Lattice Homomorphism Research Articles

Homomorphisms on Function Lattices

In this paper we study real lattice homomorphisms on a unital vector lattice \({{\cal L}} \subset {\cal C}(X)\), where X is a completely regular space. We stress on topological properties of its structure spaces and on its representation as point evaluations. These results are applied to the lattice \({{\cal L}} = {\rm Lip} (X)\) of real Lipschitz functions on a metric space. Using the automatic continuity of lattice homomorphisms with respect to the Lipschitz norm, we are able to derive a Banach-Stone theorem in this context. Namely, it is proved that the unital vector lattice structure of Lip (X) characterizes the Lipschitz structure of the complete metric space X. In the case \({{\cal L}} = {\rm Lip}^{\ast} (X)\) of bounded Lipschitz functions, an analogous result is obtained in the class of complete quasiconvex metric spaces.

Forcing extensions of partial lattices

We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let ϕ:ConcK→D be a {∨,0}-homomorphism, where ConcK denotes the {∨,0}-semilattice of all finitely generated congruences of K. Then there are a lattice L, a lattice homomorphism f:K→L, and an isomorphism α:ConcL→D such that α∘Concf=ϕ. Furthermore, L and f satisfy many additional properties, for example: (i) L is relatively complemented; (ii) L has definable principal congruences; (iii) If the range of ϕ is cofinal in D, then the convex sublattice of L generated by f[K] equals L. We mention the following corollaries, that extend many results obtained in the last decades in that area: 1. Every lattice K such that ConcK is a lattice admits a congruence-preserving extension into a relatively complemented lattice. 2. Every {∨,0}-direct limit of a countable sequence of distributive lattices with zero is isomorphic to the semilattice of compact congruences of a relatively complemented lattice with zero.

Unsolvable one-dimensional lifting problems for congruence lattices of lattices

Let S be a distributive {?, 0}-semilattice. In a previous paper, the second author proved the following result: Suppose that S is a lattice. Let K be a lattice, let $\varphi$: Con K $\to$ S be a {?, 0}-homomorphism. Then $\varphi$ is, up to isomorphism, of the form Conc f, for a lattice L and a lattice homomorphism f : K $\to$ L. In the statement above, Conc K denotes as usual the {?, 0}-semilattice of all ?nitely generated congruences of K. We prove here that this statement characterizes S being a lattice.

Homomorphisms of complete distributive lattices

The category of all complete distributive lattices and their complete homomorphisms is universal, and this is also true for the category of all complete distributive lattices whose morphisms preserve complete joins, finite meets and an additional nullary operation. A survey of analogous results on algebraic universality of categories based on finitary distributive (0,1)-lattices is included to motivate further questions about categories based on complete distributive lattices.

Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory

Let \( K^n = \{x \in {\Bbb R}^n \mid x_i \ge 0,\ 1 \le i \le n \} \) and suppose that f : K n →K n is nonexpansive with respect to the l1-norm, \( \|x\|_1 = \sum_{i=1}^n |x_i| \), and satisfies f (0) = 0. Let P3(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point \( \xi \in K^n \) of f of minimal period p. For each n≥ 1 we use the concept of 'admissible arrays on n symbols' to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P3(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear maps, in particular to periodic points of maps g : Dg→Dg, where \( D{_g} \subset {\Bbb R}^n \) is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism.

On Scott Consequence Systems

The notion of Scott consequence system (briefly, S-system) was introduced by D. Vakarelov in [32] in an analogy to a similar notion given by D. Scott in [26]. In part one of the paper we study the category SSyst of all S-systems and all their morphisms. We show that the category DLat of all distributive lattices and all lattice homomorphisms is isomorphic to a reflective full subcategory of the category SSyst. Extending the representation theory of D. Vakarelov [32] for S-systems in P-systems, we develop an isomorphism theory for S-systems and for Tarski consequence systems. In part two of the paper we prove that the separation theorem for S-systems is equivalent in ZF to some other separation principles, including the separation theorem for filters and ideals in Boolean algebras and separation theorem for convex sets in convexity spaces.

