Multiplicative Lattices Research Articles

Revisiting Baer elements

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. Additionally, we present results concerning these elements that have not been addressed in the study of $d$-ideals of rings. Furthermore, we introduce Baer closures and explore Baer maximal, prime, semiprime and meet-irreducible elements.

On z-elements of multiplicative lattices

The aim of this paper is to investigate further properties of z-elements in multiplicative lattices. We utilize z-closure operators to extend several properties of z-ideals to z-elements and introduce various distinguished subclasses of z-elements, such as z-prime, z-semiprime, z-primary, z-irreducible, and z-strongly irreducible elements, and study their properties. We provide a characterization of multiplicative lattices where z-elements are closed under finite products and a representation of z-elements in terms of z-irreducible elements in z-Noetherian multiplicative lattices.

Lower spaces of multiplicative lattices

AbstractWe 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 spaces, including lower separation axioms and compactness. We characterize lower spaces that exhibit sobriety. Introducing the concept of strongly disconnected spaces, we establish a correlation between strongly disconnected lower spaces and the presence of nontrivial idempotent elements in the corresponding multiplicative lattices. Additionally, we provide a sufficient condition for a lower space to be connected. We prove that the lower space of proper elements is a spectral space; we further explore continuous maps between lower spaces.

Multiplicative lattices, idempotent endomorphisms, and left skew braces

We show that, making use of multiplicative lattices and idempotent endomorphisms of an algebraic structure [Formula: see text], it is possible to derive several notions concerning [Formula: see text] in a natural way. The multiplicative lattice necessary here is the complete lattice of congruences of [Formula: see text] with multiplication given by commutator of congruences. Our main application is to the study of some notions concerning left skew braces.

Multiplicative lattices: Maximal implies prime and related questions

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 them the case of commutative rings with identity). This result is used to study the lattice theoretic counterpart of multiplicative closed sets, that of [Formula: see text]-systems. The notion of [Formula: see text]-system is also studied from the topological point of view.

Discrete Integrable Systems and Random Lax Matrices

We study properties of Hamiltonian integrable systems with random initial data by considering their Lax representation. Specifically, we investigate the spectral behaviour of the corresponding Lax matrices when the number N of degrees of freedom of the system goes to infinity and the initial data is sampled according to a properly chosen Gibbs measure. We give an exact description of the limit density of states for the exponential Toda lattice and the Volterra lattice in terms of the Laguerre and antisymmetric Gaussian beta -ensemble in the high temperature regime. For generalizations of the Volterra lattice to short range interactions, called INB additive and multiplicative lattices, the focusing Ablowitz–Ladik lattice and the focusing Schur flow, we derive numerically the density of states. For all these systems, we obtain explicitly the density of states in the ground states.

$\mathfrak{X}$-elements in multiplicative lattices - A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings

In this paper, we introduce the concept of an $\mathfrak{X}$-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 $\mathfrak{X}$, we define the concepts of $r$-elements, $n$-elements and $J$-elements. These elements generalize the notion of $r$-ideals, $n$-ideals and $J$-ideals of a commutative ring with identity to multiplicative lattices. In fact, we prove that an ideal $I$ of a commutative ring $R$ with identity is a $n$-ideal ($J$-ideal) of $R$ if and only if it is an $n$-element ($J$-element) of $Id(R)$, the ideal lattice of $R$.

Comaximal factorization lattices

We extend some results of Brewer and Heinzer about integral domains with certain comaximal factorization properties for ideals, to the more general setup of multiplicative lattices.

English

We study the multiplicative lattices L which satisfy the condition a = (a : (a : b))(a : b) for all a, b ∈ L. Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group ℤ or ℝ. A sharp lattice L localized at its maximal elements are totally ordered sharp lattices. The converse is true if L has finite character.

Radical factorization in commutative rings, monoids and multiplicative lattices

In this paper we study the concept of radical factorization in the context of abstract ideal theory in order to obtain a unified approach to the theory of factorization into radical ideals and elements in the literature of commutative rings, monoids and ideal systems. Using this approach we derive new characterizations of classes of rings whose ideals are a product of radical ideals, and we obtain also similar characterizations for classes of ideal systems in monoids and star ideals in integral domains.

