Related papers: On the idempotent semirings such that $\mathcal{D}…
A distributive lattice-ordered magma ($d\ell$-magma) $(A,\wedge,\vee,\cdot)$ is a distributive lattice with a binary operation $\cdot$ that preserves joins in both arguments, and when $\cdot$ is associative then $(A,\vee,\cdot)$ is an…
The Swing Lemma of the second author describes how a congruence spreads from a prime interval to another in a slim (having no $M_3$ sublattice), planar, semimodular lattice. We generalize the Swing Lemma to planar semimodular lattices.
We establish two sufficient conditions for an additively idempotent semiring to be nonfinitely based. As applications, we prove that two specific $4$-element additively idempotent semirings, $S_{(4,545)}$ and $S_{(4,634)}$, whose additive…
Motivated by a recent paper of G. Gr\"atzer, a finite distributive lattice $D$ is said to be fully principal congruence representable if for every subset $Q$ of $D$ containing $0$, $1$, and the set $J(D)$ of nonzero join-irreducible…
We prove that every finite distributive lattice $D$ can be represented as the congruence lattice of a rectangular lattice $K$ in which all congruences are principal. We verify this result in a stronger form as an extension theorem.
For any non-uniform lattice $\Gamma $ in $SL(2,R)$, we describe the limit distribution of orthogonal translates of a divergent geodesic in $\Gamma \backslash SL(2,R)$. As an application, for a quadratic form $Q$ of signature $(2,1)$, a…
For a finite lattice L, let EL denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form EL, as follows:…
We investigate endomorphism semirings of a finite semilattice with one least element and one greatest element such that all the other elements form an antichain. We construct some new finite simple semirings. Keywords: endomorphism…
In this article, we introduce a lattice congruence with respect to a nonempty ideal $I$ of a distributive lattice $L$ and a derivation $d$ on $L$ denoted by $\theta_I^d$. We investigate some necessary and sufficient conditions for the…
A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of ${\mathbb Z}^2$, as well as their determinant and minima sets. We…
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…
In the study of algebras related to non-classical logics, (distributive) semilattices are always present in the background. For example, the algebraic semantic of the $\{\rightarrow,\wedge,\top\}$-fragment of intuitionistic logic is the…
We prove a representation theorem for totally ordered idempotent monoids via a nested sum construction. Using this representation theorem we obtain a characterization of the subdirectly irreducible members of the variety of semilinear…
In this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective…
We prove that the acyclic reorientation poset of a directed acyclic graph $D$ is a lattice if and only if the transitive reduction of any induced subgraph of $D$ is a forest. We then show that the acyclic reorientation lattice is always…
The purpose of this note is describe and classify the splittable lattices in the completely solvable metabelian Lie group (semidirect product of abelian vector groups) $G:=\mathbb{R}^n\rtimes_\eta\mathbb{R}^m$, where $\eta$ is the…
Let $L$ be a finite lattice and let $I$ be an ideal of $L$. Then the restriction map is a bounded lattice homomorphism of the congruence lattice of~$L$ into the congruence lattice of $I$. In a 2009 paper, the authors proved the converse. In…
A module is said to be \textsf{distributive} if the lattice of its submodules is distributive. A direct sum of distributive modules is called a \textsf{semidistributive} module. In this paper we consider rings $A$ such that all right…
The 5-element Brandt semigroup $B_2$ admits the structure of a naturally semilattice-ordered inverse semigroup, thus becoming an additively idempotent semiring with the operation of taking greatest lower bounds as the semiring addition. For…
We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial…