On morphisms of lattice-valued formal contexts

Transcription

1 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 1/37 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Department of Mathematics and Statistics, Faculty of Science, Masaryk University Kotlarska 2, Brno, Czech Republic web page: solovjovs The 4th Novi Sad Algebraic Conference in conjunction with the workshop Semigroups and Applications 2013 Novi Sad, Serbia June 5-9, 2013

2 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 2/37 Acknowledgements The author gratefully acknowledges the support of the ESF project CZ.1.07/2.3.00/ Algebraic methods in Quantum Logic of the Masaryk University in Brno, Czech Republic.

3 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 3/37 Outline 1 Introduction 2 Preliminaries on powerset operators 3 Categories of lattice-valued formal contexts 4 Properties of the categories of lattice-valued formal contexts 5 Conclusion

4 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 4/37 Formal Concept Analysis Formal Concept Analysis Formal Concept Analysis (FCA) has taken its origin as an attempt to restructure mathematics, e.g., lattice theory. Since then, FCA has been developed as a subfield of applied mathematics, based in mathematization of concept hierarchies. The aim of FCA is to support the rational communication of humans by mathematically developing appropriate conceptual structures, which can be logically activated.

5 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 4/37 Formal Concept Analysis Formal Concept Analysis Formal Concept Analysis (FCA) has taken its origin as an attempt to restructure mathematics, e.g., lattice theory. Since then, FCA has been developed as a subfield of applied mathematics, based in mathematization of concept hierarchies. The aim of FCA is to support the rational communication of humans by mathematically developing appropriate conceptual structures, which can be logically activated.

6 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 4/37 Formal Concept Analysis Formal Concept Analysis Formal Concept Analysis (FCA) has taken its origin as an attempt to restructure mathematics, e.g., lattice theory. Since then, FCA has been developed as a subfield of applied mathematics, based in mathematization of concept hierarchies. The aim of FCA is to support the rational communication of humans by mathematically developing appropriate conceptual structures, which can be logically activated.

7 Formal Concept Analysis Formal contexts One of the main building blocks of FCA provide formal contexts. Definition 1 A formal context is a triple (G, M, I ), which comprises a set of objects G, a set of attributes M, and a binary incidence relation I between G and M, where g I m means object g has attribute m. On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 5/37

8 Formal Concept Analysis Formal contexts One of the main building blocks of FCA provide formal contexts. Definition 1 A formal context is a triple (G, M, I ), which comprises a set of objects G, a set of attributes M, and a binary incidence relation I between G and M, where g I m means object g has attribute m. On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 5/37

9 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 6/37 Formal Concept Analysis Formal context morphisms There exist at least three (different) ways of defining a morphism between two formal contexts (G 1, M 1, I 1 ) and (G 2, M 2, I 2 ). 1 The theory of FCA employs pairs of maps G 1 α G2, M 1 β M2 such that g I 1 m iff α(g) I 2 β(m) for every g G 1, m M 1. 2 The theory of Chu spaces uses pairs of maps G 1 α G2, M 2 β M 1 such that g I 1 β(m) iff α(g) I 2 m for every g G 1, m M 2.

10 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 6/37 Formal Concept Analysis Formal context morphisms There exist at least three (different) ways of defining a morphism between two formal contexts (G 1, M 1, I 1 ) and (G 2, M 2, I 2 ). 1 The theory of FCA employs pairs of maps G 1 α G2, M 1 β M2 such that g I 1 m iff α(g) I 2 β(m) for every g G 1, m M 1. 2 The theory of Chu spaces uses pairs of maps G 1 α G2, M 2 β M 1 such that g I 1 β(m) iff α(g) I 2 m for every g G 1, m M 2.

