Unique Factorization Theorem and Formal Concept Analysis Peter Mih´ ok ⋆ and Gabriel Semaniˇ sin ⋆⋆ 1 Department of Applied Mathematics, Technical University Koˇ sice, Faculty of Economics, B.Nˇ emcovej 32, 040 01 Koˇ sice, Slovak Republic and Mathematical Institute, Slovak Academy of Sciences, Greˇ s´akova 6, 040 01 Koˇ sice, Slovak Republic peter.mihok@tuke.sk 2 Institute of Computer Science, P.J. ˇ Saf´ arik University, Faculty of Science, Jesenn´a 5, 041 54 Koˇ sice, Slovak Republic, gabriel.semanisin@upjs.sk Abstract. In the theory of generalised colourings of graphs, the Unique Factorization Theorem (UFT) for additive induced-hereditary properties of graphs provides an analogy of the well-known Fundamental Theorem of Arithmetics. The purpose of this paper is to present a new, less com- plicated, proof of this theorem that is based on Formal Concept Analysis. The method of the proof can be successfully applied even for more gen- eral mathematical structures known as relational structures. 1 Introduction and motivation Formal Concept Analysis (briefly FCA) is a theory of data analysis which iden- tifies conceptual structures among data sets. It was introduced by R. Wille in 1982 and since then has grown rapidly (for a comprehensive overview see [12]). The mathematical lattices that are used in FCA can be interpreted as classifica- tion systems. Formalized classification systems can be analysed according to the consistency of their relations. Some extensions and modifications of FCA can be found e.g. in [16]. In this paper we provide a new proof of the Unique Factorization Theorem (UFT) for induced-hereditary additive properties of graphs. The problem of unique factorization of reducible hereditary properties of graphs into irreducible factors was formulated as Problem 17.9 in the book [15] of T.R. Jensen and B. Toft. Our proof is significantly shorter as the previous ones and it is based on FCA. Moreover, FCA allows us to work with concepts instead of graphs and the ⋆ Research supported in part by Slovak VEGA Grant 2/4134/24 ⋆⋆ Research supported in part by Slovak VEGA Grant 1/3004/06 and Slovak APVT grant 20-004104. 195