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A new class of the Ito integral for Brownian motion is defined and studied in the framework of Riesz spaces. The stochastic process with respect to this stochastic integral is non-adapted and it is...

In this note we prove that positive multi-valued operators on Riesz spaces which are everywhere defined are always continuous. Further, we prove that the sum of two positive relations having the sa...

We say that a Riesz space E E has the horizontal Egorov property if for every net ( f α ) (f_\alpha ) in E E , order convergent to f ∈ E f \in E with | f α | + | f | ≤ e ∈ E + |f_\alpha | + |f| \le e \in E^+ for all α \alpha , there exists a net ( e β ) (e_\beta ) of fragments of e e laterally convergent to e e such that for every β \beta , the net ( | f − f α | ∧ e β ) α \bigl (|f - f_\alpha | \wedge e_\beta \bigr )_\alpha e e -uniformly tends to zero. Our main result asserts that every Dedekind complete Riesz space which satisfies the weak distributive law possesses the horizontal Egorov property. A Riesz space E E is said to satisfy the weak distributive law if for every e ∈ E + ∖ { 0 } e \in E^+ \setminus \{0\} the Boolean algebra F e \mathfrak {F}_e of fragments of e e satisfies the weak distributive law; that is, whenever ( Π n ) n ∈ N (\Pi _n)_{n \in \mathbb N} is a sequence of partitions of F e \mathfrak {F}_e , there is a partition Π \Pi of F e \mathfrak {F}_e such that every element of Π \Pi is finitely covered by each of Π n \Pi _n (e.g., every measurable Boolean algebra is so). Using a new technical tool, we show that for every net ( f α ) (f_\alpha ) order convergent to f f in a Riesz space with the horizontal Egorov property there are a horizontally vanishing net ( v β ) (v_\beta ) and a net ( u α , β ) ( α , β ) ∈ A × B (u_{\alpha , \beta })_{(\alpha , \beta ) \in A \times B} , which uniformly tends to zero for every fixed β \beta such that | f − f α | ≤ u α , β + v β |f - f_\alpha | \le u_{\alpha , \beta } + v_\beta for all α , β \alpha , \beta .

The Cesàro operators are investigated on the class -valued sequence spaces , and with is a Riesz space. Besides, we also carry out that are order bounded operators.

Following a recent idea by Ball, we introduce the notion of strongly truncated Riesz space with a suitable spectrum. We prove that, under an extra Archimedean type condition, any strongly truncated Riesz space is isomorphic to a uniformly dense Riesz subspace of a C 0 X -space. This turns out to be a direct generalization of the classical Kakutani Representation Theorem on Archimedean Riesz spaces with strong unit. Another representation theorem on normed Riesz spaces, due to Fremlin, will be obtained as a consequence of our main result.

A sequence (a(n)) in a Riesz space E is called uo-convergent (or unbounded order convergent) to a is an element of E if vertical bar a(n) - a vertical bar Lambda u -> 0 for all u is an element of E+ and unbounded order Cauchy (uo-Cauchy) if vertical bar a(n) - a(n+p)vertical bar is uo-convergent to 0. In the first part of this study we define u(d,E)-convergence (or unbounded vectorial convergence) in vector metric spaces, which is more general than usual metric spaces, and examine relations between unbounded order convergence, unbounded vectorial convergence, vectorial convergence and order convergence. In the last part we construct the unbounded Cauchy completion of vector metric spaces by the motivation of the fact that every metric space has Cauchy completion. In this way, we have obtained a more general completion of vector metric spaces.

The paper contains a systematic study of the lateral partial order $$\sqsubseteq $$ ⊑ in a Riesz space (the relation $$x \sqsubseteq y$$ x ⊑ y means that x is a fragment of y) with applications to nonlinear analysis of Riesz spaces. We introduce and study lateral fields, lateral ideals, lateral bands and consistent subsets and show the importance of these notions to the theory of orthogonally additive operators, like ideals and bands are important for linear operators. We prove the existence of a lateral band projection, provide an elegant formula for it and prove some properties of this orthogonally additive operator. One of our main results (Theorem 7.5) asserts that, if D is a lateral field in a Riesz space E with the intersection property, X a vector space and $$T_0:D\rightarrow X$$ T 0 : D → X an orthogonally additive operator, then there exists an orthogonally additive extension $$T:E\rightarrow X$$ T : E → X of $$T_0$$ T 0 . The intersection property of E means that every two-point subset of E has an infimum with respect to the lateral order. In particular, the principal projection property implies the intersection property.