11 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 6/37 Formal Concept Analysis Formal context morphisms There exist at least three (different) ways of defining a morphism between two formal contexts (G 1, M 1, I 1 ) and (G 2, M 2, I 2 ). 1 The theory of FCA employs pairs of maps G 1 α G2, M 1 β M2 such that g I 1 m iff α(g) I 2 β(m) for every g G 1, m M 1. 2 The theory of Chu spaces uses pairs of maps G 1 α G2, M 2 β M 1 such that g I 1 β(m) iff α(g) I 2 m for every g G 1, m M 2.

12 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 7/37 Formal Concept Analysis Formal context morphisms 3 The theory of Galois connections relies on the pairs of maps P(G 1 ) α P(G 2 ), P(M 2 ) β P(M 1 ), where P(X ) stands for the powerset of X, such that the diagrams P(G 1 ) α P(G 2 ) and P(M 1 ) β P(M 2 ) H 1 H 2 P(M 1 ) P(M 2 ) β K 1 P(G 1 ) α K 2 P(G 2 ) commute, where H j (S) = {m M j s I j m for every s S} and K j (T ) = {g G j g I j t for every t T } (Birkhoff operators).

13 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 8/37 Lattice-valued Formal Concept Analysis Lattice-valued formal contexts J. T. Denniston, A. Melton, and S. E. Rodabaugh compared the approaches of items (2) and (3) by considering their respective categories of lattice-valued formal contexts (in the sense of R. Bělohlávek) over a fixed commutative quantale Q, and constructing an embedding of each category into its counterparts. They finally arrived at the conclusion that the two viewpoints on formal context morphisms were not categorically isomorphic.

14 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 8/37 Lattice-valued Formal Concept Analysis Lattice-valued formal contexts J. T. Denniston, A. Melton, and S. E. Rodabaugh compared the approaches of items (2) and (3) by considering their respective categories of lattice-valued formal contexts (in the sense of R. Bělohlávek) over a fixed commutative quantale Q, and constructing an embedding of each category into its counterparts. They finally arrived at the conclusion that the two viewpoints on formal context morphisms were not categorically isomorphic.

15 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 9/37 Lattice-valued Formal Concept Analysis Lattice-valued formal contexts This talk compares all three of the above-mentioned approaches to morphisms in the framework of lattice-valued formal contexts over a category of not necessarily commutative quantales. We construct a number of embeddings between their respective categories of formal contexts, showing that the approach of item (3) falls out of the FCA setting in the lattice-valued case.

16 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 9/37 Lattice-valued Formal Concept Analysis Lattice-valued formal contexts This talk compares all three of the above-mentioned approaches to morphisms in the framework of lattice-valued formal contexts over a category of not necessarily commutative quantales. We construct a number of embeddings between their respective categories of formal contexts, showing that the approach of item (3) falls out of the FCA setting in the lattice-valued case.

17 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 10/37 Quantales -semilattices Definition 2 CSLat( ) is the variety of -semilattices, i.e., partially ordered sets (posets), which have arbitrary joins. Every ϕ -semilattice homomorphism A 1 A2 has the upper adjoint map A 2 A 1 given by ϕ (a 2 ) = {a 1 A 1 ϕ(a 1 ) a 2 ϕ }.

18 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 10/37 Quantales -semilattices Definition 2 CSLat( ) is the variety of -semilattices, i.e., partially ordered sets (posets), which have arbitrary joins. Every ϕ -semilattice homomorphism A 1 A2 has the upper adjoint map A 2 A 1 given by ϕ (a 2 ) = {a 1 A 1 ϕ(a 1 ) a 2 ϕ }.

