, Σ) Since both a category and its dual have the same objects, the only difference between the categories of systems introduced in Definitions 22 and 26 is in the direction of the Σcomponent of their respective morphisms.In particular, with the notational conventions of Remark 23, we get that all the items of Example 24, except the second one, can be restated in terms of the category S B -TopSys  .Moreover, it is easy to provide the conditions, under which the system settings of Definitions 22 and 26 coincide.  →  2 .
As a consequence, we obtain that there is no difference between the (fixed-basis) categories  B -TopSys and  B -TopSys  .There exists, however, another obvious modification of the category of topological systems, which has never appeared in the literature before.
Definition 29.Given a variety A, a subcategory S of A  , and a reduct B of A, S B -TopSys  (notice that "" stands for "alternative") is the category, concrete over the product category Set × B × S, which comprises the following data.
Objects.Tuples  = (pt , Ω, Σ, ⊨), where (pt , Ω, Σ) It is important to emphasize that leaving (essentially) the same objects, we change the definition of morphisms dramatically.In particular, we cannot obtain a single item of Example 24 in the new framework.
Similar to the previously mentioned classical system setting, we can introduce a modified category of systems.Definition 30.Given a variety A, a subcategory S of A, and a reduct B of A, S B -TopSys  is the category, concrete over the product category Set × B × S, which comprises the following data.
Objects.Tuples  = (pt , Ω, Σ, ⊨), where (pt , Ω, Σ) The explicit restatement of Theorem 28 for the setting of the categories S B -TopSys  , S B -TopSys  is straightforward and is, therefore, left to the reader.

Lattice-Valued Formal Concept Analysis
For the required preliminaries in hand, in this section, we introduce different approaches to fuzzification of basic tools of Formal Concept Analysis (FCA) of Ganter and Wille [31].In particular, we present three possible category-theoretic machineries, the first two of which stem from the approach of Denniston et al. [29], whereas the last one comes from Ganter and Wille themselves.For the sake of brevity, we omitted every motivational reason for the need of fuzziness in FCA, which is clearly stated in, for example, [33], and also could be found at the beginning of Section 5 of [29].Our main interest is not the idea of FCA itself, but rather a better categorytheoretic framework for its successful development.

Lattice-Valued Formal Contexts as Chu Spaces.
In the last item of Example 24, we represented Chu spaces over sets of Pratt [13] as particular instances of variety-based topological systems.Moreover, we also mentioned that one of the main building blocks of FCA, that is, (many-valued) formal contexts, is, in fact, nothing else than Chu spaces (already noticed in the literature [55,56]).With these ideas in mind, we introduce the following definition.(2) S-FC    is a new notation for the category S B -TopSys  .
The notational and naming remarks for objects and morphisms of Definition 31 will apply to all the categories of (lattice-valued) formal contexts introduced in this paper.Moreover, the reader can easily apply Theorem 28, to single out the isomorphic subcategories of the categories S-FC  and S-FC   .At the moment, there is no agreement in the fuzzy community, whether to build formal contexts over L or L  , and, therefore, we just add the lower index "(−)  " in the former case.More precisely, this paper is (up to the knowledge of the author) the first attempt to develop a variable-basis approach (à la [4]) to lattice-valued FCA, the motivational article of Denniston et al. [29] being fixed-basis.Ganter and Wille.In the previous subsection, we presented a latticevalued approach to FCA based in Chu spaces.An experienced reader, however, may be well aware of many-valued contexts of Ganter and Wille themselves [31, Section 13], the theory of which is already established in the literature.In the following, we provide its category-theoretic analogue, bringing together both many-valued contexts and their morphisms.Definition 32.Let L be an extension of Quant and let B = Set.

Lattice-Valued Formal Contexts in the Sense of
(1) S-FC  (notice that "" stands for " Ganter and Wille") is a new notation for the category S B -TopSys  .
(2) S-FC   is a new notation for the category S B -TopSys  .
It should be emphasized immediately that neither Ganter nor Wille (up to the knowledge of the author) has considered a categorical approach to (many-valued) formal contexts, that is, has united both formal contexts and their morphisms in one entity.Additionally, both (many-valued) formal contexts and their morphisms are treated in an algebraic way in [31] (from where our notation for context morphisms, which is different from the system setting, is partly borrowed).There does exist some research on certain categories of contexts (e.g., [57][58][59][60][61]), which, however, does not study the categorical foundation for FCA, but rather concentrates on a fixed categorical framework and its related results.It is the main purpose of this paper, to investigate such possible categorytheoretic foundations and their relationships to each other (leaving their related results to the further study on the topic).In particular, we consider a category of contexts, whose morphisms are strikingly different from the standard framework of topological systems.

