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In this paper, we introduce and study V- and CI-semirings---semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties…
In this article we introduce theory and algorithms for learning discrete representations that take on a lattice that is embedded in an Euclidean space. Lattice representations possess an interesting combination of properties: a) they can be…
One-dimensional quasilattices are classified into mutual local-derivability (MLD) classes on the basis of geometrical and number-theoretical considerations. Most quasilattices are ternary, and there exist an infinite number of MLD classes.…
There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet's E_A, Godowski, Mayet-Godowski, and Mayet's E equations. We obtain a…
We discuss the possibility of the existence of finite algorithms that may give distinct knot classes. In particular we present two attempts for such algorithms which seem promising, one based on knot projections on a plane, the other on…
We show that the subgroup lattice of any finite group satisfies Frankl's Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common…
We consider the lattice of all the weak factorization systems on a given finite lattice. We prove that it is semidistributive, trim and congruence uniform. We deduce a graph theoretical approach to the problem of enumerating transfer…
We introduce an algorithm for computing closure systems derived from a family of implications on a set. Semilattices presentations are explored and used in conjunction with the algorithm to compute various types of lattices freely generated…
A mixed lattice is a lattice-type structure consisting of a set with two partial orderings, and generalizing the notion of a lattice. Mixed lattice theory has previously been studied in various algebraic structures, such as groups and…
We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice…
This is Part A of four Parts dedicated to modular lattices of finite length. It builds on 1992 notes of the author (available on ResearchGate), and in so doing heeds a wish of the late Gian-Carlo Rota. Part A is in fairly final form and…
We construct a family of independent sets for finite, atomic, and graded lattices, extending the well-known cryptomorphism between geometric lattices and matroids. This construction leads to an embedding theorem into geometric lattices that…
In this paper, we define a property, trimness, for lattices. Trimness is a not-necessarily-graded generalization of distributivity; in particular, if a lattice is trim and graded, it is distributive. Trimness is preserved under taking…
The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety COM of all commutative semigroups or…
This paper is concerned with realizing Lattes maps as subdivision maps of finite subdivision rules. The main result is that the Lattes maps in all but finitely many analytic conjugacy classes can be realized as subdivision maps of finite…
Let $A$, $B$, and $S$ be (v,0)-semilattices and let $f: A\to B$ be a (v,0)-embedding. Then the canonical map, $f \otimes \id\_S$, of the tensor product $A \otimes S$ into the tensor product $B \otimes S$ is not necessarily an embedding. The…
We discuss the relationship between tight and cover-to-join representations of semilattices and inverse semigroups, showing that a slight extension of the former, together with an appropriate selection of co-domains, makes the two notions…
This article is the second part of an essay dedicated to lattices freely generated by posets within a variety. The first part dealt with four easy varieties while this part is concerned with finitely generated varieties. Here we present a…
We investigate the lattice of clones that are generated by a set of functions that are induced on a finite field $\mathbb{F}$ by monomials. We study the atoms and coatoms of this lattice and investigate whether this lattice contains…
Structures involving a lattice and join-endomorphisms on it are ubiquitous in computer science. We study the cardinality of the set $\mathcal{E}(L)$ of all join-endomorphisms of a given finite lattice $L$. In particular, we show for…