19 Quantales Quantales Definition 3 1 Quant is the variety of quantales, i.e., triples (Q,, ), where (Q, ) is a -semilattice; (Q, ) is a semigroup; distributes across from both sides. 2 UQuant is the variety of unital quantales, i.e., quantales Q, which have an element 1 Q such that (Q,, 1 Q ) is a monoid. A quantale Q has two residuations, which are given by q 1 l q 2 = {q Q q q1 q 2 } and q 1 r q 2 = {q Q q 1 q q 2 }. On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 11/37

20 Quantales Quantales Definition 3 1 Quant is the variety of quantales, i.e., triples (Q,, ), where (Q, ) is a -semilattice; (Q, ) is a semigroup; distributes across from both sides. 2 UQuant is the variety of unital quantales, i.e., quantales Q, which have an element 1 Q such that (Q,, 1 Q ) is a monoid. A quantale Q has two residuations, which are given by q 1 l q 2 = {q Q q q1 q 2 } and q 1 r q 2 = {q Q q 1 q q 2 }. On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 11/37

21 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 12/37 Powerset operators Crisp forward powerset operator Definition 4 f Given a map X 1 X2, the forward powerset operator w.r.t. f is the map P(X 1 ) f P(X 2 ), which is defined by f (S) = {f (s) s S}.

22 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 13/37 Powerset operators Lattice-valued forward powerset operators I Theorem 5 1 Given a variety L, which extends CSLat( ), every subcategory S of L provides a functor Set S ( ) CSLat( ), which (f,ϕ) is defined by ((X 1, L 1 ) (X 2, L 2 )) = L X (f,ϕ) 1 1 L X 2 2, where ((f, ϕ) (α))(x 2 ) = ϕ( f (x 1 )=x 2 α(x 1 )). 2 Let L be a variety, which extends CSLat( ), and let S be a subcategory of L op ϕ such that for every S-morphism L 1 L2, ϕ the map L op 1 L 2 is -preserving. Then there exists a functor Set S ( ) CSLat( ) defined by ((X 1, L 1 ) (f,ϕ) (f,ϕ) (X 2, L 2 )) = L X 1 1 L X 2 2, where ((f, ϕ) (α))(x 2 ) = ϕ op ( f (x 1 )=x 2 α(x 1 )).

23 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 14/37 Powerset operators Lattice-valued forward powerset operators II Theorem 6 1 Given a variety L, which extends CSLat( ), every subcategory S of L op provides a functor Set op S ( ) o (CSLat( )) op (f,ϕ) ((f,ϕ) o ) op with ((X 1, L 1 ) (X 2, L 2 )) o = L X 1 1 L X 2 2, where ((f, ϕ) o (α))(x 1 ) = ϕ op ( f op (x 2 )=x 1 α(x 2 )). 2 Let L be a variety, which extends CSLat( ), and let S be ϕ a subcategory of L such that for every S-morphism L 1 ϕ L 2, the map L 2 L 1 is -preserving. Then there exists a functor Set op S ( ) o (CSLat( )) op defined by (f,ϕ) ((f,ϕ) o ) op ((X 1, L 1 ) (X 2, L 2 )) o = L X 1 1 L X 2 2, where ((f, ϕ) o (α))(x 1 ) = ϕ ( f op (x 2 )=x 1 α(x 2 )).

24 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 15/37 Galois connections Galois connections Definition 7 A tuple ((X 1, ), f, g, (X 2, )) is an order-reversing Galois connec- f tion provided that (X 1, ), (X 2, ) are posets, and X 1 X 2 are g maps with x 1 g(x 2 ) iff x 2 f (x 1 ) for every x 1 X 1, x 2 X 2.

25 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 16/37 Lattice-valued formal contexts Formal contexts as Chu spaces Definition 8 Let L be a variety, which extends Quant, and let S be a subcategory of L op. S-FC C is the category, which comprises the following data. Objects: tuples K = (G, M, L, I ) ((lattice-valued) formal contexts), where G is the set of context objects, M is the set of context attributes, L is an S-object, and G M I L is a map, which is called the context incidence relation. f Morphisms: K 1 K2 ((lattice-valued) formal context morphisms) are triples (G 1, M 1, L 1 ) f =(α,β,ϕ) (G 2, M 2, L 2 ) in Set Set op S with I 1 (g, β op (m)) = ϕ op I 2 (α(g), m) for every g G 1, m M 2.