Lattice-Valued Formal Contexts in the Sense of Denniston et al.
In [29], Denniston et al. have introduced another approach to lattice-valued FCA, which was based on the category GAL of order-preserving Galois connections of [32].In the following, we extend their approach in two respects.Firstly, we employ the variety Quant of quantales instead of CQuant of commutative ones of [29] (called there complete commutative residuated lattices).Secondly, we provide a variable-basis approach (à la [4]) to the topic, instead of the fixed-based one of [29].
To begin with, we introduce some new notions and notations (induced by the respective ones of [29]), which will be used in the subsequent developments of the paper.
Given a lattice-valued context K, the notation  for the elements of   (resp.,  for the elements of   ), that is, latticevalued sets of context objects (resp., attributes), will be used throughout the paper.
To give the reader more intuition, we provide an example of the just introduced notions.Proof.For convenience of the reader, we provide the proof, which is different from that of [29].
We emphasize immediately that Theorem 35 is an analogue of the already obtained results of [38,Lemma 3] (for  a complete residuated lattice) and of [40, Proposition 3.9] (for  a unital quantale) in the framework of fuzzy Galois connections, which is different (more demanding) from the setting of Galois connections of the current paper (see additionally the remark after Theorem 42).There also exists an extension of Theorem 35 to a more general context of (commutative, right-distributive, complete) extended-order algebras  (whose multiplication in some cases need not be associative) [62, Propositions 3.9, 5.5] and, additionally, to even more general case, which involves relations instead of maps (the so-called -Galois triangles) [63].
Everything is now in hand, to define another latticevalued extension of formal contexts of FCA.
Definition 36.Given a variety L, which extends Quant, and a subcategory S of L, S-FC  (notice that "DMR" stands for "Denniston, Melton and Rodabaugh") is the category, concrete over the product category Set × Set  , which comprises the following data.
Objects.Lattice-valued formal contexts K (in the sense of Definition 31) such that  is an object of S.
2 ), which make the diagrams It should be emphasized that Definition 36 is inspired by the respective one of [29], which, in its turn, comes from the definition of the category GAL of order-preserving Galois connections of [32].Moreover, two important features of the category S-FC  are worth mentioning.Firstly, it never depends on the morphisms of the category S, mentioned at the beginning of its definition (S-objects, however, are employed), which is reflected well enough in its truncated ground category.Secondly, its underlying functor to the ground category Set × Set  is defined on objects as |K| = (  ,   ), which is a huge difference from the already introduced categories of formal contexts (based on the machinery of topological systems), the underlying functor of which is just |K| = (, , ).
By analogy with the already presented categories of lattice-valued formal contexts, we introduce another version of the category of Definition 36.Unlike the case of the categories S-FC  and S-FC  , where the change concerns the direction of the algebraic part of their morphisms, this time we change the set-theoretic morphism part and, therefore, use the notation "(−)  " ("alternative") instead of "(−)  " ("modified").
Definition 37. Given a variety L, which extends Quant, and a subcategory S of L, S-FC   is the category, concrete over the product category Set×Set, which comprises the following data.
Objects.Lattice-valued formal contexts K (in the sense of Definition 31) such that  is an object of S.
2 ), which make the diagrams commute.
In the following, we adapt the technique of Theorem 28, to single out the possible isomorphic subcategories of the categories S-FC The reader should be aware that the fixed-basis approach (i.e., the case of S = S  ) will (in general) provide nonisomorphic categories -FC  and -FC   , since both of them are not dependant on S-morphisms.More precisely, it is possible to obtain the following result.
Theorem 40.Suppose that  is not a singleton, and, additionally, its quantale multiplication is not a constant map to ⊥  .Then the categories -FC  and -FC   are nonisomorphic.
Proof.It is easy to see that the context K = (, , ), where  is the empty set and  ×    →  is the unique possible map, provides a terminal object in the category -FC   .The category -FC  , however, does not have a terminal object, which can be shown as follows.
Suppose the context K  = (  ,   ,   ) is terminal in -FC  .Define a context K  = (  ,   ,   ), where   is a nonempty set and   = ⊤  (the constant map with value ⊤  ).One gets two equal constant maps      =  =⊤    →    .
Since there exists an -FC  -morphism K  =(,)   → K  , the diagrams commute.It follows that the map   in the right-hand diagram is constant (since   has that property).Thus, if         →    maps everything to ⊤  , then every map       →    , taking ⊤  to the constant value of the map   , gives another -FC  -morphism K      → K  .The uniqueness of such morphism necessitates    to be a singleton, and then   is the empty set (since  is not a singleton).As a consequence, we get that the map   is constant as well (with the value ⊤  ).Define now another context K  = (  ,   ,   ), where   is a nonempty set and   = ⊥  .Since the quantale multiplication of  is not the constant map to ⊥  , we get a nonconstant map and, therefore, a contradiction, since the left-hand side of the previous two diagrams factorizes then (with the help of some -FC  -morphism K      → K  ) the nonconstant map   through the constant map   .
The case when  is a singleton clearly provides the isomorphic categories -FC  and -FC   .The case when  is equipped with the constant ⊥  -valued quantale multiplication results in all context maps ,  being constant (with the value ⊤  , which should additionally be preserved by the corresponding maps  and   (resp., )), which does not fit the machinery of Theorem 40, and will be treated in our subsequent articles on lattice-valued FCA.We should emphasize, however, immediately that the just mentioned two cases are not that interesting (i.e., are usually skipped by the researchers), and, therefore, we can rightly conclude that the approaches of Definitions 36 and 37 are fundamentally different, which provides a (partial) justification for our alternative setting of the category -FC   .The difference between the categories S-FC  and S-FC   is similar to that between the categories S-FC  and S-FC  (which served as our main motivation for introducing the approach of S-FC   , which was mentioned, but never studied, in [29]).

