Lattice Of Ideals Research Articles

Relations and Radicals in Abstract Lattices and in Lattices of Subspaces of Banach Spaces and of Ideals of Banach Algebras. Amitsur’s Theory Revisited

We refine Amitsur's theory of radicals in complete lattices and apply the obtained results to the theory of radicals in the lattices of subspaces of Banach spaces and in the lattices of ideals of Banach and C*-algebras and of Banach Lie algebras.

Some combinatorial properties of hexagonal lattices

In this paper, we consider the combinatorial properties of the hexagonal lattice. Let $e(n)$ be the number of $n$-element order ideals in a hexagonal lattice. We give the enumeration of $e(n)$ by showing a bijection between the order ideals and Schroder paths. Further, we get formulae for the flag $f$- and $h$-vectors of the hexagonal lattice.

Lattices and quantales of ideals of semigroups and their preservation under Morita contexts

We study properties of the lattice of unitary ideals of a semigroup. In particular, we show that it is a quantale. We prove that if two semigroups are connected by an acceptable Morita context then there is an isomorphism between the quantales of unitary ideals of these semigroups. Moreover, factorisable ideals corresponding to each other under this isomorphism are strongly Morita equivalent.

Residuated lattice of L-fuzzy ideals of a ring

In 1988, given a complete Brouwerian lattice $${\mathbb {L}}:=(L;~\wedge ,~\vee ;~0,~1)$$ and a ring $${\mathcal {A}}:=(A;~+,~\cdot ;~-;~0)$$ with unity 1, Swamy and Swamy (J Math Anal Appl 134:94–103, 1988) built a lattice structure, on the set of L-fuzzy ideals of $${\mathcal {A}}$$, and investigated some of its arithmetic properties. Since the residuation theory is richer than the lattice theory [see, Ciungu (Non-commutative multiple-valued logic algebras, Springer monographs in mathematics, Springer, Berlin, 2014), Galatos et al. (An algebraic glimpse at substructural logics, volume 151 of studies in logic and the foundations of mathematics, Elsevier, Amsterdam, 2007), Jipsen and Tsinakis (in: Martinez (ed) Ordered algebraic structures, Kluwer Academic Publisher, Dordrecht, 2002), Piciu (Algebras of fuzzy logic, Editura Universitaria Craiova, Craiova, 2007)], in this paper, we consider the notion of fuzzy ideals rather under a complete Brouwerian residuated lattice $${\mathcal {L}}:=(L;~\wedge ,~\vee ,~\ominus ,~\twoheadrightarrow ,~\multimap ;~0,~1)$$. A residuated lattice $${\mathcal {F}}id({\mathcal {A}},L):=\big ( Fid({\mathcal {A}},L);~\wedge ,~+,~\otimes ,~\hookrightarrow ,~\looparrowright ; ~\chi _0,~{\underline{1}}\big )$$ is built on the set $$Fid({\mathcal {A}},L)$$ of L-fuzzy ideals of $${\mathcal {A}}$$ and it is shown that the latter is both an extension of $${\mathcal {L}}$$ and the residuated lattice $${\mathcal {I}}d({\mathcal {A}}):=\big (Id({\mathcal {A}});~\cap ,~+,~\odot ,~\rightarrow ,~\rightsquigarrow ;~\{0\},~A \big )$$ on the set $$Id({\mathcal {A}})$$ of ideals of $${\mathcal {A}}$$.

A characterization of large Dedekind domains

Let D be a commutative domain with identity, and let $${\mathcal {L}}(D)$$ be the lattice of nonzero ideals of D. Say that D is ideal upper finite provided $${\mathcal {L}}(D)$$ is upper finite, that is, every nonzero ideal of D is contained in but finitely many ideals of D. Now let $$\kappa >2^{\aleph _0}$$ be a cardinal. We show that a domain D of cardinality $$\kappa $$ is ideal upper finite if and only if D is a Dedekind domain. We also show (in ZFC) that this result is sharp in the sense that if $$\kappa $$ is a cardinal such that $$\aleph _0\le \kappa \le 2^{\aleph _0}$$ , then there is an ideal upper finite domain of cardinality $$\kappa $$ which is not Dedekind.

Rough sets based on fuzzy ideals in distributive lattices

Abstract In this paper, we present a rough set model based on fuzzy ideals of distributive lattices. In fact, we consider a distributive lattice as a universal set and we apply the concept of a fuzzy ideal for definitions of the lower and upper approximations in a distributive lattice. A novel congruence relation induced by a fuzzy ideal of a distributive lattice is introduced. Moreover, we study the special properties of rough sets which can be constructed by means of the congruence relations determined by fuzzy ideals in distributive lattices. Finally, the properties of the generalized rough sets with respect to fuzzy ideals in distributive lattices are also investigated.

Boolean lifting property in quantales

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings, maximal rings, etc. Inspired by LIP, lifting properties were also defined for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congruence distributive universal algebras, etc. In this paper, we define a lifting property (LP) in commutative coherent integral quantales, structures that are a good abstraction for lattices of ideals, filters and congruences. LP generalizes all the lifting properties existing in the literature. The main tool in the study of LP in a quantale A is the reticulation of A, a bounded distributive lattice whose prime spectrum is homeomorphic to the prime spectrum of A. The principal results of the paper include a characterization theorem for quantales with LP and a characterization theorem for hyperarchimedean quantales.

Ideal structure and factorization properties of the regular kernel operators

We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on L^p-spaces. We show, in particular, that for any L^p(mu ) space, with mu sigma -finite and 1<p<infty , the norm-closure of the ideal of finite-rank operators on L^p(mu ), is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the L^p setting is the fact that every regular kernel operator on an L^p(mu ) space (mu and p as before) factors with regular factors through ell _p. We show that a similar but weaker factorization property, where ell _p is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit.

The talented monoid of a Leavitt path algebra

There is a tight relation between the geometry of a directed graph and the algebraic structure of a Leavitt path algebra associated to it. In this note, we show a similar connection between the geometry of the graph and the structure of a certain monoid associated to it. This monoid is isomorphic to the positive cone of the graded K0-group of the Leavitt path algebra which is naturally equipped with a Z-action. As an example, we show that a graph has a cycle without an exit if and only if the monoid has a periodic element. Consequently a graph has Condition (L) if and only if the group Z acts freely on the monoid. We go on to show that the algebraic structure of Leavitt path algebras (such as simplicity, purely infinite simplicity, or the lattice of ideals) can be described completely via this monoid. Therefore an isomorphism between the monoids (or graded K0's) of two Leavitt path algebras implies that the algebras have similar algebraic structures. These all bolster the claim that the graded Grothendieck group could be a sought-after complete invariant for the classification of Leavitt path algebras.

$d$-fuzzy ideals and injective fuzzy ideals in distributive lattices

$d$-fuzzy ideals and injective fuzzy ideals in distributive lattices

A Residuated Lattice of L-Fuzzy Subalgebras of a Mono-Unary Algebra

Given a complete residuated lattice [Formula: see text] and a mono-unary algebra [Formula: see text], it is well known that [Formula: see text] and the residuated lattice [Formula: see text] of [Formula: see text]-fuzzy subsets of [Formula: see text] satisfy the same residuated lattice identities. In this paper, we show that [Formula: see text] and the residuated lattice [Formula: see text] of [Formula: see text]-fuzzy subalgebras of [Formula: see text] satisfy the same residuated lattice identities if and only if the Heyting algebra [Formula: see text] of subuniverses of [Formula: see text] is a Boolean algebra. We also show that [Formula: see text] is a Boolean algebra (respectively, an [Formula: see text]-algebra) if and only if [Formula: see text] is a Boolean algebra (respectively, an [Formula: see text]-algebra) and [Formula: see text] is a Boolean algebra.

α -ideals in a 0-distributive lattice

α -ideals in a 0-distributive lattice

Fuzzy Ideals and Fuzzy Filters of Pseudocomplemented Semilattices

In this paper, we introduce the concept of kernel fuzzy ideals and ⁎-fuzzy filters of a pseudocomplemented semilattice and investigate some of their properties. We observe that every fuzzy ideal cannot be a kernel of a ⁎-fuzzy congruence and we give necessary and sufficient conditions for a fuzzy ideal to be a kernel of a ⁎-fuzzy congruence. On the other hand, we show that every fuzzy filter is the cokernel of a ⁎-fuzzy congruence. Finally, we prove that the class of ⁎-fuzzy filters forms a complete lattice that is isomorphic to the lattice of kernel fuzzy ideals.

Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras

In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and the point functor are introduced for finitely semi-graded rings. Finitely semi-graded algebras and rings include many important examples of non- N -graded algebras coming from mathematical physics that play a very important role in mirror symmetry problems, and for these concrete examples, the Koszulity will be established, as well as the explicit computation of its Hilbert and Poincaré series.

Minimal set of binomial generators for certain Veronese 3-fold projections

The goal of this paper is to explicitly describe a minimal binomial generating set of a class of lattice ideals, namely the ideal of certain Veronese 3-fold projections. More precisely, for any integer d≥4 and any d-th root e of 1 we denote by Xd the toric variety defined as the image of the morphism φTd:P3⟶Pμ(Td)−1 where Td are all monomials of degree d in k[x,y,z,t] invariant under the action of the diagonal matrix M(1,e,e2,e3). In this work, we describe a Z-basis of the lattice Lη associated to I(Xd) as well as a minimal binomial set of generators of the lattice ideal I(Xd)=I+(η).

Rowmotion in slow motion

Rowmotion is a simple cyclic action on the distributive lattice of order ideals of a poset: it sends the order ideal x to the order ideal generated by the minimal elements not in x. It can also be computed in ‘slow motion’ as a sequence of local moves. We use the setting of trim lattices to generalize both definitions of rowmotion, proving many structural results along the way. We introduce a flag simplicial complex (similar to the canonical join complex of a semidistributive lattice) and relate our results to recent work of Barnard by proving that extremal semidistributive lattices are trim. As a corollary, we prove that if A is a representation finite algebra and mod A has no cycles, then the torsion classes of A ordered by inclusion form a trim lattice.

On the lattice of biorder ideals of regular rings

The study of biordered set plays a significant role in describing the structure of a regular semigroup and since the definition of regularity involves only the multiplication in the ring, it is natural that the study of semigroups plays a significant role in the study of regular rings. Here, we extend the biordered set approach to study the structure of the regular semigroup [Formula: see text] of a regular ring [Formula: see text] by studying the idempotents [Formula: see text] of the regular ring and show that the principal biorder ideals of the regular ring [Formula: see text] form a complemented modular lattice and certain properties of this lattice are studied.

α-fuzzy ideals and space of prime α-fuzzy ideals in distributive lattices

α-fuzzy ideals and space of prime α-fuzzy ideals in distributive lattices

Erratum to: Congruences and Ideals in a Distributive Lattice with Respect to a Derivation

The present note is an Erratum for the two theorems of the paper "Congruences and ideals in a distributive lattice with respect to a derivation" by M. Sambasiva Rao.

On EMV-algebras

The paper deals with an algebraic extension of MV-algebras based on the definition of generalized Boolean algebras. We introduce a new class of structures, not necessarily with a top element, which are called EMV-algebras, in a way that every EMV-algebra contains an MV-algebra. First, we present basic properties of EMV-algebras. We give some examples, introduce and investigate congruence relations, ideals and filters on these algebras. We establish a basic representation result saying that each EMV-algebra can be embedded into an EMV-algebra with top element and we characterize EMV-algebras either as structures which are termwise equivalent to MV-algebras or as maximal ideals of EMV-algebras with top element. We study the lattice of ideals of an EMV-algebra and prove that every EMV-algebra has at least one maximal ideal. We define an EMV-clan of fuzzy sets as a special EMV-algebra where all operations are defined by points. We show that any semisimple EMV-algebra is isomorphic to an EMV-clan of fuzzy functions on a set. The set of EMV-algebras is neither a variety nor a quasivariety, but rather a special class of EMV-algebras which we call a q-variety of EMV-algebras. We present an equational base for each proper q-subvariety of the q-variety of EMV-algebras. We establish categorical equivalencies among the category of proper EMV-algebras, the category of MV-algebras with a fixed special maximal ideal, and a special category of Abelian unital ℓ-groups.