26 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 17/37 Lattice-valued formal contexts Modified formal contexts as Chu spaces Definition 9 Let L be a variety, which extends Quant, and let S be a subcategory of L. S-FC C m is the category, which comprises the following data. Objects: (lattice-valued) formal contexts. f =(α,β,ϕ) f Morphisms: K 1 K2 are triples (G 1, M 1, L 1 ) (G 2, M 2, L 2 ) in Set Set op S with ϕ I 1 (g, β op (m)) = I 2 (α(g), m) for every g G 1, m M 2.

27 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 18/37 Lattice-valued formal contexts Formal contexts of B. Ganter and R. Wille Definition 10 Let L be a variety, which extends Quant, and let S be a subcategory of L op. S-FC GW is the category, which comprises the following data. Objects: (lattice-valued) formal contexts. f =(α,β,ϕ) f Morphisms: K 1 K2 are triples (G 1, M 1, L 1 ) (G 2, M 2, L 2 ) in Set Set S with I 1 (g, m) = ϕ op I 2 (α(g), β(m)) for every g G 1, m M 1.

28 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 19/37 Lattice-valued formal contexts Modified formal contexts of B. Ganter and R. Wille Definition 11 Let L be a variety, which extends Quant, and let S be a subcategory of L. S-FC GW m is the category, which comprises the following data. Objects: (lattice-valued) formal contexts. f =(α,β,ϕ) f Morphisms: K 1 K2 are triples (G 1, M 1, L 1 ) (G 2, M 2, L 2 ) in Set Set S with ϕ I 1 (g, m) = I 2 (α(g), β(m)) for every g G 1, m M 1.

29 Lattice-valued formal contexts Lattice-valued Birkhoff operators Definition 12 Every lattice-valued formal context K provides the following (latticevalued) Birkhoff operators: 1 L G H L M given by (H(s))(m) = g G (s(g) l I (g, m)); 2 L M K L G given by (K(t))(g) = m M (t(m) r I (g, m)). Theorem 13 For every lattice-valued context K, (L G, H, K, L M ) is an orderreversing Galois connection. On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 20/37

30 Lattice-valued formal contexts Lattice-valued Birkhoff operators Definition 12 Every lattice-valued formal context K provides the following (latticevalued) Birkhoff operators: 1 L G H L M given by (H(s))(m) = g G (s(g) l I (g, m)); 2 L M K L G given by (K(t))(g) = m M (t(m) r I (g, m)). Theorem 13 For every lattice-valued context K, (L G, H, K, L M ) is an orderreversing Galois connection. On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 20/37

31 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 21/37 Lattice-valued formal contexts Crisp Birkhoff operators Example 14 Every crisp context K provides the maps 1 P(G) H P(M), H(S) = {m M s I m for every s S}; 2 P(M) K P(G), K(T ) = {g G g I t for every t T }; which are the classical Birkhoff operators of a binary relation.

32 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 22/37 Lattice-valued formal contexts Formal contexts of J. T. Denniston et al. Definition 15 Given a variety L, which extends Quant, and a subcategory S of L, S-FC DMR is the category, concrete over the product category Set Set op, which comprises the following data. Objects: lattice-valued formal contexts K with L an object of S. f =(α,β) Morphisms: K 1 K 2 are Set Set op -morphisms (L G 1 1, LM 1 1 ) (α,β) (L G 2 2, LM 2 2 ), making the next diagrams commute L G 1 1 α L G 2 2 L M 1 1 β op L M 2 2 H 1 H 2 L M 1 1 β op L M 2 2 K 1 L G 1 1 α L G 2 K 2 2.

