Ordered set of concepts in n-ary relation
by
V.E. Novikov
Department of Mathematics, Saratov State University, Astrachanskaya 83, Saratov 410026 Russia.

Based on algebraic theory of binary relation due by V.Wagner we generalize ideas of the formal Concept Analysis introduced by Wille . The main idea is to consideration the notion of multivalued object together with multivalued attributes.

This approach permits us to construct order set of notions based on the well-known database operators without interpretation of the database in form of a binary relation.

Let \rho subset or equal M₁× ... ×M_n be an n-ary relation. For arbitrary elements x_i1 in M_i1, x_i2 in M_i2, ... , x_ik in M_ik, 1 <= i₁ < i₂ < ... < i_k\leqn, the k-system (x_i1, x_i2, ... , x_ik) is said to enter into \rho if their is an n-system (x₁, x₂, ... , x_n), such that x_i1, x_i2, ... , x_ik are its corresponding components. For arbitrary 1 <= i₁ < ... < i_k, we denote [`(\iota)]<sub>k</sub> :=(i<sub>1</sub>, i<sub>2</sub>, ... , i<sub>k</sub>). In particular, [`(\iota)]₁:=i₁, [`n]:=(1, 2, ... , n), x<sub>[`(\iota)]k</sub>:=(x_i1, x_i2, ... , x_ik), M_[`(\iota)]k:=M_i1×M_i2× ... ×M_ik.

For any n-ary relation \rho subset or equal M_{[`]</sub>n, 1 <= k, s <= n and a<sub>[`(\iota)]k} in M_{[`(\iota)]k</sub>, X subset or equal M<sub>[`(\iota)]s}, we denote

\pi[`(\iota)]k(\rho) : = {x[`(\iota)]k in M[`(\iota)]k | x[`(\iota)]k  enters into\\rho};

\sigma{a[`(\iota)]k} (\rho):={(x1, ... , xn) in \rho|  a[`(\iota)]k enters into (x1, ... , xn)};

Date received: April 7, 2004