Related papers: Generalizing Galvin and J\'onsson's Classification…
We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive $\exists \forall$-sentence true in…
We investigate the lattice of machine invariant classes. This is an infinite completely distributive lattice but it is not a Boolean lattice. We show the subword complexity and the growth function create machine invariant classes. So the…
Lorenzen's ``Algebraische und logistische Untersuchungen \"uber freie Verb\"ande'' appeared in 1951 in The Journal of Symbolic Logic. These ``Investigations'' have immediately been recognised as a landmark in the history of infinitary proof…
The systematic study of planar semimodular lattices started in 2007 with a series of papers by G. Gr\"atzer and E. Knapp. These lattices have connections with group theory and geometry. A planar semimodular lattice $L$ is {\it slim} if…
A flat solvmanifold is a compact quotient $\Gamma\backslash G$ where $G$ is a simply-connected solvable Lie group endowed with a flat left invariant metric and $\Gamma$ is a lattice of $G$. Any such Lie group can be written as…
A resolution of the intersection of a finite number of subgroups of an abelian group by means of their sums is constructed, provided the lattice generated by these subgroups is distributive. This is used for detecting singularities of…
A distributive lattice $L$ with minimum element $0$ is called decomposable if $a$ and $b$ are not comparable elements in $L$ then there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b),…
A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. Examples of finitely universal varieties of semigroups have been available since the early 1970s, but it is unknown if there…
We continue our work on the model theory of free lattices, solving two of the main open problems from our first paper on the subject. Our main result is that the universal (existential) theory of infinite free lattices is decidable. Our…
Diversities are a generalization of metric spaces, where instead of the non-negative function being defined on pairs of points, it is defined on arbitrary finite sets of points. Diversities have a well-developed theory. This includes the…
Concept lattices are well-known conceptual structures that organise interesting patterns-the concepts-extracted from data. In some applications, such as software engineering or data mining, the size of the lattice can be a problem, as it is…
A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups…
An important and long-standing open problem in universal algebra asks whether every finite lattice is isomorphic to the congruence lattice of a finite algebra. Until this problem is resolved, our understanding of finite algebras is…
Let L be a join-distributive lattice with length n and width(Ji L) \leq k. There are two ways to describe L by k-1 permutations acting on an n-element set: a combinatorial way given by P.H. Edelman and R.E. Jamison in 1985 and a recent…
An integral lattice which is generated by some vectors of norm $q$ is called $q$-lattice. Classification of 3-lattices of dimension at most four is given by Mimura (On 3-lattice, 2006). As a expansion, we give a classification of 3-lattices…
In the second edition of the congruence lattice book, Problem 22.1 asks for a characterization of subsets $Q$ of a finite distributive lattice $D$ such that there is a finite lattice $L$ whose congruence lattice is isomorphic to $D$ and…
We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean's condition (D) which arises from Dean's solution to…
In his 1910 paper, \'Elie Cartan gave a tour-de-force solution to the (local) equivalence problem for generic rank 2 distributions on 5-manifolds, i.e. $(2,3,5)$-distributions. From a modern perspective, these structures admit equivalent…
The concept of a $\lambda$-lattice was introduced by V. Sn\'a\v sel in order to generalize some lattice concepts for directed posets whose elements need not have suprema or infima. We extend the concept of semimodularity from lattices to…
Motivated by Gentzen disjunction elimination rule in his Natural Deduction calculus and reading inequalities with meet in a natural way, we conceive a notion of distributivity for join-semilattices. We prove that it is equivalent to a…