33 Formal contexts as Galois connections Relations versus Birkhoff operators There is a one-to-one correspondence between relations I G M and order-reversing Galois connections on (P(G), P(M)). What about the lattice-valued case? Definition 16 Given a -semilattice L and a set X, every S X and every a L provide the map X χa S L, which is defined by { χ a S (x) = a, x S L, otherwise. On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 23/37

34 Formal contexts as Galois connections Relations versus Birkhoff operators There is a one-to-one correspondence between relations I G M and order-reversing Galois connections on (P(G), P(M)). What about the lattice-valued case? Definition 16 Given a -semilattice L and a set X, every S X and every a L provide the map X χa S L, which is defined by { χ a S (x) = a, x S L, otherwise. On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 23/37

35 Formal contexts as Galois connections Relations versus Birkhoff operators There is a one-to-one correspondence between relations I G M and order-reversing Galois connections on (P(G), P(M)). What about the lattice-valued case? Definition 16 Given a -semilattice L and a set X, every S X and every a L provide the map X χa S L, which is defined by { χ a S (x) = a, x S L, otherwise. On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 23/37

36 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 24/37 Formal contexts as Galois connections Lattice-valued relations versus Birkhoff operators Theorem 17 Let G, M be sets and let L be a unital quantale. For every orderreversing Galois connection (L G, α, β, L M ), equivalent are: 1 There exists a map G M I L such that α = H and β = K. 2 For every g G, m M, a L, it follows that (a) (α(χ 1 L {g} ))(m) = (β(χ1 L {m} ))(g); (b) (α(a χ 1 L {g} ))(m) = a l (α(χ 1 L {g} ))(m); (c) (β(χ 1 L {m} a))(g) = a r (β(χ 1 L {m} ))(g). 3 For every g G, m M, a L, it follows that (a) (α(a χ 1 L {g} ))(m) = a l (β(χ 1 L {m} ))(g); (b) (β(χ 1 L {m} a))(g) = a r (α(χ 1 L {g} ))(m).

37 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 25/37 Formal contexts as Galois connections Consequences Every map G M I L gives rise to an order-reversing Galois connection, but the converse way needs additional requirements. Counterexample Let L be the unit interval I = ([0, 1],,, 1), and let both G and M be singletons. One can assume that both I G and I M is I. The order-reversing involution map I α I, α(a) = 1 a is a part of the order-reversing Galois connection (I, α, α, I). The condition of, e.g., Theorem 17(3)(a) gives α(a) = a α(1) for every a I. However, for a = 1 2, one obtains that α( 1 2 ) = = = 1 2 α(1).

38 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 25/37 Formal contexts as Galois connections Consequences Every map G M I L gives rise to an order-reversing Galois connection, but the converse way needs additional requirements. Counterexample Let L be the unit interval I = ([0, 1],,, 1), and let both G and M be singletons. One can assume that both I G and I M is I. The order-reversing involution map I α I, α(a) = 1 a is a part of the order-reversing Galois connection (I, α, α, I). The condition of, e.g., Theorem 17(3)(a) gives α(a) = a α(1) for every a I. However, for a = 1 2, one obtains that α( 1 2 ) = = = 1 2 α(1).

39 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 26/37 Relationships between the categories of lattice-valued formal contexts From S-FC C to S-FC DMR Definition 18 S-FC C is a subcategory of S-FC C, with the same objects, and f α whose morphisms K 1 K2 have surjective maps G 1 G2, β op ϕ M 1, and an S-isomorphism L 1 L2. M 2 Let L extend UQuant. S-FC C (resp. S-FC C ) is a full subcategory of S-FC C, whose objects K = (G, M, L, I ) have non-empty G (resp. M) and, moreover, 1 L L. Theorem 19 There exists a functor S-FC C H CD S-FC DMR, which is given by f ((α,ϕ) H CD (K 1 K2 ) = K,((β,ϕ) o ) op ) 1 K 2. Its restriction to S-FC C (resp. S-FC C ) is a (non-full) embedding.

