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In this paper, we introduce the notions of radical filters and extended filters of Intuitionistic Linear algebras (IL-algebras for short) and give some of their properties. The notion of closure operation on an IL-algebra is also introduced as well as the study of some of their main properties. The radical of filters and extended filters are examples of closure operations among several others provided. The class of stable closure operations on an IL-algebra L is used to study the unifying properties of some subclasses of the lattice of filters of L. In particular, we obtain that for a stable closure operation c on an IL-algebra, the collection of c-closed elements of its lattice of filters forms a complete Heyting algebra. Hyperarchimedean IL-algebras are also characterized using closure operations. It is shown that the image of a semi-prime closure operation on an IL-algebra is isomorphic to a quotient IL-algebra. Some properties of the quotients induced by closure operations on an IL-algebra are explored.

There are several ways to convert a closure or interior operation to a different operation that has particular desirable properties. In this paper, we axiomatize three ways to do so, drawing on disparate examples from the literature, including tight closure, basically full closure, and various versions of integral closure. In doing so, we explore several such desirable properties, including hereditary , residual , and cofunctorial , and see how they interact with other properties such as the finitistic property. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study connections between ideals of the semisimple EMV-algebra M and filters on some nonempty set Ω. We show that there is a bijection between the set of all closed ideals of M and the set of all filters on Ω. We prove that the topological space of all closed prime ideals of M and the topological space of all weak ultrafilters on Ω are homeomorphic.

As a basic notion in algebra, closure operations have been successfully applied to many fields of computer science. In this paper we study dense family in the closure operations. In particular, we prove some families to be dense in any closure operation, in which the greatest and smallest dense families, including the collection of the whole closed sets and the minimal generator of the closed sets, are also pointed out. More important, a necessary and sufficient condition for an arbitrary family to be dense is provided in our paper. Then we use these dense families to characterize minimal keys of the closure operation under the viewpoint of transversal hypergraphs and construct an algorithm for determining the minimal keys of a closure operation.

Exploiting the interior-closure duality developed by Epstein and R.G. [10] , we show that for the class of Matlis dualizable modules M over a Noetherian local ring, when cl is a Nakayama closure and i its dual interior, there is a duality between cl-reductions and i-expansions that leads to a duality between the cl-core of modules in M and the i-hull of modules in M ∨. We further show that many algebra and module closures and interiors are Nakayama and describe a method to compute the interior of ideals using closures and colons. We use our methods to give a unified proof of the equivalence of F-rationality with F-regularity, and of F-injectivity with F-purity, in the complete Gorenstein local case. Additionally, we give a new characterization of the finitistic tight closure test ideal in terms of maps from R 1 / p e . Moreover, we show that the liftable integral spread of a module exists. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the connection between ideals of semisimple MV-algebras and set-theoretic filters. In particular, we obtain that there is a bijection between closed ideals (with respect to a specified closure operation) and all filters on some associated set X. Finally, we establish that the space of closed prime ideals is homeomorphic to a well-described space having the Stone-Čech compactification βX as a subspace. [ABSTRACT FROM AUTHOR]

We construct some types of universal closure operations induced by certain collection of morphisms. For this purpose, we use Lawvere-Tierney topologies and universal closure operations that correspond to each other to establish the equivalent conditions over the collection of morphisms. In this way we use multiple sieves instead of principal sieves for constructing results. Examples are also given to illustrate the established results. [ABSTRACT FROM AUTHOR]

