Theoretical foundations#
Relations#
Hasse diagram of algebras#
\(\times\) The Cartesian product is a way to combine two sets to form a new set of ordered pairs. Each pair consists of one element from the first set and one element from the second set.
A binary relation between two sets is a way of associating elements from one set with
elements from another set.
An \(n\)-ary relation is a generalization of the concept of a binary relation,
extending it to involve more than two sets. While a binary relation involves
pairs of elements from two sets, an \(n\)-ary relation involves tuples of elements
from \(n\) sets.
An endo-relation is a special type of binary relation where the elements being related
come from the same set. In other words, it’s a relation that a set has with itself.
Posets#
\(\leq\) A poset is a set where some elements are related to each other in a specific way, but not all elements need to be comparable. This relationship is a binary relation on the set itself, meaning it pairs elements from the set with other elements from the same set.
\(\prec\) A covering relation is a specific type of relation in a partially ordered set (poset) that describes the immediate or direct successors of an element. It’s like identifying the very next step or level up from a given position.
Lattices#
A lattice is a special type of partially ordered set (poset) where every pair of elements
has both a least upper bound (supremum) and a greatest lower bound (infimum).
A semi-lattice is a partial order where any two elements have either a least upper bound
(join) or a greatest lower bound (meet), but not necessarily both. It’s like having a
structure where you can always find a common “top” or “bottom” for any pair of elements,
but not both.
The minimum, the maximum, the atoms, the co-atoms, the join and meet irreducibles are
special elements that help in understanding the structure and properties of a lattice,
providing insights into its foundational components.
Set theoretic operators#
A closure operator is a function that takes a subset of a universe and returns a
closed version of that subset. This closed version includes the original subset
and possibly additional elements needed to satisfy certain conditions or properties.
A Moore family, also known as a closure system, is a collection of sets that is
closed under intersection. This means that if you take any number of sets from
the collection and find their intersection (the set of elements they all share),
that intersection is also a set in the collection.
A Galois connection is a pair of mappings between two partially ordered sets (posets)
that reverses the order. It’s a way to relate elements from one set to elements in
another set, while preserving certain structural properties.
Concepts#
In FCA, A formal context is a structured way of organizing and analyzing data that
consists of objects and their attributes.
In FCA, a concept is a fundamental unit that pairs a set of objects with a set of
attributes that those objects share. It’s a way to group and understand data by
identifying common characteristics.
A concept lattice is a special type of lattice used in formal concept analysis (FCA)
to represent the relationships between objects and their attributes. It’s a way to
organize and visualize data by grouping objects that share common attributes.
An AOC-poset, or Attribute-Object-Concept Poset, is a structure in formal concept analysis
(FCA) that focuses on concepts where a specific object or attribute appears. It highlights
the relevance of particular objects or attributes within the lattice of concepts.
Description lattices#
A description space is a framework used in formal concept analysis (FCA) to represent
and analyze the attributes or features of objects in a structured way.
It provides a way to describe objects based on their characteristics and to explore
the relationships between these descriptions.
A generalized convex hull is a concept that extends the idea of a traditional convex hull
to more complex or abstract spaces. In traditional geometry, the convex hull of a set of
points is the smallest convex shape that can enclose all the points—like wrapping a rubber
band around a set of nails.
Description lattices are an extension of concept lattices in which the intensions of each
concept are described by the boundaries of generalized convex hulls.