Lattice Homomorphism Research Articles

A note on the representation of lattice homomorphisms

A note on the representation of lattice homomorphisms

Simpler characterizations of total orderization invariant maps

Given a finite subset [Formula: see text] of a distributive lattice, its total orderization [Formula: see text] is a natural transformation of [Formula: see text] into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms.

Character rigidity for lattices and commensurators

abstract: We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, from certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.

Extremal structure of cones of positive homogeneous polynomials

It was proved by Anthony Wickstead that the cone of positive linear operators between Banach lattices E and F coincides with the strongly closed convex hull of the set of lattice homomorphisms from E to F if and only if the cone of positive elements on E is the weakly closed convex hull of the union of extremal rays of the cone of positive liner functionals on E. This note aims to show that this result extends to a wider class of operators, namely, orthogonally additive homogeneous polynomials acting between vector lattices.

Congruence filter pairs, equational filter pairs and adjoints

Abstract Filter pairs are a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is substitution invariant. Every substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattice homomorphism is a lattice of congruences for some quasivariety. We show that the class of logics admitting a presentation by such a filter pair is exactly the class of logics having an algebraic semantics. We study the properties of a certain Galois connection coming with such filter pairs. We give criteria for a congruence filter pair to present a logic in some classes of the Leibniz hierarchy by means of this Galois connection, and its interplay with the Leibniz operator.

Direct limits in categories of normed vector lattices and Banach lattices

After collecting a number of results on interval and almost interval preserving linear maps and vector lattice homomorphisms, we show that direct systems in various categories of normed vector lattices and Banach lattices have direct limits, and that these coincide with direct limits of the systems in naturally associated other categories. For those categories where the general constructions do not work to establish the existence of general direct limits, we describe the basic structure of those direct limits that do exist. A direct system in the category of Banach lattices and contractive almost interval preserving vector lattice homomorphisms has a direct limit. When the Banach lattices in the system all have order continuous norms, then so does the Banach lattice in a direct limit. This is used to show that a Banach function space over a locally compact Hausdorff space has an order continuous norm when the topologies on all compact subsets are metrisable and (the images of) the continuous compactly supported functions are dense.

Free complex Banach lattices

The construction of the free Banach lattice generated by a real Banach space is extended to the complex setting. It is shown that for every complex Banach space E there is a complex Banach lattice FBLC[E] containing a linear isometric copy of E and satisfying the following universal property: for every complex Banach lattice XC, every operator T:E→XC admits a unique lattice homomorphic extension Tˆ:FBLC[E]→XC with ‖Tˆ‖=‖T‖. The free complex Banach lattice FBLC[E] is shown to have analogous properties to those of its real counterpart. However, examples of non-isomorphic complex Banach spaces E and F can be given so that FBLC[E] and FBLC[F] are lattice isometric. The spectral theory of induced lattice homomorphisms on FBLC[E] is also explored.

A New Methodology on Rough Lattice Using Granular Concepts

Rough set theory has a vital role in the mathematical field of knowledge representation problems. Hence, a Rough algebraic structure is defined by Pawlak. Mathematics and Computer Science have many applications in the field of Lattice. The principle of the ordered set has been analyzed in logic programming for crypto-protocols. Iwinski extended an approach towards the lattice set with the rough set theory whereas an algebraic structure based on a rough lattice depends on indiscernibility relation which was established by Chakraborty. Granular means piecewise knowledge, grouping with similar elements. The universe set was partitioned by an indiscernibility relation to form a Granular. This structure was framed to describe the Rough set theory and to study its corresponding Rough approximation space. Analysis of the reduction of granular from the information table is based on object-oriented. An ordered pair of distributive lattices emphasize the congruence class to define its projection. This projection of distributive lattice is analyzed by a lemma defining that the largest and the smallest elements are trivial ordered sets of an index. A Rough approximation space was examined to incorporate with the upper approximation and analysis with various possibilities. The Cartesian product of the distributive lattice was investigated. A Lattice homomorphism was examined with an equivalence relation and its conditions. Hence the approximation space exists in its union and intersection in the upper approximation. The lower approximation in different subsets of the distributive lattice was studied. The generalized lower and upper approximations were established to verify some of the results and their properties.

Bipolar Fuzzy Sublattices and Ideals

In this article, we introduce and study the theory of bipolar fuzzy sublattices (BFLs) and bipolar fuzzy ideals (BFIs) of a lattice, and some interesting properties of these BFLs and BFIs are established. Moreover, we study the properties of BFIs under lattice homomorphisms and also an application of BFLs.