40 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 26/37 Relationships between the categories of lattice-valued formal contexts From S-FC C to S-FC DMR Definition 18 S-FC C is a subcategory of S-FC C, with the same objects, and f α whose morphisms K 1 K2 have surjective maps G 1 G2, β op ϕ M 1, and an S-isomorphism L 1 L2. M 2 Let L extend UQuant. S-FC C (resp. S-FC C ) is a full subcategory of S-FC C, whose objects K = (G, M, L, I ) have non-empty G (resp. M) and, moreover, 1 L L. Theorem 19 There exists a functor S-FC C H CD S-FC DMR, which is given by f ((α,ϕ) H CD (K 1 K2 ) = K,((β,ϕ) o ) op ) 1 K 2. Its restriction to S-FC C (resp. S-FC C ) is a (non-full) embedding.

41 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 27/37 Relationships between the categories of lattice-valued formal contexts From S-FC C m to S-FC DMR Definition 20 S-FC C m is a subcategory of S-FC C m, with the same objects, and f α whose morphisms K 1 K2 have surjective maps G 1 G2, β op ϕ M 1, and an S-isomorphism L 1 L2. M 2 Let L extend UQuant. S-FC C m (resp. S-FC C m ) is a full subcategory of S-FC C m, whose objects K = (G, M, L, I ) have non-empty G (resp. M) and, moreover, 1 L L. Theorem 21 There exists a functor S-FC C H CmD m S-FC DMR, which is given by f ((α,ϕ) H CmD (K 1 K2 ) = K,((β,ϕ) o ) op ) 1 K 2. Its restriction to S-FC C m (resp. S-FC C m ) is a (non-full) embedding.

42 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 27/37 Relationships between the categories of lattice-valued formal contexts From S-FC C m to S-FC DMR Definition 20 S-FC C m is a subcategory of S-FC C m, with the same objects, and f α whose morphisms K 1 K2 have surjective maps G 1 G2, β op ϕ M 1, and an S-isomorphism L 1 L2. M 2 Let L extend UQuant. S-FC C m (resp. S-FC C m ) is a full subcategory of S-FC C m, whose objects K = (G, M, L, I ) have non-empty G (resp. M) and, moreover, 1 L L. Theorem 21 There exists a functor S-FC C H CmD m S-FC DMR, which is given by f ((α,ϕ) H CmD (K 1 K2 ) = K,((β,ϕ) o ) op ) 1 K 2. Its restriction to S-FC C m (resp. S-FC C m ) is a (non-full) embedding.

43 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 28/37 Relationships between the categories of lattice-valued formal contexts Formal concepts, protoconcepts, and preconcepts Definition 22 Let K be a lattice-valued formal context, and let s L G, t L M. The pair (s, t) is called a (lattice-valued) formal concept of K provided that H(s) = t and K(t) = s; (lattice-valued) formal protoconcept of K provided that K H(s) = K(t) (equivalently, H K(t) = H(s)); (lattice-valued) formal preconcept of K provided that s K(t) (equivalently, t H(s)).

44 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 29/37 Relationships between the categories of lattice-valued formal contexts From S-FC DMR to S-FC C Definition 23 Given an L-algebra L, L-FC DMR i is a subcategory of L-FC DMR, f with the same objects, and whose morphisms K 1 K2 have injective maps L G α 1 L G 2, L M β op 2 L M 1. An L-algebra L is called quasi-strictly right-sided (qsrs-algebra) provided that a ( L l a) L for every a L. Theorem 24 There exists a functor L-FC DMR H i DC i S-FC C, which is given by H i DC (K 1 f K 2 ) = (L G 1, L M 1, L, Î 1 ) (α,β,1 L) (L G 2, L M 2, L, Î 2 ), where Î j (s, t) = L if (s, t) is a formal concept of K j, and L otherwise. If L is a qsrs-algebra, then H i DC is a (non-full) embedding.