We show that any continuous orthosymmetric multilinear map from an Archimedean Riesz space into a Hausdorff topological vector space is symmetric. Then, we establish a linear representation for continuous orthogonally additive homogeneous polynomials. This representation will be used to introduce and describe a new class of homogeneous polynomials, namely that of polyorthomorphisms. In particular, we prove that, for a Riesz space E and a natural number $$n\ge 2$$ , the space $${{\mathcal{P}}}_{orth}(^nE)$$ of all polyorthomorphisms of degree n is a Riesz space.

In this paper we introduce a new concept for Riesz almost lacunary $\Gamma^{3}$ sequence spaces strong $P-$ convergent to zero with respect to an Musielak-Orlicz function and examine some properties of the resulting sequence spaces. We also introduce and study statistical convergence of Riesz almost lacunary $\Gamma^{3}$ sequence spaces and also some inclusion theorems are discussed.

In this paper, we define ordered vectorial quasi contractions. We show that ordered quasi contractions are ordered vectorial quasi contractions, but the reverse is not true. We also define ordered vectorial almost contractions and present fixed point theorems for this type of contractions. Hence, we disclose many results in the literature. With the help of examples, we illustrate the relationship between these two types of contractions and some others in the literature.

Every Dedekind complete Riesz space X has a unique sup-completion X^{s}, which is a Dedekind complete lattice cone. This paper aims to present a systematic study this cone by extending several known results to general setting, proving new results and, in particular, introducing for elements of X^{s} finite and infinite parts. This enuables us to get a satisfactory abstract formulation of some classical results in the setting of Riesz spaces. We prove, in pareticular, a Riesz space version of Borel-Cantelli Lemma and present some applications to it.29*, 29 pages

We explore inverses of disjointness preserving bijections in infinite dimensional normed pre-Riesz spaces by several methods. As in the case of Banach lattices, our aim is to show that such inverses are disjointness preserving. One method is extension of the operator to the Riesz completion, which works under suitable denseness and continuity conditions. Another method involves a condition on principle bands. Examples illustrate the differences to the Riesz space theory.

We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space E is an ideal in $$E^{\sim \sim }$$ and F is a Dedekind complete Riesz space then for each ideal A of E, T is strongly order bounded on A if and only if $$T_A$$ is strongly order bounded. We show that the class of strongly order bounded operators satisfies the domination problem. On the other hand, we present two ways for decomposition of strongly order bounded operators, and we give some of their properties. Also, it is shown that E has order continuous norm or F has the b-property whenever each pre-regular operator form E into F is order bounded.

We establish an interpolation result for sublinear operators which map into Riesz spaces. This result applies to all interpolation functors including the real interpolation and the complex interpolation. One component of our proof which may be of independent interest is the perhaps already known fact that the generalized versions of the Hahn-Banach theorem due to L. V. Kantorovich and M. M. Day also hold for complex vector spaces.

We prove, in particular, that if E is a Dedekind complete atomless Riesz space and X is a Banach space then the sum of a narrow and a C-compact laterally continuous orthogonally additive operators from E to X is narrow. This generalizes in several directions known results on narrowness of the sum of a narrow and a compact operators for the settings of linear and orthogonally additive operators defined on Kothe function spaces and Riesz spaces.

In this paper we formulate and proof Girsanov’s theorem in vector lattices. To reach this goal, we develop the theory of cross-variation processes, derive the cross-variation formula and the Kunita–Watanabe inequality. Also needed and derived are properties of exponential processes, Ito’s rule for multi-dimensional processes and the integration by parts formula for martingales. After proving Girsanov’s theorem for the one-dimensional case, we also discuss the multi-dimensional case.

An ideal $I$ is a family of positive integers $\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. In this paper, we introduce the notions of ideal versions of weighted lacunary statistical $\tau$-convergence, statistical $\tau$-Cauchy, weighted lacunary $\tau$-boundedness of sequences in locally solid Riesz spaces endowed with the topology $\tau$. We also prove some topological results related to these concepts in locally solid Riesz space.

