# Optimal decompositions of matrices with entries from residuated lattices

```@article{Belohlvek2012OptimalDO,
title={Optimal decompositions of matrices with entries from residuated lattices},
journal={J. Log. Comput.},
year={2012},
volume={22},
pages={1405-1425}
}```
• R. Belohlávek
• Published 1 December 2012
• Mathematics, Computer Science
• J. Log. Comput.
We describe optimal decompositions of matrices whose entries are elements of a residuated lattice L, such as L=[0, 1]. Such matrices represent relationships between objects and attributes with the entries representing degrees to which attributes represented by columns apply to objects represented by rows. Given such an n × m object-attribute matrix I, we look for a decomposition of I into a product A ° B of an n × k object-factor matrix A and a k × m factor-attribute matrix B with entries from…
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## References

SHOWING 1-10 OF 35 REFERENCES
• Computer Science
Annals of Mathematics and Artificial Intelligence
• 2010
A theorem is presented which shows that decompositions which use particular formal concepts of I as factors for the decomposition are optimal in that the number of factors involved is the smallest possible.
Factor Analysis of Incidence Data via Novel Decomposition of Matrices
• Computer Science
ICFCA
• 2009
This work presents a novel approach to decomposition and factor analysis of matrices with incidence data, based on a geometric insight provided by a theorem identifying particular rectangular-shaped submatrices as optimal factors for the decompositions.
The Discrete Basis Problem
• Computer Science
IEEE Transactions on Knowledge and Data Engineering
• 2008
This paper describes a matrix decomposition formulation for Boolean data, the Discrete Basis Problem, and gives a simple greedy algorithm for solving it and shows how it can be solved using existing methods.
Computing the Lattice of All Fixpoints of a Fuzzy Closure Operator
• Computer Science
IEEE Transactions on Fuzzy Systems
• 2010
We present a fast bottom-up algorithm to compute all fixpoints of a fuzzy closure operator in a finite set over a finite chain of truth degrees, along with the partial order on the set of all
Concept lattices and order in fuzzy logic
Concept Equations
The question ofsolvability, structure of solutions, and how solvability of non-solvable systems may be attained by so-called decrease of logical precision are answered.
Fuzzy Closure Operators
The aim of this paper is to outline a general theory of fuzzy closure operators and fuzzy closure systems, and to introduce the necessary concepts.