Article in volume
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Title:
Filters, ideals and power of double Boolean algebras
PDFSource:
Discussiones Mathematicae - General Algebra and Applications 44(2) (2024) 451-478
Accepted: 2024-10-16 , Available online: 2024-10-17 , https://doi.org/10.7151/dmgaa.1466
Abstract:
Double Boolean algebras (dBas) are algebras $\underline{D}=(D;\sqcap,\sqcup,
\neg,\lrcorner,\bot,\top)$ 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 $\underline{D}$,
$\underline{M}$ and the lattices of filters (resp. ideals) of these two dBas.
Giving a dBa $\underline{D}$ and a non-emptyset $X,$ we study some relationship
between $\underline{D}$ and $\underline{L}=\underline{D}^{X}$ by showing that
$\underline{D}$ is contextual, fully contextual (resp. trivial) if and only
if $\underline{L}$ is contextual, fully contextual (resp. trivial). In addition,
we show that $\underline{D}$ embeds into $\underline{L}$ and the lattice of
filters $\mathcal{F}(\underline{D})$ (resp. of ideals $\mathcal{I}(\underline{D})$)
is algebraic and embeds in the lattice $\mathcal{F}(\underline{L})$ (resp.
$\mathcal{I}(\underline{L})$). We finish this paper by showing that some sets
of polynomial functions of $\underline{D}$ form a Boolean algebra isomorphic to
the set of principal filters (resp. principal ideals) of $\underline{D}$.
Keywords:
double Boolean algebra, protoconcepts algebra, concept algebra, formal concept
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