Congruences and Spectral Theory of Double Boolean Algebras
Structures arised from defining negations on concepts have a more expressive power compare to classical propositional logic. They are crucial in Concept Logic. How close are these structures to Boolean algebras?
Factsheet
- Lead school Business School
- Institute Institute for Applied Data Science & Finance
- Research unit Applied Data Science
- Funding organisation SNSF
- Duration (planned) 01.08.2023 - 31.12.2023
- Project management Prof. Dr. Léonard Kwuida
- Head of project Prof. Dr. Léonard Kwuida
- Keywords Formal concept, Negation, Proto-concept, Boolean algebra, Double Boolean algebra, Simplicity and decomposition
Situation
Concepts are well defined by their intent (common properties) and extent (entities belonging to the concept). They can be combined with logical operators "and" (conjunction) and "or" (disjunction). To define a negation of a concept, we need to take the complement of the intent or of the extent. This leads to two operations called respectively "negation" and "opposition". However, the negation of a concept is not more a concept, but a so-called proto-concept. The algebraic structure obtained is called a proto-concept algebra. Rudolf Wille found the set of equations that are satisfied by all proto-concept algebras and get the abstract algebra called double Boolean algebra, which is a relatively new structure, and is of interest in several point of vue: structural and substructural properties as well as duality representation theory.
Course of action
Congruences are equivalence relations compatible with the algebra operations, and are the main ingredient for the decomposition of algebraic structures. Therefore, a good description of congruences of double Boolean algebras will be a substantial contribution to understand their structure theory, and will be apply to characterize directly and sub-directly irreducible double Boolean algebras. Filters and ideals are usually closely related to congruences classes in many algebras such as Boolean algebras or rings, and play a key role in topological representation theory or natural dualities. In the cases of double Boolean algebras, there are many notions of filters and ideals. We will characterize these and investigate the topologies arising from these different filters.
Result
Characterisation of congruences of double Boolean algebras, simple, directly and sub-directly irreducible double Boolean algebras. Topological representation of double Boolean algebras.
Looking ahead
Duality theory