A nucleus on a meet-semilattice A is a closure operation that preserves binary meets. The nuclei form a semilattice N A that is isomorphic to the system N A of all nuclear ranges, ordered by dual inclusion. The nuclear ranges are those closure ranges which are total subalgebras (l-ideals). Nuclei have been studied intensively in the case of complete Heyting algebras. We extend, as far as possible, results on nuclei and their ranges to the non-complete setting of implicative semilattices (whose unary meet translations have adjoints). A central tool are so-called r-morphisms, that is, residuated semilattice homomorphisms, and their adjoints, the l-morphisms. Such morphisms transport nuclear ranges and preserve implicativity. Certain completeness properties are necessary and sufficient for the existence of a least nucleus above a prenucleus or of a greatest nucleus below a weak nucleus. As in pointfree topology, of great importance for structural investigations are three specific kinds of l-ideals, called basic open, boolean and basic closed. [ABSTRACT FROM AUTHOR]

Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempotent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the prime k-spectrum of an idempotent semiring. In fact, we enrich Hochster's theorem by constructing a subcategory of idempotent semirings which is antiequivalent to the category of spectral spaces. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the space of valuations and the space of prime congruences on an idempotent semiring are spectral, and there is a natural bijection of sets between the two; this shows a stark difference between rings and idempotent semirings. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of closure operations for semirings, and provide several examples. In particular, we introduce an integral closure operation and a Frobenius closure operation for idempotent semirings. [ABSTRACT FROM AUTHOR]

The extended plus (epf) closure and the rank 1 (r1f) closure are two closure operations introduced by Raymond C. Heitmann for rings of mixed characteristic. Recently, he and Linquan Ma proved that the epf closure satisfies the usual colon-capturing property under mild conditions. In this paper, we extend their result and prove that the epf closure satisfies what we call the p -colon-capturing property. Based on that, we define a new closure notion called "weak epf closure," and prove that it satisfies the generalized colon-capturing property and some other colon-capturing properties. This gives a new proof of the existence of big Cohen-Macaulay algebras in the mixed characteristic case. We also show that any module-finite extension of a complete local domain is epf -phantom, which generalizes a result of Mel Hochster and Craig Huneke about "phantom extensions." Finally, we prove some related results in characteristic p. [ABSTRACT FROM AUTHOR]

Two different kinds of multiple-conclusion consequence relations taken from Shoesmith and Smiley (Multiple-conclusion logic, Cambridge University Press, Cambridge, 1978) and Galatos and Tsinakis (J Symb Logic 74:780–810, 2009) or Nowak (Bull Sect Logic 46:219–232, 2017), called here disjunctive and conjunctive, respectively, defined on a formal language, are considered. They are transferred into a bounded lattice and a complete lattice, respectively. The properties of such abstract consequence relations are presented. [ABSTRACT FROM AUTHOR]

We present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. For this purpose, our main tools are finite type closure operations and continuous valuations on monoids which we introduce in this work. In the process, we make a detailed study of different closure operations on monoids. We prove that the collection of continuous valuations on a topological monoid with topology determined by any finitely generated ideal is a spectral space. [ABSTRACT FROM AUTHOR]

The concept of multiple-conclusion consequence relation from [8] and [7] is con- sidered. The closure operation C assigning to any binary relation r (defined on the power set of a set of all formulas of a given language) the least multiple- conclusion consequence relation containing r, is defined on the grounds of a nat- ural Galois connection. It is shown that the very closure C is an isomorphism from the power set algebra of a simple binary relation to the Boolean algebra of all multiple-conclusion consequence relations. [ABSTRACT FROM AUTHOR]

In this article, we show that almost Cohen-Macaulay algebras are solid. Moreover, we seek for the conditions when (a) an almost Cohen-Macaulay algebra is a phantom extension and (b) when it maps into a balanced big Cohen-Macaulay module. [ABSTRACT FROM AUTHOR]

In the paper, tracing the traditional Hilbert-style syntactic account of logics, a syntactic characteristic of a closure operation defined on a complete lattice follows. The approach is based on observation that the role of rule of inference for a given consequence operation may be played by an ordinary binary relation on the complete lattice on which the closure operation is defined. [ABSTRACT FROM AUTHOR]