45 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 29/37 Relationships between the categories of lattice-valued formal contexts From S-FC DMR to S-FC C Definition 23 Given an L-algebra L, L-FC DMR i is a subcategory of L-FC DMR, f with the same objects, and whose morphisms K 1 K2 have injective maps L G α 1 L G 2, L M β op 2 L M 1. An L-algebra L is called quasi-strictly right-sided (qsrs-algebra) provided that a ( L l a) L for every a L. Theorem 24 There exists a functor L-FC DMR H i DC i S-FC C, which is given by H i DC (K 1 f K 2 ) = (L G 1, L M 1, L, Î 1 ) (α,β,1 L) (L G 2, L M 2, L, Î 2 ), where Î j (s, t) = L if (s, t) is a formal concept of K j, and L otherwise. If L is a qsrs-algebra, then H i DC is a (non-full) embedding.

46 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 30/37 Relationships between the categories of lattice-valued formal contexts From S-FC DMR to S-FC C Definition 25 Given an L-algebra L, L-FC DMR rfp is a subcategory of L-FC DMR, with f the same objects, and whose morphisms K 1 K2 have maps L G α 1 β op L G 2, L M 2 L M 1 such that K 2 H 2 α(s) = α(s) implies K 1 H 1 (s) = s, and H 1 K 1 β op (t) = β op (t) implies H 2 K 2 (t) = t, for every s L G 1 1, t LM 2 2. Theorem 26 There exists a functor L-FC DMR rfp H rfp DC S-FC C, which is given by H rfp DC (K 1 f K 2 ) = (L G 1, L M 1, L, Î 1 ) (α,β,1 L) (L G 2, L M 2, L, Î 2 ), where Î j (s, t) = L if (s, t) is a formal concept of K j, and L otherwise. If L is a qsrs-algebra, then the functor is a (non-full) embedding.

47 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 30/37 Relationships between the categories of lattice-valued formal contexts From S-FC DMR to S-FC C Definition 25 Given an L-algebra L, L-FC DMR rfp is a subcategory of L-FC DMR, with f the same objects, and whose morphisms K 1 K2 have maps L G α 1 β op L G 2, L M 2 L M 1 such that K 2 H 2 α(s) = α(s) implies K 1 H 1 (s) = s, and H 1 K 1 β op (t) = β op (t) implies H 2 K 2 (t) = t, for every s L G 1 1, t LM 2 2. Theorem 26 There exists a functor L-FC DMR rfp H rfp DC S-FC C, which is given by H rfp DC (K 1 f K 2 ) = (L G 1, L M 1, L, Î 1 ) (α,β,1 L) (L G 2, L M 2, L, Î 2 ), where Î j (s, t) = L if (s, t) is a formal concept of K j, and L otherwise. If L is a qsrs-algebra, then the functor is a (non-full) embedding.

48 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 31/37 Relationships between the categories of lattice-valued formal contexts From S-FC DMR to S-FC C Definition 27 Given an L-algebra L, L-FC DMR orp is a subcategory of L-FC DMR, with f the same objects, and whose morphisms K 1 K2 have orderpreserving maps L G α 1 L G 2, L M β op 2 L M 1. Theorem 28 There exists a functor L-FC DMR orp H orp DC S-FC C, which is given by H orp DC (K 1 f K 2 ) = (L G 1, L M 1, L, Î1) (α,β,1 L) (L G 2, L M 2, L, Î 2 ), where Î j (s, t) = L if (s, t) is a formal preconcept of K j, and L otherwise. If L is a qsrs-algebra, then the functor is a (non-full) embedding.

49 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 31/37 Relationships between the categories of lattice-valued formal contexts From S-FC DMR to S-FC C Definition 27 Given an L-algebra L, L-FC DMR orp is a subcategory of L-FC DMR, with f the same objects, and whose morphisms K 1 K2 have orderpreserving maps L G α 1 L G 2, L M β op 2 L M 1. Theorem 28 There exists a functor L-FC DMR orp H orp DC S-FC C, which is given by H orp DC (K 1 f K 2 ) = (L G 1, L M 1, L, Î1) (α,β,1 L) (L G 2, L M 2, L, Î 2 ), where Î j (s, t) = L if (s, t) is a formal preconcept of K j, and L otherwise. If L is a qsrs-algebra, then the functor is a (non-full) embedding.