We consider the notion of generalized density, namely, the natural density of weight 1 recently introduced in [4] and primarily study some sufficient and almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Also we consider similar types of results for the case of generalized statistically bounded sequence. Some results are further obtained in a more general form by using the notion of ideals. The entire investigation is performed in the setting of Riesz spaces extending the recent results in [13].

This paper deals with a study on classes of non linear operators. Let $SL(X,Y)$ be the set of all sublinear operators between two Riesz spaces $X$ and $Y$. It is a convex cone of the space $H(X,Y)$ of all positively homogeneous operators. In this paper we study some spaces generated by this cone, therefore we study several properties, which are well known in the theory of Riesz spaces, like order continuity, order boundedness etc. Finally, we try to generalise the concept of adjoint operator. First, by using the analytic form of Hahn-Banach theorem, we adapt the notion of adjoint operator to the category of positively homogeneous operators. Then we apply it to the class of operators generated by the sublinear operators.

In this paper, we prove some theorems and a common fixed point theorem in vector metric spaces for generalized contraction mappings and give an example.

We show that the paper in which P.J. Daniell introduced his well-known integral, used modern Riesz space techniques to derive the properties of the integral and to prove a fundamental decomposition result for the integral. The latter result was proved a decade later by F. Riesz and was considered to be the origin of Riesz space theory. After a survey of Daniell’s paper, we generalize P.E. Protter’s version of the Lp-valued (0 ≤ p ≤∞) Daniell integral to a vector lattice-valued Daniell integral, following closely Daniell’s original method. A.C.M van Rooij and W.B. van Zuijlen also introduced integrals for functions with values in a partially ordered vector space a more general setting than the one we use.

The ergodic theorems of Hopf, Wiener and Birkhoff were extended to the context of Riesz spaces with a weak order unit and conditional expectation operator by Kuo, Labuschagne and Watson in [Ergodic Theory and the Strong Law of Large Numbers on Riesz Spaces. J Math Anal Appl 325 (2007), 422–437]. However, the precise concept of what constitutes ergodicity in Riesz spaces was not considered. In this short paper we fill in this omission and give some explanations of the choices made. In addition, we consider the interplay between mixing and ergodicity in the Riesz space setting.

A mixed lattice group is a generalization of a lattice ordered group. The theory of mixed lattice semigroups dates back to the 1970s, but the corresponding theory for groups and vector spaces has been relatively unexplored. In this paper we investigate the basic structure of mixed lattice groups, and study how some of the fundamental concepts in Riesz spaces and lattice ordered groups, such as the absolute value and other related ideas, can be extended to mixed lattice groups and mixed lattice vector spaces. We also investigate ideals and study the properties of mixed lattice group homomorphisms and quotient groups. Most of the results in this paper have their analogues in the theory of Riesz spaces.

We consider radially symmetric m-subharmonic functions on the unit ball. We study their convexity and their relation with a solution of the complex Hessian equations. Furthermore, we consider the ordered vector space of \(\delta \) radially symmetric m-subharmonic functions which is a Riesz space. Moreover, we shall define a space of m-subharmonic functions along with the Mabuchi metric. We also introduce a geodesic between two points in this space and give an equivalent condition when a curve on the space is geodesics.

We study multiorthomorphisms on Riesz spaces. As an application, we introduce and we describe a new class of order bounded orthogonally additive homogeneous polynomials.

The full lattice convergence on a locally solid Riesz space is an abstraction of the topological, order, and relatively uniform convergences. We investigate four modifications of a full convergence $\mathbb{c}$ on a Riesz space. The first one produces a sequential convergence $\mathbb{sc}$. The second makes an absolute $\mathbb{c}$-convergence and generalizes the absolute weak convergence. The third modification makes an unbounded $\mathbb{c}$-convergence and generalizes various unbounded convergences recently studied in the literature. The last one is applicable whenever $\mathbb{c}$ is a full convergence on a commutative $l$-algebra and produces the multiplicative modification $\mathbb{mc}$ of $\mathbb{c}$. We study general properties of full lattice convergence with emphasis on universally complete Riesz spaces and on Archimedean $f$-algebras. The technique and results in this paper unify and extend those which were developed and obtained in recent literature on unbounded convergences.