We develop a duality for operations on nested pairs of modules that generalizes the duality between absolute interior operations and residual closure operations from [5] , extending our previous results to the expanded context. We apply this duality in particular to integral and basically full closures and their respective cores to obtain integral and basically empty interiors and their respective hulls. We also dualize some of the known formulas for the core of an ideal to obtain formulas for the hull of a submodule of the injective hull of the residue field. The article concludes with illustrative examples in a numerical semigroup ring. [ABSTRACT FROM AUTHOR]

We prove that the space of radical ideals of a ring R , endowed with the hull-kernel topology, is a spectral space, and that it is canonically homeomorphic to the space of the non-empty Zariski closed subspaces of Spec ( R ) , endowed with a Zariski-like topology. [ABSTRACT FROM AUTHOR]

In [1] , the dagger closure is extended over finitely generated modules over Noetherian local domain ( R , m ) and it is proved to be a Dietz closure. In this short note we show that it also satisfies the ‘Algebra axiom’ of [9] and this leads to the following result of this paper: For a complete Noetherian local domain, if it is contained in an almost Cohen–Macaulay domain, then there exists a balanced big Cohen–Macaulay algebra over it. [ABSTRACT FROM AUTHOR]

The theory of star and semistar operations has been developed mostly over domains. Recently, Epstein defined star and semistar operations over more general commutative rings. We will compare some of the known results over domains with the non-domain setting. We conclude with a classification of the standard closures on the nodal curve. [ABSTRACT FROM AUTHOR]

Let $$G$$ be a $$K_{1,3}$$ -free graph. A circuit of $$G$$ is essential if it contains a non-locally connected vertex $$v$$ and passes through both components of $$N(v)$$ . The essential girth of $$G$$ , denoted by $$g_e(G)$$ , is the length of a shortest essential circuit. It can be seen easily that, by Ryjáček closure operation, the essential girth of $$G$$ is closely related to the girth of $$H$$ where $$H$$ is the Ryjáček closure of $$G$$ and is a line graph. A generalized net, denoted by $$N_{i_1,i_2,i_3}$$ , is a graph obtained from a triangle $$C_3$$ and three disjoint paths $$P_{i_\mu +1}$$ ( $$\mu =1,2,3$$ ), by identifying each vertex $$v_\mu $$ of $$C_3=v_1v_2v_3v_1$$ with an end vertex of the path $$P_{i_\mu +1}$$ , for every $$\mu = 1,2,3$$ . In this paper, we prove that every $$2$$ -connected $$\{ K_{1,3}, N_{1,1,g_e(G)-4}\}$$ -free (and $$\{ K_{1,3}, N_{1,0,g_e(G)-3}\}$$ -free) graph $$G$$ contains a Hamilton circuit. With the additional parameter $$g_e$$ , these results extend some earlier theorems about Hamilton circuits in $$\{ K_{1,3}, N_{a,b,c}\}$$ -free graphs (for some small integers $$a, b$$ and $$c$$ ). [ABSTRACT FROM AUTHOR]

In this note we present some remarks on big Cohen-Macaulay algebras and almost Cohen-Macaulay algebras via closure operations on the ideals and the modules. Our methods for doing these are inspired by the notion of the tight closure and the dagger closure and by the ideas of Northcott on dropping of the Noetherian assumption of certain homological properties. [ABSTRACT FROM AUTHOR]

Abstract: Forming closures of subsets of a set X is a technique that plays an important role in many scientific disciplines and there are many cryptomorphic mathematical structures that describe closures and their construction. One of them was introduced by Aumann in the year 1970 under the name contact relation. Using relation algebra, we generalize Aumann’s notion of a contact relation between X and its powerset and that of a closure operation on from powersets to general membership relations and their induced partial orders. We also investigate the relationship between contacts and closures in this general setting and present some applications. In particular, we investigate the connections between contacts, closures and topologies and use contacts to establish a one-to-one correspondence between the column intersections space and the row intersections space of arbitrary relations. [Copyright &y& Elsevier]