Some Properties of Lattice-Valued Birkhoff Operators.
It is a well-known (and easy to establish) fact that all orderreversing Galois connections between crisp powersets arise from a pair of crisp Birkhoff operators of Example 34.In the following, we show that the lattice-valued case destroys partly this essential property (the reader is advised to recall the notational remarks concerning particular maps of the powerset algebras, mentioned at the beginning of Section 2.2).
Theorem 41.If L extends UQuant, then every lattice-valued context K satisfies the following.
Theorem 42.Let ,  be sets and let  be a unital quantale.For every order-reversing Galois connection (  , , ,   ), the following statements are equivalent.
In particular, (  ,  ⊗ ,  → − ,   ) is an order-preserving Galois connection (recall Definition 18).It is the main purpose of [64, Proposition 7.1] to clarify the conditions, when an orderpreserving Galois connection (  , , ,   ) has the previous form, that is, is generated by a map  ×    → .We end the subsection with another simple (but useful) property of lattice-valued Birkhoff operators.Lemma 43.Given a lattice-valued formal context K such that  = ⊤  , it follows that  = ⊤  and  = ⊤  .In particular, in the crisp case,  =  and  = .
Using the properties of Galois connections, mentioned in Theorem 19, one can easily get that every concept is a protoconcept, and every protoconcept is a preconcept, whereas the converse implications, in general, are not true.Moreover, the next result provides simple properties of the notions of Definition 44 with respect to the morphisms of the previously mentioned categories S-FC  and S-FC   . ( ( ) is injective, then  (resp., ) reflects fixed points in the sense of (2).
Ad (3).It follows from ( 2) and the last claim of the theorem.

Ad (3). "⇒": If
The two previous theorems provide crucial tools for defining certain functors between the categories of latticevalued formal contexts, which is the main topic of the next section.

Functorial Relationships between the Categories of Lattice-Valued Formal Contexts
The previous section introduced several categories (partly motivated by the respective ones of [29]), the objects of which are lattice-valued formal contexts.It is the purpose of this section to consider functorial relationships between the new categories, the functors in question being again partly inspired by the respective approach of [29].As a result, we not only extend (the change from commutative quantales to noncommutative ones and the shift from fixed-basis to variable-basis) and streamline the machinery of Denniston et al., but also provide several new functors (induced by new categories), thereby clarifying the relationships between different frameworks for doing lattice-valued FCA and bringing to light their respective (dis)advantages.