Aydin, Abdullah/0000-0002-0769-5752 The multiplicative convergence on Riesz algebras introduced and investigated with respect to norm and order convergences. If is a Riesz space E and is a Riesz algebra then the vector norm mu: X -> E+ can be considered. Then (X, mu, E) is called algebraic lattice normed spaces. A net (x(alpha))(alpha is an element of A) in an (X, mu, E) is said to be multiplicative mu-convergent to x is an element of X if mu(x(alpha) - x).u ->(o) 0 holds for all u is an element of E+. In this paper, the general properties of this convergence are studied.

In this paper, using the concept of unbounded order convergence in Riesz spaces, we define new classes of operators, named unbounded order continuous (uo-continuous, for short) and boundedly unbounded order continuous operators. We give some conditions under which uo-continuity will be equivalent to order continuity of some operators on Riesz spaces. We show that the collection of all uo-continuous linear functionals on a Riesz space E is a band of $$E^\sim $$ .

In this paper, the extent to which the Burkholder Inequalities in classical Stochastic Analysis can be generalized to the new Theory of Stochastic Analysis in Riesz spaces.

This paper aims to study fuzzy order bounded linear operators between two fuzzy Riesz spaces. Two lattice operations are defined to make the set of all bounded linear operators as a fuzzy Riesz space when the codomain is fuzzy Dedekind complete. As a special case, separation property in fuzzy order dual is studied. Furthermore, we studied fuzzy norms compatible with fuzzy ordering (fuzzy norm Riesz space) and discussed the relation between the fuzzy order dual and topological dual of a locally convex solid fuzzy Riesz space.

The notion of distribution function with respect to a conditional expectation is defined and studied in the framework of Riesz spaces.Keywords: Riesz space, conditional expectation operator, distribution function, Daniell integral

Various properties of tropical convex sets of the positive cone of a Riesz space are established. Under suitable, but standard assumptions, we show that they have fixed points properties, for single valued and multivalued maps, analogous to those associated to the usual convex sets in topological vector spaces. Selection theorems and extension theorems similar to those of Michael and Dugundji also hold and, in the appropriate setting, tropical convex sets are absolute extensors for the class of metric spaces and absolute retracts. We also give some applications to standard problems of mathematical economics: Nash equilibria, existence of elements for non transitive preference relations and existence of equilibria for abstract economies in Riesz spaces under tropical convexity assumptions. This work can be conceived as a contribution to infinite dimensional idempotent analysis.

In the theory of operators on a Riesz space (vector lattice), an important result states that the Riesz homomorphisms (lattice homomorphisms) on C(X) are exactly the weighted composition operators. We extend this result to Riesz* homomorphisms on order dense subspaces of C(X). On those subspace we consider and compare various classes of operators that extend the notion of a Riesz homomorphism. Furthermore, using the weighted composition structure of Riesz* homomorphisms we obtain several results concerning bijective Riesz* homomorphisms. In particular, we characterize the automorphism group for order dense subspaces of C(X). Lastly, we develop a similar theory for Riesz* homomorphisms on subspace of \(C_0(X)\), for a locally compact Hausdorff space X, and apply it to smooth manifolds and Sobolev spaces.

In this paper, we introduce an order ϕ-function on a Riesz space. Further, we construct a Riesz- Valued Sequence spaces using the ϕ-function and obtain the condition to characterize these spaces.

We introduce and investigate the Henstock-Kurzweil (HK) integral for Riesz-space-valued functions on time scales. Some basic properties of the HK delta integral for Riesz-space-valued functions are proved. Further, we prove uniform and monotone convergence theorems., Comment: This is a preprint of a paper whose final and definite form is with 'J. Nonlinear Sci. Appl.', ISSN 2008-1898 (Print) ISSN 2008-1901 (Online). Article Submitted 17-Jan-2017; Revised 17-Apr-2017; Accepted for publication 19-Apr-2017. See [http://www.tjnsa.com]

We consider relatively uniform convergence of nets in a partially ordered vector space. We give an example of a set V in a space X where adding the limits of nets in V that converge in X does not produce the closure of V in X. The closure of a set can be constructed by adding limits if that process is repeated by transfinite induction. We also consider closed sets and complete spaces and show that they coincide with sequentially closed sets and sequentially complete spaces, respectively.

