•We present a new describing a hierarchy of concept lattices by a hierarchy-matrix.•We generate the hierarchy-matrix endowing concept lattices with Scott topology.•The hierarchy-matrix embodies all ...the information of a Hasse diagram.•The hierarchy-matrix is well adapted to diverse software based on FCA.•We present a category of finite lattices endowed with Scott topology.
Concept lattices (also called Galois lattices) are complete ones with the hierarchical order relation of the formal concepts defined by a formal context or Galois connection. In this paper, we present a new of method describing a hierarchy of a finite concept lattice by using a matrix. Given a finite concept lattice L, we introduce Scott topology σ(L) on L and choose an order of a unique minimal base for σ(L). Then, there is a one-to-one correspondence between the finite topological space (L, σ(L)) and a proper square matrix with integral entries; thus we obtain a hierarchy-matrix describing the hierarchy of the concept lattice. We explain how to get the information of the hierarchy from the hierarchy-matrix and discuss the relation between the hierarchy-matrix and the Hasse diagram. Since the hierarchy-matrix allowed us to store the information of hierarchy of the concept lattice, we believe that any software autonomously understand the information of hierarchy of the concepts from the hierarchy-matrix.
For a lattice
L
with 0 and 1, let Princ(
L
) denote the set of principal congruences of
L
. Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Grätzer proved that every bounded ...ordered set is representable as Princ(
L
); in fact, he constructed
L
as a lattice of length 5. For {0, 1}-sublattices
A
⊆
B
of
L
, congruence generation defines a natural map Princ(
A
)
⟶
Princ(
B
). In this way, every family of {0, 1}-sublattices of
L
yields a small category of bounded ordered sets as objects and certain 0-separating {0, 1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: every small category of bounded ordered sets with these properties is representable by principal congruences of selfdual lattices of length 5 in the above sense. As a corollary, we can construct a selfdual lattice
L
in G. Grätzer's above-mentioned result.
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