The prime spectrum of $$\ell $$-groups and MV-algebras

Lattices, spectral spaces, and closure operations on idempotent semirings

The theory of bounded, distributive lattices provides the appropriate language for describing directionality and asymptotics in dynamical systems. For bounded, distributive lattices the general notion of ‘set-difference’ taking values in a semilattice is introduced, and is called the Conley form. The Conley form is used to build concrete, set-theoretic models of spectral spaces, or Priestley spaces, of bounded, distributive lattices and their finite coarsenings. Such representations formulate and compute order-theoretic models of dynamical systems such as Morse decompositions and Morse representations, which may be regarded as global characteristics of a dynamical system.

Localic Priestley duality

In this paper we study prime spectra of commutative BCK-algebras. We give a new construction for commutative BCK-algebras using rooted trees, and determine both the ideal lattice and prime ideal lattice of such algebras. We prove that the spectrum of any commutative BCK-algebra is a locally compact generalized spectral space which is compact if and only if the algebra is finitely generated as an ideal. Further, we show that if a commutative BCK-algebra is involutory, then its spectrum is a Priestley space. Finally, we consider the functorial properties of the spectrum and define a functor from the category of commutative BCK-algebras to the category of distributive lattices with zero. We give a partial answer to the question: what distributive lattices lie in the image of this functor?

Let $M$ be a lattice module over a $C$-lattice $L$. A proper element $P$ of $M$ is said to be classical prime if for $a ,b\in L$ and $X\in M, abX\leq P$ implies that $aX\leq P$ or $bX\leq P$. The set of all classical prime elements of $M$, $Spec^{cp}(M)$ is called as classical prime spectrum. In this article, we introduce and study a topology on $Spec^{cp}(M)$, called as Zariski-like topology of $M$. We investigate this topological space from the point of view of spectral spaces. We show that if $M$ has ascending chain condition on classical prime radical elements, then $Spec^{cp}(M)$ with the Zariski-like topology is a spectral space.

We introduce the classical Zariski topology on the prime spectrum of a semimodule and study this topological space from the viewpoint of spectral spaces. First, we prove that every irreducible closed subset has a generic point in the prime spectrum of a semimodule and that the prime spectrum of a semimodule is a spectral space if it is a Noetherian topological space. Then, we prove that the prime spectrum of a semimodule is a Noetherian topological space if and only if the semimodule satisfies the ascending chain condition on intersection of prime subsemimodules. Also, we show that each subsemimodule in a semimodule satisfying the ascending chain condition on intersection of prime subsemimodules has at most finitely many minimal prime divisors. As consequences, we give positive and complete answers to the Conjecture in [Int. Electron. J. Algebra 4 (2008) 104] and Question 3.5 in [Int. Electron. J. Algebra 4 (2008) 131].

In the present survey paper, we present several new classes of Hochster’s spectral spaces “occurring in nature,” actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The general setting is the space of the semistar operations (of finite type), endowed with a Zariski-like topology, which turns out to be a natural topological extension of the space of the overrings of an integral domain, endowed with a topology introduced by Zariski. One of the key tool is a recent characterization of spectral spaces, based on the ultrafilter topology, given in Finocchiaro, Commun Algebra, 42:1496–1508, 2014, [15]. Several applications are also discussed.

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces – the duals of Boolean algebras – and show that they are exactly the bitopological duals of bounded distributive lattices. The categoryPStoneof pairwise Stone spaces is isomorphic to the categorySpecof spectral spaces and to the categoryPriesof Priestley spaces. In fact, the isomorphism ofSpecandPriesis most naturally seen throughPStoneby first establishing thatPriesis isomorphic toPStone, and then showing thatPStoneis isomorphic toSpec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia's duality.

In memory of my father. Let X be the prime spectrum of a ring. In Fontana and Loper [5] the authors define a topology on X by using ultrafilters and show that this topology is precisely the constructible topology. In this paper we generalize the construction given in Fontana and Loper [5] and, starting from a set X and a collection of subsets ℱ of X, we define by using ultrafilters a topology on X in which ℱ is a collection of clopen sets. We use this construction for giving a new characterization of spectral spaces and several examples of spectral spaces.

An MV-space is a topological space X such that there exists an MV-algebra A whose prime spectrum Spec A is homeomorphic to X. The characterization of the MV-spaces is an important open problem. We shall prove that any projective limit of MV-spaces in the category of spectral spaces is an MV-space. In this way, we obtain new classes of MV-spaces related to some preservation properties of the Belluce functor.