50 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 32/37 Relationships between the categories of lattice-valued formal contexts From S-FC DMR to S-FC C Theorem 29 There exists a functor L-FC DMR H DC S-FC C, which is given by f H DC (K 1 K2 ) = (L G 1, L M 1, L, Î 1 ) (α,β,1 L) (L G 2, L M 2, L, Î 2 ), where Î j (s, t) = L if (s, t) is a formal protoconcept of K j, and L otherwise. If L is a qsrs-algebra, then H DC is a (non-full) embedding.

51 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 33/37 Conclusion Final remarks This talk considered some approaches to morphisms of latticevalued formal contexts of Formal Context Analysis (FCA). We constructed several categories, whose objects are latticevalued analogues of formal contexts of FCA, and whose morphisms reflect the crisp setting of Chu spaces, the lattice-valued setting of J. T. Denniston, A. Melton, and S. E. Rodabaugh, as well as the many-valued setting of B. Ganter and R. Wille. We considered a number of functors between the categories of formal contexts, embedding each of them into its counterparts.

52 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 33/37 Conclusion Final remarks This talk considered some approaches to morphisms of latticevalued formal contexts of Formal Context Analysis (FCA). We constructed several categories, whose objects are latticevalued analogues of formal contexts of FCA, and whose morphisms reflect the crisp setting of Chu spaces, the lattice-valued setting of J. T. Denniston, A. Melton, and S. E. Rodabaugh, as well as the many-valued setting of B. Ganter and R. Wille. We considered a number of functors between the categories of formal contexts, embedding each of them into its counterparts.

53 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 33/37 Conclusion Final remarks This talk considered some approaches to morphisms of latticevalued formal contexts of Formal Context Analysis (FCA). We constructed several categories, whose objects are latticevalued analogues of formal contexts of FCA, and whose morphisms reflect the crisp setting of Chu spaces, the lattice-valued setting of J. T. Denniston, A. Melton, and S. E. Rodabaugh, as well as the many-valued setting of B. Ganter and R. Wille. We considered a number of functors between the categories of formal contexts, embedding each of them into its counterparts.

54 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 34/37 Open problem FCA without relations The difference between the settings of relations and Galois connections in the lattice-valued case, motivates the following problem. Problem 30 Is it possible to build a lattice-valued approach to FCA, which is based in 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?

55 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 34/37 Open problem FCA without relations The difference between the settings of relations and Galois connections in the lattice-valued case, motivates the following problem. Problem 30 Is it possible to build a lattice-valued approach to FCA, which is based in 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?

56 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 35/37 References References I J. Adámek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories: The Joy of Cats, Dover Publications, G. Birkhoff, Lattice Theory, American Mathematical Society Publications, R. Bělohlávek, Fuzzy Relational Systems. Foundations and Principles, Kluwer Academic Publishers, J. T. Denniston, A. Melton, and S. E. Rodabaugh, Formal concept analysis and lattice-valued Chu systems, Fuzzy Sets Syst. 216 (2013), B. Ganter and R. Wille, Formale Begriffsanalyse. Mathematische Grundlagen, Berlin: Springer, 1996.

57 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 36/37 References References II D. Kruml and J. Paseka, Algebraic and Categorical Aspects of Quantales, Handbook of Algebra (M. Hazewinkel, ed.), vol. 5, Elsevier, 2008, pp J. M. McDill, A. C. Melton, and G. E. Strecker, A category of Galois connections, Lect. Notes Comput. Sci. 283 (1987), V. Pratt, Chu spaces, Coimbra: Universidade de Coimbra, Departamento de Matemática. Textos Mat., Sér. B. 21 (1999), S. Solovyov, Lattice-valued topological systems as a framework for lattice-valued Formal Concept Analysis, Journal of Mathematics 2013 (2013), 33 pages, Article ID ,

58 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 37/37 Thank you for your attention!