S-FC 𝐶 and S-FC 𝐶
versus S-FC  and S-FC   .In this subsection, we consider possible functorial links between the categories S-FC  and S-FC   , on one side, and the categories S-FC  and S-FC   , on the other.

From S-FC 𝐶
to S-FC   .In this subsection, we construct a functorial embedding of a particular subcategory of the category S-FC    into the category S-FC   .We begin with singling out the subcategory in question.Notice that the notation (−) * will usually be used for the constructed subcategories in this paper, the scope of each (−) * being limited to its respective subsection.More important constructions, however, will be distinguished, respectively.
With the category of Definition 47 in hand, we can construct the following functor (the reader is advised to recall our powerset operator notations from Section 2.2).

Theorem 48. There exists the functor S-FC
Proof.To show that the functor is correct on morphisms, we verify commutativity of the diagrams For the left-hand diagram, notice that given  ∈ where ( †) relies on Theorem 6(2), ( † †) uses the fact that For the right-hand diagram, notice that given  ∈ where ( †), ( † †) use the definition of the category S-FC   * , whereas ( † † †) employs the fact that K 1   → K 2 is a morphism of the category S-FC   .In the notation "  ", "" shows its domain, and "" (short for "") stands for its codomain.Similar notational conventions will be used in the remainder of this paper but will not be mentioned explicitly again.
In the following, we make the new functor into an embedding.The result depends on a particular subcategory of its domain.
Definition 49.Suppose that the variety L, of which S is a subcategory, extends the variety UQuant of unital quantales.S-FC    * * (resp., S-FC   * • ) is the full subcategory of the category S-FC   * , whose objects K = (, , , ) are such that  is nonempty (resp.,  is nonempty) and, moreover, The notation (−) * * (resp., (−) * • ) will be used for many of the subcategories constructed in this paper, the scope of each being limited to its respective subsection.Additionally, more important constructions will be distinguished in a different way.It is also important to emphasize that the algebra requirement of Definition 49 is satisfied by the crisp case (see Definition 31 (1)).Proof.We consider the case of the functor  *  first and show its faithfulness.Given To show that  1 =  2 (which we will refer to as ), notice that given  1 ∈  1 , → ( for every  2 ∈  2 .As a consequence, we obtain that ← (

Since
(( 1 ,  1 ) ← ( Turning the attention to the functor  •  , one follows the same steps to obtain  1 =  2 =  and For the nonfullness claim, we restrict the setting to the crisp case (L = CBAlg and S = S  and  •  , we consider the case of the former one.Suppose there exists some K ({⊥}) = , which is an obvious contradiction.

From S-FC 𝐶
to S-FC  .In this subsection, we construct a functorial embedding of a particular subcategory of the category S-FC    into the category S-FC  .We begin again with singling out the subcategory in question.

Definition 51. S-FC 𝐶
* is the (nonfull) subcategory of the category S-FC    , with the same objects, whose morphisms With the new definition in hand, we can construct the following functor.

Theorem 52. There exists the functor S-FC
Proof.To show that the functor is correct on morphisms, we verify commutativity of the diagrams For the left-hand diagram, notice that given  ∈ where ( †) relies on the fact that K 1   → K 2 is a morphism of the category S-FC   , whereas ( † †) employs the definition of the category S-FC    * (notice that the inverse of  is easily seen to be  ⊢ ).
For the right-hand diagram, notice that given  ∈ where ( †) uses the definition of the category S-FC   * , whereas ( † †) relies on the fact that K 1   → K 2 is a morphism of the category S-FC   .
The category S-FC   * * (resp., S-FC   * • ) of Definition 49 provides the following result.For the new category in hand, in the following, we construct its respective functor.
Proof.To show that the functor is correct on morphisms, we verify commutativity of the diagrams For the left-hand diagram, notice that given  ∈ where ( †) uses Definition 54 (notice that   ⊢ is a left inverse to   ), whereas ( † †) relies on the fact that K 1   → K 2 is a morphism of S-FC  .
For the right-hand diagram, notice that given  ∈ where ( †) uses Definition 54, whereas ( † †) relies on the fact In the following, we make the new functor into an embedding.
Definition 56.Suppose that the variety L, of which S is a subcategory, extends UQuant.S-FC  * * (resp., S-FC  * • ) is the (nonfull) subcategory of the category S-FC  * , comprising the following data.
Morphisms.S-FC  * -morphisms K 1   → K 2 , for which The reader should notice the essential difference in the setting of Definition 49, namely, an additional condition on the morphisms of the new category.
Proof.To show that the functor is correct on morphisms, we verify commutativity of the diagrams For the left-hand diagram, notice that given  ∈

