Adam Mata
supervisor: Anna Zamojska-Dzienio
Lattices are algebraic structures, frequently denoted as `(L, ⋁, ⋀)`, where:
• `L` is an arbitrary, nonempty set,
• `⋁` and `⋀` binary operations on `L` which are associative, commutative and idempotent,
• `⋁` and `⋀` satisfy absorption laws, namely (for all elements `a, b ∊ L`):
o `a ⋁(a⋀b) = a`
o `a ⋀ (a⋁b) = a`
If it does not lead to disambiguation we call `L` itself the lattice without mentioning the binary operations. We say that `S` is a sublattice of the lattice `L` if `S` ⊆ `L` and `S` is closed under binary operations inherited from `L`. A sublattice `S` is considered maximal if `S`≠`L` and there is no sublattice `R` of lattice `L` such that `S ⊊ R ⊊ L`.
The algebraic definition of lattices may be equivalently transformed to order-wise definition. We can introduce partial order on the set `L` by the following rule (for all elements `a, b ∊ L`):
• `a` ≤ `b` if and only if `a = a ⋀ b`, or equivalently
• `a` ≤ `b` if and only if `b = a ⋁ b`,
Convex geometry is a pair (`C`, α) where `C` is an arbitrary, nonempty set and α is a closure operator with an extra condition involved called anti-exchange property which generalizes the notion of convexity in affine spaces. The lattice of all closed sets in `C`, I.e. `B`⊆`C` such that `B = `α`(B)`, ordered by inclusion is called shortly a convex geometry, too.
It is known that for many classes of lattices the complements of their maximal sublattices are intervals, e.g. it holds for distributive lattices. It is not the case for the convex geometries but our hypothesis states that the complement has has a minimal element and is a convex set (for any elements `a,b` in the elements between them are in the complement, too). There are many properties of the complements which confirm the conjecture.
In parallel to search for the formal proof, we are investigating computer-generated examples which are on their own interests due to the one-to-one correspondence of convex geometries and anti-matroids widely studied in combinatorics. There were positive verification of smaller cases carried out, namely the vast implemented on the computer. The implementation of algorithms has been performed in Haskell programming language what enabled us exploiting recursive character of this language to find sublattices of a particular convex geometry. The automated check, in all cases, returned the positive answer to the hypothesis. This gives a solid background and good motive to continue the research on that matter.
This is a joint work with Kira Adaricheva (Hofstra University, (Hofstra University) and Anna Zamojska-Dzienio (Warsaw University of Technology).