Varvec{varepsilon }-Shading Operator on Riesz Spaces and Order Continuity of Orthogonally Additive Operators

Given a Riesz space E and 0 < e in E, we introduce and study an order continuous orthogonally additive operator which is an varepsilon -approximation of the principal lateral band projection Q_e (the order discontinuous lattice homomorphism Q_e :E rightarrow E which assigns to any element x in E the maximal common fragment Q_e(x) of e and x). This gives a tool for constructing an order continuous orthogonally additive operator with given properties. Using it, we provide the first example of an order discontinuous orthogonally additive operator which is both uniformly-to-order continuous and horizontally-to-order continuous. Another result gives sufficient conditions on Riesz spaces E and F under which such an example does not exist. Our next main result asserts that, if E has the principal projection property and F is a Dedekind complete Riesz space then every order continuous regular orthogonally additive operator T :E rightarrow F has order continuous modulus |T|. Finally, we provide an example showing that the latter theorem is not true for E = C[0,1] and some Dedekind complete F. The above results answer two problems posed in a recent paper by O. Fotiy, I. Krasikova, M. Pliev and the second named author.

Categorical Dualities for Some Two Categories of Lattices: An Extended Abstract

The categorical dualities presented are: (first) for the category of bi-algebraic lattices that belong to the variety generated by the smallest non-modular lattice with complete (0,1)-lattice homomorphisms as morphisms, and (second) for the category of non-trivial (0,1)-lattices belonging to the same variety with (0,1)-lattice homomorphisms as morphisms. Although the two categories coincide on their finite objects, the presented dualities essentially differ mostly but not only by the fact that the duality for the second category uses topology. Using the presented dualities and some known in the literature results we prove that the Q-lattice of any non-trivial variety of (0,1)-lattices is either a 2-element chain or is uncountable and non-distributive.

Lattice embeddings in free Banach lattices over lattices

In this article we deal with the free Banach lattice generated by a lattice and its behavior with respect to subspaces. In general, any lattice embedding $i\colon \mathbb{L} \longrightarrow \mathbb{M}$ between two lattices $\mathbb{L} \subseteq \mathbb{M}$ induces a Banach lattice homomorphism $\hat \imath\colon FBL \langle \mathbb{L} \rangle \longrightarrow FBL \langle \mathbb{M}\rangle$ between the corresponding free Banach lattices. We show that this mapping $\hat \imath$ might not be an isometric embedding neither an isomorphic embedding. In order to provide sufficient conditions for $\hat \imath$ to be an isometric embedding we define the notion of locally complemented lattices and prove that, if $\mathbb L$ is locally complemented in $\mathbb M$, then $\hat \imath$ yields an isometric lattice embedding from $FBL\langle\mathbb L\rangle$ into $FBL\langle\mathbb M\rangle$. We provide a wide number of examples of locally complemented sublattices and, as an application, we obtain that every free Banach lattice generated by a lattice is lattice isomorphic to an AM-space or, equivalently, to a sublattice of a $C(K)$-space. Furthermore, we prove that $\hat \imath$ is an isomorphic embedding if and only if it is injective, which in turn is equivalent to the fact that every lattice homomorphism $x^*\colon \mathbb{L} \longrightarrow [-1,1]$ extends to a lattice homomorphism $\hat x^*\colon \mathbb{M} \longrightarrow [-1,1]$. Using this characterization we provide an example of lattices $\mathbb{L} \subseteq \mathbb{M}$ for which $\hat \imath$ is an isomorphic not isometric embedding.

On unbounded order continuous operators

Let $U$ and $V$ be two Archimedean Riesz spaces. An operator $S:U\rightarrow V$ is said to be unbounded order continuous ($uo$-continuous), if $r_{\alpha }\overset{uo}{\rightarrow }0$ in $U$ implies $Sr_{\alpha }\overset{uo}{% \rightarrow }0$ in $V$. In this paper, we give some properties of the $uo$% -continuous dual $U_{uo}^{\sim }$ of $U$. We show that a nonzero linear functional $f$ on $U$ is $uo$-continuous if and only if $f$ is a linear combination of finitely many order continuous lattice homomorphisms. The result allows us to characterize the $uo$-continuous dual $U_{uo}^{\sim }.$ In general, by giving an example that the $uo$-continuous dual $U_{uo}^{\sim }$ is not a band in $U^{\sim }$, we obtain the conditions for the $uo$% -continuous dual of a Banach lattice $U$ to be a band in $U^{\sim }$. Then, we examine the properties of $uo$-continuous operators. We show that $S$ is an order continuous operator if and only if $S$ is an unbounded order continuous operator when $S$ is a lattice homomorphism between two Riesz spaces $U$ and $V$. Finally, we proved that if an order bounded operator $S:U\rightarrow V$ between Archimedean Riesz space $U$ and atomic Dedekind complete Riesz space $V$ is $uo$-continuous, then $\left\vert S\right\vert $ is $uo$-continuous.

