Related papers: On $z$-elements of multiplicative lattices
The aim of this series of papers is to study $z$-ideals of semirings. In this article, we introduce some distinguished classes of $z$-ideals of semirings, which include $z$-prime, $z$-semiprime, $z$-irreducible, and $z$-strongly irreducible…
The aim of this article is to extend the notions of strongly hollow and completely strongly hollow ideals of commutative rings to multiplicative lattices. We investigate their basic structural properties and prove several characterizations…
Idempotent elements are a well-studied part of ring theory, with several identities of the idempotents in $\mathbb{Z}/m\mathbb{Z}$ already known. Although the idempotents are not closed under addition, there are still interesting additive…
The objective of this paper is to extend certain properties observed in $d$-ideals of rings and $d$-elements of frames to Baer elements in multiplicative lattices introduced in D. D. Anderson, C. Jayaram, and P. A. Phiri, Baer lattices,…
In this paper, we introduce a concept of $\mathfrak{X}$-element with respect to an $M$-closed set $\mathfrak{X}$ in multiplicative lattices and study properties of $\mathfrak{X}$-elements. For a particular $M$-closed subset $\mathfrak{X}$,…
In this paper, we introduce $S$-prime elements in $V$-lattices, where $S$ is a multiplicatively closed subset of a $V$-lattice $L$. In addition, we introduce the $S$-Prime Element Principle to prove that certain elements in $V$-lattices are…
In this paper, we introduce $\phi$-$\delta$-primary elements in a compactly generated multiplicative lattice $L$ and obtain its characterizations. We prove many of its properties and investigate the relations between these structures. By a…
We consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower…
Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $\ell$-algebras, preferably in $f$-algebras, and in product algebras. The additional structure of an associative multiplication leads…
In this paper, we introduce an element $\phi$-$\delta$-primary to another element in a compactly generated multiplicative lattice $L$ and obtain its characterizations. We prove many of its properties and investigate the relations between…
M.S. Rao recently investigated some sorts of special filters in distributive pseudocomplemented lattices. In our paper we extend this study to lattices which need neither be distributive nor pseudocomplemented. For this sake we define a…
In this study, we introduce the concept of quasi n-absorbing elements of multiplicative lattices. A proper element q is said to be a quasi n-absorbing element of L if whenever $a^nb\le q$ implies that either $a^n\le q$ or $a^{n-1}b\le q$.…
The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among…
We survey results concerning special elements of nine types (modular, lower-modular, upper-modular, cancellable, distributive, codistributive, standard, costandard and neutral elements) in the lattice of all semigroup varieties and certain…
In this paper, we introduce the expansion function $\delta$ on an $L$-module $M$. We define and investigate a $\delta$-primary element in an $L$-module $M$. Its characterizations and many of its properties are obtained. $\delta_0$-primary…
In this paper, we investigate zero-divisor, nilpotent, idempotent, unit, small, and irreducible elements in semiring extensions such as amount, content, and monoid semialgebras. We also introduce new concepts such as the prime avoidance…
We introduce and study $\mu$-elements, that generalize a lattice-theoretic abstraction (namely, essential elements) of essential ideals of rings, essential submodules of modules, and dense subsets of topological spaces. Exploring several…
Let $R$ be a commutative ring with nonzero identity and $M$ be an $R$-module. Quasi-prime submodules of $M$ and the developed Zariski topology on $q\Spec(M)$ are introduced. We also, investigate the relationship between the algebraic…
In this paper, we introduce the notion of pseudo-primary elements and pseudo-classical primary elements in an $L$-module $M$ and obtain their characterizations. The aim of the paper is to show $rad(N)\in M$, the radical of $N\in M$ is prime…
In this paper we study the set of prime ideals in vector lattices and how the properties of the prime ideals structure the vector lattice in question. The different properties that will be considered are firstly, that all or none of the…