S-FC 𝐶
) to the category S-FC  (resp., S-FC   ).In particular, assuming that L extends the variety UQuant of unital quantales, whereas S-FC  † * is the domain of the newly constructed functor (see the results of following subsections), where † is ether "" or ", " we can define the following two subcategories.
Definition 73.S-FC  † * * (resp., S-FC  † * • ) is the full subcategory of the category S-FC  † * , whose objects K are such that  is nonempty (resp.,  is nonempty) and, moreover, Employing the technique of Section 5.1, we can easily get that the restriction of the respective constructed functor to the previously mentioned two categories provides two embeddings.To save the space, therefore, we will not mention the just discussed developments explicitly.
Proof.To show that the functor is correct on morphisms, we verify commutativity of the diagrams For the left-hand diagram, notice that given  ∈ where ( †) uses Definition 74, whereas ( † †) employs the fact that K 1   → K 2 is a morphism of S-FC   .For the right-hand diagram, notice that given  ∈ where ( †) uses Definition 74, whereas ( † †) employs the fact that K 1   → K 2 is a morphism of S-FC   .

From S-FC 𝐺𝑊
to S-FC  .In this subsection, we construct a functor from a particular subcategory of the category S-FC    to the category S-FC  .To begin with, we provide the definition of the subcategory in question.

Definition 76. S-FC
Proof.To show that the functor is correct on morphisms, we verify commutativity of the diagrams For the left-hand diagram, notice that given  ∈ where ( †) uses the fact that K Proof.We have to show that the functor is correct on morphisms.Given  1 ∈   Proof.Follow the steps of the proof of Theorem 64.
As a last remark, we notice that the case of L extending the variety UQuant simplifies the previously mentioned results (see the remark at the end of Section 5.1.5).

S-FC 𝐺𝑊 and S-FC 𝐺𝑊
versus S-FC  and S-FC   .In this subsection, we consider possible functorial links between the categories S-FC  and S-FC   , on one side, and the categories S-FC  and S-FC   , on the other.

From S-FC
Proof.To show that the functor   is correct on morphisms, notice that given Additionally, to show that the functor   is correct on morphisms, notice that given

5.3.2.
From S-FC  to S-FC  .In this subsection, we construct an isomorphism between particular subcategories of the categories S-FC  and S-FC  , respectively.

𝐿-FC
Due to the already considerable size of the paper, the respective fixed-basis restrictions of the variable-basis setting of the previous subsections are left to the reader.

Conclusion: Open Problems
In this paper, we provided different approaches to a possible lattice-valued generalization of Formal Concept Analysis (FCA) of Ganter and Wille [31], motivated by the recent study of Denniston et al. [29] on this topic.In particular, we have constructed several categories, whose objects are latticevalued extensions of formal contexts of FCA and whose morphisms reflect the crisp setting of Pratt [13], the latticevalued setting of Denniston et al. [29], and the many-valued setting of Ganter and Wille [31] themselves.In the next step, we considered many possible functors between the newly defined categories.As a consequence, we embedded each of the constructed categories into its respective counterparts.The crucial difference of this paper from the motivating one of Denniston et al. [29] can be briefly formulated as follows.
(i) The underlying lattices of lattice-valued formal contexts are extensions of quantales, instead of restricted to precisely commutative quantales as in [29].
(ii) All the categories of lattice-valued formal contexts and almost all of their respective functors are made variable-basis (in the sense of [4]), instead of being fixed-basis as in [29].
(iii) Unlike [29], we consider the setting of many-valued formal contexts of Ganter and Wille [31].
In the wake of the constructed functors of this paper, we can single out the following important properties of the approach of Denniston et al. [29], which is based on the category of Galois connections of [32].
Firstly, their approach falls rather out of the standard settings of Pratt as well as Ganter and Wille.On one hand, there exists an obvious isomorphism between the subcategory 2-PGAL of the category GAL [32] of Galois connections on crisp powersets and the crisp category 2-FC   (cf.Definitions 31(1) and 37) of this paper (essentially due to Denniston et al. [29]).On the other hand, the result of Theorem 42 shows that given a unital quantale , the categories -PGAL and -FC   are, in general, nonisomorphic.The variablebasis approach complicates the case even more.Speaking metamathematically, while the notion of formal concept of FCA does involve an order-reversing Galois connection between the respective powersets in its definition, its main essence appears to come from a binary relation on two sets, that is, the set of objects () and the set of their respective attributes (), which says whether a given object  ∈  has a given property  ∈ .Thus, these are not the Galois connections, but their generating relations, which seem to play the main role.Moreover, Theorem 99 of the paper states that while a particular subcategory of the category of latticevalued formal contexts in the sense of Pratt is isomorphic to a nonfull subcategory of the category of Denniston et al., it does not provide a (co)reflective subcategory of the latter category with respect to the constructed functors (at the moment though, we do not know whether one can define different functors, which do give rise to a (co)reflective subcategory in question).
Secondly, the category of lattice-valued formal contexts, based on the category of Galois connections, tends to have the nature of a fixed-basis category.More precisely, while making the setting of Denniston et al. variable-basis in the current paper, we were unable to involve the morphisms of the underlying algebras of lattice-valued formal contexts in its definition, thereby providing a quasi-variable-basis setting.
We would like to end the section with several open problems related to the setting of this paper.Problem 1.Is it possible to build a lattice-valued approach to FCA, which is based on order-reversing Galois connections on lattice-valued powersets, which are not generated by lattice-valued relations on their respective sets of objects and their attributes?
This problem seems especially important because we have established that using order-reversing Galois connections is more general than using relations.It may even seem that using order-reversing Galois connections with the corresponding Birkhoff operators is more robust and better than using relations.Therefore, it seems important to be able to understand how to use the Birkhoff operators when a corresponding relation may not exist.The answer may not depend on defining a relation.The answer may depend on being able to do formal concept analysis without relations.
Notice that although the notions of formal concept, protoconcept, and preconcept (cf.Definition 44) can be easily defined with the help of just a Galois connection in hand, it is their respective interpretation that poses the main problem.More precisely, what will be then in place of the crucial term "an object  has an attribute "?It is our opinion that such an approach can still be developed, substituting the required single lattice-valued relation  ×  An ad hoc approach could be to read  1 (, ) as the degree to which "an object  has an attribute " and to read  2 (, ) as the degree to which "an attribute  is shared by an object , " that is, distinguishing between the classically indistinguishable expressions "to have an attribute" and "an attribute is being shared."

Adjoint Situations between Possible Approaches to Lattice-
Valued FCA.In the paper, we have constructed functorial embeddings of several categories of lattice-valued formal contexts into each other.However, we failed to construct a single adjoint situation between the categories in question, being able just to obtain a negative result in this respect (Theorem 99).In view of the discussion, our next problem could be formulated as follows.
Problem 3. Construct (if possible) an adjoint situation between the categories of lattice-valued formal contexts introduced in this paper.
Notice that the problem has a close relation to the discussion of the previous subsection.More precisely, an adjoint situation between the categories of lattice-valued formal contexts in the sense of Pratt and Denniston et al., respectively, can provide a partial answer to the problem of dispensing with binary relations in lattice-valued FCA and substituting them by order-reversing Galois connections.

Proper Morphisms of Lattice-Valued Formal Contexts.
In the paper, we have dealt with both lattice-valued formal contexts and their respective morphisms.Whereas the former have already been treated in the fuzzy literature, the case of the latter is still not sufficiently clear.In particular, we have provided three possible approaches to (lattice-valued) formal context morphisms, that is, that of Pratt, that of Denniston et al., and that of Ganter and Wille.Taking apart the metamathematical discussion on the fruitfulness of any of the previously mentioned approaches, the last problem of the paper can be stated then as follows.
Problem 4. What is the most suitable (if any) way of defining the morphisms of lattice-valued FCA?
All the previously mentioned open problems will be addressed in our subsequent articles on the topic of latticevalued FCA.An interested reader is kindly invited to share the effort.

Example 34 .
Every crisp context K (recall our crispness remark from Definition 31(1)) provides the maps (1) P()   → P(), which is given by () = { ∈  |    for every  ∈ }, (2) P()   → P(), which is given by () = { ∈  |    for every  ∈ }, which are the classical Birkhoff operators generated by a binary relation.An important property of the maps of Definition 33 is contained in the following result.Theorem 35.For every lattice-valued context K, (  , , ,   ) is an order-reversing Galois connection.
a morphism of S-FC   , whereas ( † † †) employs the definition of the category S-FC   * .