A Godefroy—Kalton principle for free Banach lattices

Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Banach lattices and lattice homomorphisms. Namely, a Banach lattice X satisfies the lattice-lifting property if every lattice homomorphism to X having a bounded linear right-inverse must have a lattice homomorphism right-inverse. In terms of free Banach lattices, this can be rephrased into the following question: which Banach lattices embed into the free Banach lattice which they generate as a lattice-complemented sublattice? We will provide necessary conditions for a Banach lattice to have the lattice-lifting property, and show that this property is shared by Banach spaces with a 1-unconditional basis as well as free Banach lattices. The case of C(K) spaces will also be analyzed.

Congruences on glrac semigroups (I)

The theory of congruences on semigroups is an important part in the theory of semigroups. The aim of this paper is to study (2,1)-congruences on a glrac semigroup. It is proved that the (2,1)-congruences on a glrac semigroup become a complete sublattice of its lattice of congruences. Especially, the structures of left restriction semigroup (2,1)-congruences and the projection-separating (2,1)-congruences on a glrac semigroup are established. Also, we demonstrate that they are both complete sublattice of (2,1)-congruences and consider their relations with respect to complete lattice homomorphism.

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

Kolmogorov–Sinai type logical entropy for generalized simultaneous measurements

The present paper aims at developing a theory of logical entropy for p -preserving systems ( L, p, ψ ), where p is a Bayesian gs -map on a bounded lattice L and ψ is a p -preserving lattice homomorphism . Having proved several desirable properties , we construct a Rokhlin-type metric involving logical conditional entropy of p -partitions of a Bayesian gs -space ( L,p ), and use it to establish a logical entropy version of Kolmogorov–Sinai theorem in the context. A formal mathematical foundation for the corresponding information theory is also presented in the end.

Norm-attaining lattice homomorphisms

In this paper we study the structure of the set ${\\rm Hom}(X,\\mathbb{R})$ of all lattice homomorphisms from a Banach lattice $X$ into $\\mathbb{R}$. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice ${\\rm FBL}(A)$ generated by a set $A$ contains a disjoint family of cardinality $2^{|A|}$, answering a question of B. de Pagter and A. W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as $c_0$, $L_p$- and $C(K)$-spaces, every lattice homomorphism on it attains its norm, which shows, in particular, that there is no James theorem for this class of functions. We prove that, indeed, every lattice homomorphism on $X$ and $C(K,X)$ attains its norm whenever $X$ has order continuous norm. On the other hand, we provide what seems to be the first example in the literature of a lattice homomorphism which does not attain its norm. In general, we study the existence and characterization of lattice homomorphisms not attaining their norm in free Banach lattices. As a consequence, it is shown that no Bishop–Phelps type theorem holds true in the Banach lattice setting, i.e., not every lattice homomorphism can be approximated by norm-attaining lattice homomorphisms.

On the class of almost (L) limited operators

In this paper, we introduce and study the class of almost (L) limited operators, these are operators from a Banach space into a Banach lattice whose adjoint carries disjoint (L) and weak* null sequences to norm null ones. We establish some characterizations of this class of operators, and present some connections between this class of operators and almost limited operators. After that, we prove that almost (L) limited operators from a Banach lattice into a σDedekind complete one which are lattice homomorphism are exactly operators whose adjoint carries almost (L) sets into L weakly compact ones. Finally, we derive some characterizations of a σ-Dedekind complete Banach lattice whose dual has order continuous norm.

Large deviations built on max-stability

In this paper, we show that the basic results in large deviations theory hold for general monetary risk measures, which satisfy the crucial property of max-stability. A max-stable monetary risk measure fulfills a lattice homomorphism property, and satisfies under a suitable tightness condition the Laplace Principle (LP), that is, admits a dual representation with affine convex conjugate. By replacing asymptotic concentration of probability by concentration of risk, we formulate a Large Deviation Principle (LDP) for max-stable monetary risk measures, and show its equivalence to the LP. In particular, the special case of the asymptotic entropic risk measure corresponds to the classical Varadhan–Bryc equivalence between the LDP and LP. The main results are illustrated by the asymptotic shortfall risk measure.