# Soft concept lattice for formal concept analysis based on soft sets: theoretical foundations and applications

## Abstract

Formal concept analysis (FCA) is an important theory on the study of hierarchical structures caused by a binary relation between objects and attributes. And it is well known that a soft set always induces a binary relationship between objects and attributes for a formal context. Based on these facts, we study a new system called a soft context, which is induced using a soft set instead of a binary relation in the formal context, and investigate its theoretical foundations. First, we introduce the notions of soft context and soft concept for objects and study some basic properties. In particular, for the purpose of studying soft concepts, we introduce and study the notions of independent and dependent soft concepts. And we obtain the important things: (1) the set of all soft concepts can be decomposed into the set of independent soft concepts and the set of dependent soft concepts; (2) every dependent soft concept is generated by some independent soft concepts. Second, we obtain important properties about the relationship between formal context and soft contexts as the following: (1) each formal context always induces an associated soft context; (2) for a given formal context, there exists a bijection between the set of all formal concepts and the set of all soft concepts of the associated soft context induced by the given formal context. And we apply the above-mentioned properties to get the set of all formal concepts of a given formal context and explain that this method is more effective than the traditional way of computing all formal concepts in formal contexts using the example. Finally, we introduce the notion of soft concept lattice. Also, for a given formal context, we show that the formal concept lattice is order-isomorphic to the soft concept lattice of the associated soft context.

## Keywords

Formal context Formal concept Concept lattice Soft set Soft context Soft concept Soft concept lattice## Notes

### Compliance with ethical standards

### Conflict of interest

W. K. Min and Y. K. Kim declare that they have no conflict of interest.

### Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.

### Informed consent

Informed consent was obtained from all individual participants included in the study.

