suggested to represent patterns using deterministic finite automata. The results obtained experimentally show that minimum automata provide a compact representation. Nevertheless, Mielik¨ainen considered collections of itemsets and not of concepts. Moreover he does not provide any generic framework, based on some algebraic operators.

The next section recalls some basic definitions useful for the understanding of the paper. In Section 3, we introduce the labeled graph representation of concepts collections while Section 4 presents a basic algorithm to build **this** graph. In Section 5 we define operators that allow to query the graph and that can be combined using an algebra (in some sense, this section is related to [6]).

## 2 Definitions

A database Db is a relation between a set of attributes A = {a1, a2, ...} and a set of objects O = {o1, o2**, ...**}.

Such a database can be represented as a boolean matrix where the **columns**
are attributes and the rows are objects.

![0_image_0.png](0_image_0.png)

For instance, this database can be the result of gene expression measures.

In this case, the columns represent genes and the rows represent biological situations. There is a relation between a gene and a situation if the gene **is overexpressed in the given situation. Mining formal concepts in this kind of data has**
been shown to be interesting for biologists [1].

A bi-set is a pair (X, Y ) where X ⊆ A and Y ⊆ O. A 1-rectangle **is a bi-set**
(X, Y ) such that all the attributes of X **are in relation with all the objects of** Y .

In the matrix, a 1-rectangle thus defines a sub-matrix containing only ones.