We investigate when the regular operators from one Banach lattice into another form a Riesz space. We give complete results when the domain is either separable or has an order continuous norm. In these two settings, at least, the lattice operators are given by the Riesz-Kantorovich formulae, in contrast with Elliott’s negative result for the general setting.

We prove that ifEis a Dedekind complete atomless Riesz space andXis a Banach space, then the sum of two laterally continuous orthogonally additive operators fromEtoX, one of which is strictly narrow and the other one is hereditarily strictly narrow with finite variation (in particular, has finite rank), is strictly narrow. Similar results were previously obtained for narrow operators by different authors; however, no theorem of the kind was known for strictly narrow operators.

A sequence $(x_n)$ in a locally solid Riesz space $(E,\tau)$ is said to be statistically unbounded $\tau$-convergent to $x\in E$ if, for every zero neighborhood $U$, $\frac{1}{n}\big\lvert\{k\leq n:\lvert x_k-x\rvert\wedge u\notin U\}\big\rvert\to 0$ as $n\to\infty$. In this paper, we introduce this concept and give the notions $st$-$u_\tau$-closed subset, $st$-$u_\tau$-Cauchy sequence, $st$-$u_\tau$-continuous and $st$-$u_\tau$-complete locally solid vector lattice. Also, we give some relations between the order convergence and the $st$-$u_\tau$-convergence., Comment: 10

The aim of this article is to describe a class of -algebras that allows to treat well-behaved algebras of unbounded operators independently of a representation. To this end, Archimedean ordered *-algebras (*-algebras whose real linear subspace of Hermitian elements are an Archimedeanordered vector space with rather weak compatibilities with the algebraic structure) are examined. The order induces a translation-invariant uniform metric which comes from a C*-norm in the bounded case. It will then be shown that uniformly complete Archimedean ordered -algebras have good order properties (like existence of infima, suprema or absolute values) if and only if they have good algebraic properties (like existence of inverses or square roots). This suggests the definition of Su*-algebras as uniformly complete Archimedean ordered *-algebras which have allthese equivalent properties. All methods used are completely elementary and do not require any representation theory and not even any assumptions of boundedness, so Su*-algebras generalize some important properties of C*-algebras to algebras of unbounded operators. Similarly, they generalize some properties of Φ-algebras (certain lattice-ordered commutative real algebras) to non-commutative ordered *-algebras. As an example, Su*-algebras of unbounded operators on a Hilbert space are constructed. They arise e.g. as *-algebras of symmetries of a self-adjoint (not necessarily bounded) Hamiltonian operator of a quantum mechanical system., 26 pages, info:eu-repo/semantics/published

This paper approaches the construction of the universal completion of the Riesz space C ( L ) of continuous real functions on a completely regular frame L in two different ways. Firstly as the space of continuous real functions on the Booleanization of L. Secondly as the space of nearly finite Hausdorff continuous functions on L. The former has no counterpart in the classical theory, as the Booleanization of a spatial frame is not spatial in general, and it offers a lucid way of representing the universal completion as a space of continuous real functions. As a corollary we obtain that C ( L ) and C ( M ) have isomorphic universal completions if and only if the Booleanization of L and M are isomorphic and we characterize frames L such that C ( L ) is universally complete as almost Boolean frames. The application of this last result to the classical case C ( X ) of the space of continuous real functions on a topological space X characterizes those spaces X for which C ( X ) is universally complete. Finally, we present a pointfree version of the Maeda-Ogasawara-Vulikh representation theorem and use it to represent the universal completion of an Archimedean Riesz space with weak unit as a space of continuous real functions on a Boolean frame.

In this paper we introduce an independence measure between a vector f in a Riesz space E and a conditional expectation operator T on E. We show that this measure satisfies some postulates corresponding to these formulated by Renyi (1959) for dependence measure between random variables. The resulting dependence measure in E is also investigated. Some examples of application are given.

In the present paper we introduce the notion of a positive linear relation and we investigate the class of such operators. As well as placing the theory of positive operators in a natural setting, this structure seems to be interesting for the study of abstract boundary value problems.