Theorem 53 . 1 𝑓
The restriction  *  (resp.,  •  ) of the functor S-FC   *     → S-FC  to S-FC   * * (resp., S-FC   * • ) gives an (in general, nonfull) embedding of the latter category into S-FC  .Proof.Follow the steps of the proof of Theorem 50, where the backward powerset operator (, ) ← is substituted by the forward powerset operator (, ) ⊢[ .5.1.3.From S-FC  to S-FC .In this subsection, we construct a functorial embedding of a particular subcategory of the category S-FC  into the category S-FC   .We start by defining the subcategory in question.Definition 54.S-FC * is the (nonfull) subcategory of the category S-FC  , with the same objects, whose morphisms Journal of Mathematics K  → K 2 are such that the maps  1   →  2 ,  2     →  1 are surjective, whereas the S  -morphism  2     →  1 is injective, and, moreover, | 1 |   ⊢   → | 2 | preserves ⋁, →  and →  .

Theorem 55 .
There exists the functor S-FC  *     → S-FC   , which is given by

Theorem 57 . 1 𝑓Theorem 59 .
The restriction  *  (resp.,  •  ) of the functor S-FC  *     → S-FC   to the category S-FC  * * (resp., S-FC  * • ) gives an (in general nonfull) embedding of the latter category into S-FC   .Proof.Follow the proof of Theorem 50, with the respective changes in the powerset operators.5.1.4.From S-C  to S-FC .In this subsection, we construct a functorial embedding of a particular subcategory of the category S-FC  into the category S-FC  .As before, we begin with singling out the subcategory in question.Definition 58.S-FC * is the (nonfull) subcategory of the category S-FC  , with the same objects, and whose morphismsK  → K 2 are such that the maps  1   →  2 ,  2     →  1 are surjective, whereas the S  -morphism  2     →  1 is an isomorphism.With the new category in hand, we are ready to define a new functor as follows.There exists the functor S-FC  *     → S-FC  , which is given by

5. 2 . 1 . 1 𝑓
From S-FC   to S-FC   .In this subsection, we construct a functor from a particular subcategory of the category S-FC   to the category S-FC   .To begin with, we provide the definition of the subcategory in question.Definition 74.S-FC   * is the (nonfull) subcategory of the category S-FC   , with the same objects, whose morphisms K  → K 2 are such that the maps  1   →  2 and  1   →  2 are surjective, whereas the S-morphism  1   →  2 preserves ⋀, →  and →  .For the new category in hand, we can construct the following functor.Theorem 75.There exists the functor S-FC   *     → S-FC   , which is defined by the formula   (K 1   →

Definition 93 .
S-FC  * (resp., S-FC  * ) is the (nonfull) subcategory of the category S-FC  (resp., S-FC  ), with the same objects, whose morphisms K 1   → K 2 have the property that the map  1   →  2 (resp.,  2     →  1 ) is bijective.The categories of Definition 93 give rise to the following functors.Theorem 94.There exist two functors (1) S-FC  *     → S-FC  * , which is given by

K 2 ) = K 1 (
versus -FC   .The reader should notice that the results of Section 5.1 can be easily restricted to the case of the category -FC   .In particular, each of Theorems 48 and 55 provides the following result.Definition 100.Given an L-algebra , -FC  * is the (nonfull) subcategory of the category -FC  , with the same objects and with morphisms K 1   → K 2 such that the maps  1   →  2 ,  2     →  1 are surjective.Theorem 101.There exists the functor -FC  *     → -FC   , which is defined by the formula   (K 1   →  →  ,(  ) ←  )

6. 1 .
Proper Way of Building Lattice-Valued FCA.The previously mentioned discussion on the difference between the settings of binary relations and Galois connections in the lattice-valued case motivates the following open problem.

Problem 2 .
by a pair of lattice-valued relations  1 (, ) = (( 1  {} ))() and  2 (, ) = (( 1  {} ))(), where        is a given orderreversing Galois connection, which violates the conditions of Theorem 42 (notice that the case of a commutative quantale, i.e., ⋅ →  ⋅ = ⋅ →  ⋅, simplifies, but does not save the situation).In view of the discussion, a particular instance of Problem 1 is as follows.What is the interpretation of the relations  1 and  2 and what kind of lattice-valued FCA can be developed through them?