Filters, ideals and power of double Boolean algebras

Tenkeu Jeufack, Yannick Léa  ; Tenkeu Kembang, Gael ; Temgoua Alomo, Etienne Romuald ; Kwuida, Léonard

Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 2, pp. 451-478

Résumé

Double Boolean algebras (dBas) are algebras D=(D; ⊓, ⊔, ¬, y, ⊥, ⊤) of type (2,2,1,1,0,0), introduced by Rudolf Wille to capture the equational theory of the algebra of protoconcepts. Boolean algebras form a subclass of dBas. Our goal is an algebraic investigation of dBas, based on similar results on Boolean algebras. In this paper, we describe filters, ideals, homomorphisms and powers of dBas. We show that principal filters as well as principal ideals of dBas form (non necessary isomorphic) Boolean algebras. We also show that, a primary ideal (resp. primary filter) is exactly maximal ideal (resp. ultrafilter) in dBas and primary ideal (resp. filter) needs not be a prime ideal (resp. filter). For a finite dBa, a primary filters (resp. ideals) are principal filter (resp. ideals) generated by atom (resp. co-atom). Some properties of homomorphisms of dBas are investigated and the relationship between the homomorphism of dBas D, M and the lattices of filters (resp. ideals) of these two dBas. Giving a dBa D and a non-emptyset X, we study some relationship between D and L=D^X by showing that D is contextual, fully contextual (resp. trivial) if and only if L is contextual, fully contextual (resp. trivial). In addition, we show that D embeds into L and the lattice of filters ℱ(D) (resp. of ideals ℐ(D)) is algebraic and embeds in the lattice ℱ(L) (resp. ℐ(L)). We finish this paper by showing that some sets of polynomial functions of D form a Boolean algebra isomorphic to the set of principal filters (resp. principal ideals) of D.

Détail

Keywords: double Boolean algebra, protoconcepts algebra, concept algebra, formal concept