In this paper we have introduced an order relation on convergent double sequences and have constructed an ordered vector space, Riesz space, order complete vector space in case of double sequences. We have verified the Archimedean property.

In this paper we generalize the Hajek-Renyi-Chow maximal inequality for submartingales to L p type Riesz spaces with conditional expectation operators. As applications we obtain a submartingale convergence theorem and a strong law of large numbers in Riesz spaces. Along the way we develop a Riesz space variant of the Clarkson's inequality for 1 ≤ p ≤ 2 .

For any real number $p\in [1,+\infty)$, we characterise the operations $\mathbb{R}^I \to \mathbb{R}$ that preserve $p$-integrability, i.e., the operations under which, for every measure $\mu$, the set $\mathcal{L}^p(\mu)$ is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind $\sigma$-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that $\mathbb{R}$ generates this variety. From this, we exhibit a concrete model of the free Dedekind $\sigma$-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve $p$-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind $\sigma$-complete Riesz spaces with weak unit, $\mathbb{R}$ is proved to generate this variety, and a concrete model of the free Dedekind $\sigma$-complete Riesz spaces with weak unit is exhibited., Comment: Changed the definition of "conditionally partitionable measure space", results unchanged; minor changes

Frolik's theorem says that a homeomorphism from a certain kind of topological space to itself decomposes the space into the clopen set of xed points together with three clopen sets, each of whose images is disjoint from the original set. Stone's theorem translates this result to a corresponding theorem about the Riesz space of continuous functions on the topological space. We prove a theorem analogous to that for Riesz spaces in the much more general setting of (possibly noncommutative) lattice-ordered groups and group-endomorphisms. The groups to which our result applies satisfy a weak condition, introduced by Abramovich and Kitover, on the polars; the images of our endomorphisms have a kind of order- density on their polars; the double polars of the images are cardinal summands; and the endomorphisms themselves are disjointness-preserving in both directions. We explain how to extend our result to larger groups to which it does not apply, and, to give additional insight, we provide many examples.Mathematics Subject Classication (2010): 06F15, 46A40, 06F20, 06E30, 03G05, 54C05, 47B60.Key words: Lattice-ordered group, vector lattice, Frolks theorem, disjointnesspreserving, decomposition, polar, disjoint complement.

This is the first part of a series of two abstract, the second one being by Daniel McNeill.If X is any topological space, its collection of opens sets O(X) is a complete distributive lattice and also a Heyting algebra. When X is equipped with a distinguished basis D(X) for its topology, closed under finite meets and joins, one can investigate situations where D(X) is also a Heyting subalgebra of O(X).Recall that X is a spectral space if it is compact and T0, its collection D(X) of compact open subsets forms a basis which is closed under finite intersections and unions, and X is sober. By Stone duality, spectral spaces are precisely the spaces arising as sets of prime ideals of some distributive lattice, topologised with the Stone or hull-kernel topology. Specifically, given such a spectral space X, its collection of compact open sets D(X) is (naturally isomorphic to) the distributive lattice dual to X under Stone duality.We are going to exhibit a significant class of such spaces for which D(X) is a Heyting subalgebra of O(X).We work with lattice-ordered Abelian groups and vector spaces. Using Mundici’s Gamma-functor the results can be rephrased in terms of MV-algebras, the algebraic semantics of Lukasiewicz infinite-valued propositional logic.Let (G,u) be a finitely presented vector lattice (or Q-vector lattice, or l-group) G equipped with a distinguished strong order unit u. It turns out that Spec(G,u), i.e. the the space of prime ideals of (G,u) topologised with the hull-kernel topology, is a compact spectral space. Our first main result states that the collection D(Spec(G,u)) of compact open subsets of Spec(G,u) is a Heyting subalgebra of the Heyting algebra of open subsets O(Spec(G,u)).As a consequence, we also prove that the subspace Min(G,u) of minimal prime ideals of G is a Boolean space, i.e. a compact Hausdorff space whose clopen sets form a basis for the topology.Further, for any fixed maximal ideal m of G, the set l(m) of prime ideals of G contained in m, equipped with the subspace topology, is a spectral space, and the subspace Min(l(m)) of l(m) is a Boolean space.