Diameter and girth of the multiplicative zero-divisor graph of multiplicative lattices

In this paper, we study the multiplicative zero-divisor graph [Formula: see text] of a multiplicative lattice [Formula: see text]. Under certain conditions, we prove that for a reduced multiplicative lattice [Formula: see text] having more than two minimal prime elements, [Formula: see text] contains a cycle and [Formula: see text]. This essentially settles the conjecture of Behboodi and Rakeei [The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10(4) (2011) 741–753]. Further, we have characterized the diameter of [Formula: see text].

A GCD and LCM-like inequality for multiplicative lattices

Let $A_1,\ldots,A_n$ $(n\ge 2)$ be elements of an commutative multiplicative lattice. Let $G(k)$ (resp., $L(k)$) denote the product of all the joins (resp., meets) of $k$ of the elements. Then we show that $$L(n)G(2)G(4)\cdots G(2\lfloor n/2 \rfloor ) \leq G(1)G(3)\cdots G(2\lceil n/2 \rceil -1).$$ In particular this holds for the lattice of ideals of a commutative ring. We also consider the relationship between $$G(n)L(2)L(4)\cdots L(2\lfloor n/2 \rfloor ) \text{ and } L(1)L(3)\cdots L(2\lceil n/2 \rceil -1)$$and show that any inequality relationships are possible.

Open Set Lattices of Subspaces of Spectrum Spaces

AbstractWe take a unified approach to study the open set lattices of various subspaces of the spectrum of a multiplicative lattice L. The main aim is to establish the order isomorphism between the open set lattice of the respective subspace and a sub-poset of L. The motivating result is the well known fact that the topology of the spectrum of a commutative ring R with identity is isomorphic to the lattice of all radical ideals of R. The main results are as follows: (i) for a given nonempty set S of prime elements of a multiplicative lattice L, we define the S-semiprime elements and prove that the open set lattice of the subspace S of Spec(L) is isomorphic to the lattice of all S-semiprime elements of L; (ii) if L is a continuous lattice, then the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m-semiprime elements of L; (iii) we define the pure elements, a generalization of the notion of pure ideals in a multiplicative lattice and prove that for certain types of multiplicative lattices, the sub-poset of pure elements of L is isomorphic to the open set lattice of the subspace Max(L) consisting of all maximal elements of L.

Zariski topology on lattice modules

Let [Formula: see text] be a lattice module over a [Formula: see text]-lattice [Formula: see text] and [Formula: see text] be the set of all prime elements in lattice modules [Formula: see text]. In this paper, we study the generalization of the Zariski topology of multiplicative lattices [N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: Minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. 52 (1988) 53–67; N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, in Algebra and Its Applications (Marcel Dekker, New York, 1984), pp. 265–276.] to lattice modules. Also we investigate the interplay between the topological properties of [Formula: see text] and algebraic properties of [Formula: see text].

Improvement on asymptotic density of packing families derived from multiplicative lattices

Improvement on asymptotic density of packing families derived from multiplicative lattices

On ϕ-Absorbing Primary Elements in Lattice Modules

Let L be a C-lattice and let M be a lattice module over L. Let ϕ:M→M be a function. A proper element P∈M is said to be ϕ-absorbing primary if, for x1,x2,…,xn∈L and N∈M, x1x2⋯xnN≤P and x1x2⋯xnN≰ϕ(P) together imply x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM, for some i∈{1,2,…,n}. We study some basic properties of ϕ-absorbing primary elements. Also, various generalizations of prime and primary elements in multiplicative lattices and lattice modules as ϕ-absorbing elements and ϕ-absorbing primary elements are unified.

Uniformly primary elements of multiplicative lattices

Uniformly primary elements of multiplicative lattices

Beck’s conjecture and multiplicative lattices

Beck’s conjecture and multiplicative lattices

2-Absorbing and Weakly 2-Absorbing Elements in Multiplicative Lattices

In this article we study weakly 2-absorbing elements and 2-absorbing elements in C-lattices. Next we characterize C-lattices in which every nonzero proper element is a (weakly) 2-absorbing element. We also establish a new characterization for principal element domains in terms of 2-absorbing elements.

On generalization of prime, weakly prime and almost prime elements in multiplicative lattices

On generalization of prime, weakly prime and almost prime elements in multiplicative lattices