## References

- Aktas H, Cagman N (2007) Soft sets and soft groups. Inf Sci 177:2726–2735MathSciNetCrossRefGoogle Scholar
- Ali MI (2011) A note on soft sets, rough sets and fuzzy soft sets. Appl Soft Comput 11:3329–3332CrossRefGoogle Scholar
- Ali MI (2012) Another view on reduction of parameters in soft sets. Appl Soft Comput 12:1814–1821CrossRefGoogle Scholar
- Ali MI, Feng F, Liu XY, Min WK, Shabir M (2009) On some new operations in soft set theory. Comput Math Appl 57:1547–1553MathSciNetCrossRefGoogle Scholar
- Ali MI, Shabir M, Naz M (2011) Algebraic structures of soft sets associated with new operations. Comput Math Appl 61:2647–2654MathSciNetCrossRefGoogle Scholar
- Belohlavek R (2004) Concept lattices and order in fuzzy logic. Ann Pure Appl Log 128:277–298MathSciNetCrossRefGoogle Scholar
- Cgman N, Karatas S, Enginoglu S (2011) Soft topology. Comput Math Appl 62:351–358MathSciNetCrossRefGoogle Scholar
- Chen L, Huang T, Song Z, Pei Z (2008) Formal concept analysis based on set-valued mapping. Chin Q J Math 23(3):390–396MathSciNetzbMATHGoogle Scholar
- Deoguna JS, Saquer J (2004) Monotone concepts for formal concept analysis. Discrete Appl Math 144:70–78MathSciNetCrossRefGoogle Scholar
- Feng F, Jun YB, Zhao X (2008) Soft semirings. Comput Math Appl 56(10):2621–2628MathSciNetCrossRefGoogle Scholar
- Feng F, Li CX, Davvaz B, Irfan Ali M (2010a) Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Comput 14:899–911CrossRefGoogle Scholar
- Feng F, Jun YB, Liu X, Li L (2010b) An adjustable approach to fuzzy soft set based decision making. J Comput Appl Math 234(1):10–20MathSciNetCrossRefGoogle Scholar
- Feng F, Li YM, Leoreanu-Fotea V (2010c) Application of level soft sets in decision making based on interval-valued fuzzy soft sets. Comput Math Appl 60:1756–1767MathSciNetCrossRefGoogle Scholar
- Feng F, Liu X, Leoreanu-Fotea V, Jun YB (2011) Soft sets and soft rough sets. Inf Sci 181:1125–1137MathSciNetCrossRefGoogle Scholar
- Ganter B, Wille R (1999) Formal concept analysis: mathematical foundations. Springer, BerlinCrossRefGoogle Scholar
- Jin J, Qin K, Pei Z (2006) Reduction-based approaches towards constructing Galois (concept) lattices, vol 4062. Lecture notes in artificial intelligence. Springer, BerlinzbMATHGoogle Scholar
- Jun YB (2008) Soft BCK/BCI-algebras. Comput Math Appl 56:1408–1413MathSciNetCrossRefGoogle Scholar
- Jun YB, Park CH (2008) Applications of soft sets in ideal theory of BCK/BCI algebras. Inf Sci 178:2466–2475MathSciNetzbMATHGoogle Scholar
- Lai H, Zhang D (2009) Concept lattices of fuzzy contexts: formal concept analysis vs. rough set theory. Int J Approx Reason 50(5):695–707MathSciNetCrossRefGoogle Scholar
- Li F (2010) Soft lattices. Glob J Sci Front Res 10(4):56–58Google Scholar
- Maji PK, Biswas R, Roy R (2003) Soft set theory. Comput Math Appl 45:555–562MathSciNetCrossRefGoogle Scholar
- Medina J, Ojeda-Aciego M, Ruiz-Calvino J (2008) Formal concept analysis via multi-adjoint concept lattices. Fuzzy Sets Syst 160(2):130–144MathSciNetCrossRefGoogle Scholar
- Min WK (2011) A note on soft topological spaces. Comput Math Appl 62:3524–3528MathSciNetCrossRefGoogle Scholar
- Min WK (2014) Soft sets over a common topological universe. J Intell Fuzzy Syst 26(5):2099–2106MathSciNetzbMATHGoogle Scholar
- Molodtsov D (1999) Soft set theory first results. Comput Math Appl 37:19–31MathSciNetCrossRefGoogle Scholar
- Nagarajan R, Meenambigai G (2011) An application of soft sets to lattices. Kragujev J Math 35(1):61–73MathSciNetzbMATHGoogle Scholar
- Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356CrossRefGoogle Scholar
- Pei Z, Qin K (2007) Topological space for attributes set of a formal context. Lecture notes in artificial intelligence (RSKT2007), 4481, Canada, pp 460–467Google Scholar
- Pei Z, Ruan D, Meng D, Liu Z (2013) Formal concept analysis based on the topology for attributes of a formal context. Inf Sci 236:66–82MathSciNetCrossRefGoogle Scholar
- Tanay B, Burc Kandemir M (2011) Topological structure of fuzzy soft sets. Comput Math Appl 61(10):2952–2957MathSciNetCrossRefGoogle Scholar
- Wang L, Liu X, Cao J (2010) A new algebraic structure for formal concept analysis. Inf Sci 180:4865–4876CrossRefGoogle Scholar
- Wille R (1982) Restructuring the lattice theory: an approach based on hierarchies of concepts. In: Rival I (ed) Ordered sets. Reidel, Dordrecht, pp 445–470CrossRefGoogle Scholar
- Wille R (1992) Concepe lattices and conceptual knowledge systems. Comput Math Appl 23(69):493–515CrossRefGoogle Scholar
- Yao YY, Zhao Y (2008) Attribute reduction in decision-theoretic rough set models. Inf Sci 178:3356–3373MathSciNetCrossRefGoogle Scholar
- Zhang W, Wei L, Qi J (2005) Attribute reduction theory and approach to concept lattice. Sci China Ser F Inf Sci 48(6):713–726MathSciNetCrossRefGoogle Scholar