32 research outputs found
On Nontrivial Weak Dicomplementations and the Lattice Congruences that Preserve Them
We study the existence of nontrivial and of representable (dual) weak
complementations, along with the lattice congruences that preserve them, in
different constructions of bounded lattices, then use this study to determine
the finite (dual) weakly complemented lattices with the largest numbers of
congruences, along with the structures of their congruence lattices. It turns
out that, if is a natural number, then the four largest numbers of
congruences of the --element (dual) weakly complemented lattices are:
, , and . For smaller
numbers of elements, several intermediate numbers of congruences appear between
the elements of this sequence. After determining these numbers, along with the
structures of the (dual) weakly complemented lattices having these numbers of
congruences, we derive a similar result for weakly dicomplemented lattices.Comment: 28 page
Finite Distributive Concept Algebras
Concept algebras are concept lattices enriched by a weak negation and a weak opposition. In Ganter and Kwuida (Contrib. Gen. Algebra, 14:63-72, 2004) we gave a contextual description of the lattice of weak negations on a finite lattice. In this contribution1 we use this description to give a characterization of finite distributive concept algebra
On the isomorphism problem of concept algebras
Weakly dicomplemented lattices are bounded lattices equipped with two unary operations to encode a negation on concepts. They have been introduced to capture the equational theory of concept algebras (Wille 2000; Kwuida 2004). They generalize Boolean algebras. Concept algebras are concept lattices, thus complete lattices, with a weak negation and a weak opposition. A special case of the representation problem for weakly dicomplemented lattices, posed in Kwuida (2004), is whether complete weakly dicomplemented lattices are isomorphic to concept algebras. In this contribution we give a negative answer to this question (Theorem4). We also provide a new proof of a well known result due to M.H. Stone(Trans Am Math Soc 40:37-111, 1936), saying that each Boolean algebra is a field of sets (Corollary4). Before these, we prove that the boundedness condition on the initial definition of weakly dicomplemented lattices (Definition1) is superfluous (Theorem1, see also Kwuida (2009)
On the homomorphism order of labeled posets
Partially ordered sets labeled with k labels (k-posets) and their
homomorphisms are examined. We give a representation of directed graphs by
k-posets; this provides a new proof of the universality of the homomorphism
order of k-posets. This universal order is a distributive lattice. We
investigate some other properties, namely the infinite distributivity, the
computation of infinite suprema and infima, and the complexity of certain
decision problems involving the homomorphism order of k-posets. Sublattices are
also examined.Comment: 14 page
Prime ideal theorem for double Boolean algebras
Double Boolean algebras are algebras (D,⊓,⊔,⊲,⊳,⊥,⊤) of type (2,2,1,1,0,0). They have been introduced to capture the equational theory of the algebra of protoconcepts. A filter (resp. an ideal) of a double Boolean algebra D is an upper set F (resp. down set I) closed under ⊓ (resp. ⊔). A filter F is called primary if F ≠ ∅ and for all x ∈ D we have x ∈ F or . In this note we prove that if F is a filter and I an ideal such that F ∩ I = ∅ then there is a primary filter G containing F such that G ∩ I = ∅ (i.e. the Prime Ideal Theorem for double Boolean algebras)
Interpretable Concept-Based Classification with Shapley Values
Among the family of rule-based classification models, there
are classifiers based on conjunctions of binary attributes. For example,
JSM-method of automatic reasoning (named after John Stuart Mill) was
formulated as a classification technique in terms of intents of formal concepts as classification hypotheses. These JSM-hypotheses already represent interpretable model since the respective conjunctions of attributes
can be easily read by decision makers and thus provide plausible reasons
for model prediction. However, from the interpretable machine learning
viewpoint, it is advisable to provide decision makers with importance
(or contribution) of individual attributes to classification of a particular
object, which may facilitate explanations by experts in various domains
with high-cost errors like medicine or finance. To this end, we use the
notion of Shapley value from cooperative game theory, also popular in
machine learning. We provide the reader with theoretical results, basic
examples and attribution of JSM-hypotheses by means of Shapley value
on real data