Lattice homomorphism inequalities for vector valued functions

Lattice homomorphism inequalities for vector valued functions

Lattice homomorphism --- Korovkin type inequalities for vector valued functions

Lattice homomorphism --- Korovkin type inequalities for vector valued functions

Stone algebra extensions with bounded dense sets

Stone algebras have been characterized by Chen and Gratzer in terms of triples \( (B, D, \varphi) \), where D is a distributive lattice with 1, B is a Boolean algebra, and \( \varphi \) is a bounded lattice homomorphism from B into the lattice of filters of D. If D is bounded, the construction of these characterizing triples is much simpler, since the homomorphism \( \varphi \) can be replaced by one from B into D itself. The triple construction leads to natural embeddings of a Stone algebra into ones with bounded dense set. These embeddings correspond to a complete sublattice of the distributive lattice of lattice congruences of S. In addition, the largest embedding is a reflector to the subcategory of Stone algebras with bounded dense sets and morphisms preserving the zero of the dense set.

Maximal sublattices of finite distributive lattices

Algebraic properties of lattices of quotients of finite posets are considered. Using the known duality between the category of all finite posets together with all order-preserving maps and the category of all finite distributive (0, 1)-lattices together with all (0, 1)-lattice homomorphisms, algebraic and arithmetic properties of maximal proper sublattices and, in particular, Frattini sublattices of finite distributive (0, 1)-lattices are thereby obtained.

On extensions of compact positive operators on Banach lattices

Extension properties of compact positive operators on Banach lattices are investigated. The following results are obtained: 1. (1) Any compact positive operator (any compact lattice homomorphism, resp.) from a majorizing sublattice G of a Banach lattice E into another Banach lattice F can be extended to a compact positive operator (a compact lattice homomorphism, resp.) from E into F; 2. (2) Any compact positive operator defined on a closed majorizing sublattice G of a Banach lattice E has a compact positive extension on E that preserves the spectrum (a necessary modification is needed). Related extension problems are also studied.

Distinguished extensions of a lattice-ordered group

Any lattice-ordered group (l-group for short) is essentially extended by its lexicographic product with a totally ordered group. That is, anl-homomorphism (i.e., a group and lattice homomorphism) on the extension which is injective on thel-group must be injective on the extension as well. Thus nol-group has a maximal essential extension in the categoryIGp ofl-groups withl-homomorphisms. However, anl-group is a distributive lattice, and so has a maximal essential extension in the categoryD of distributive lattices with lattice homomorphisms. Adistinguished extension of onel-group by another is one which is essential inD. We characterize such extensions, and show that everyl-groupG has a maximal distinguished extensionE(G) which is unique up to anl-isomorphism overG.E(G) contains most other known completions in whichG is order dense, and has mostl-group completeness properties as a result. Finally, we show that ifG is projectable then E(G) is the α-completion of the projectable hull ofG.

Bands in lattices of operators

We consider the lattice of regular operators on a Dedekind complete Banach lattice. We show that in general the projection onto a band generated by a lattice homomorphism need not be continuous and that the principal bands need not be closed for the operator norm. In fact it is possible to find a convergent sequence of operators all the members of which are disjoint from the limit.

Construction of Homomorphisms of M-Continuous Lattices

Construction of Homomorphisms of M-Continuous Lattices

A CHARACTERIZATION OF COMPLEX LATTICE HOMOMORPHISMS ON BANACH LATTICE ALGEBRAS

Abstract In this paper we present a characterization of complex lattice homomorphisms on Banach lattice algebras.

Construction of homomorphisms of 𝑀-continuous lattices

We present a direct approach to constructing homomorphisms of M M -continuous lattices, a generalization of continuous lattices, into the unit interval, and show that an M M -continuous lattice has sufficiently many homomorphisms into the unit interval to separate the points.

Extension properties of order continuous functionals and applications to the theory of Banach lattices

Let E be a Banach lattice and E 0 an ideal of E. Let f be a positive norm-bounded order continuous functional on E 0. Then f has a norm preserving positive order continuous extension to E. Moreover, there exists a minimum extension among all such extensions. This result enables us to obtain an isometric lattice homomorphism from ( E 0) ∗ n into E ∗ n. As an application, we prove that if F is a Banach lattice and if E 0 is the ideal generated by F in E (= F ∗∗), then ( E 0) ∗ n can be identified with F ∗ in the canonical sense. Certain extension properties of order continuous functionals thus follow. Banach lattices with strictly monotone norm are also investigated.

A representation lattice isomorphism for the peripherical spectrum

In this paper we construct a representation isometric lattice isomorphism for the peripherical spectrum of a positive operator on a Banach lattice. By a representation lattice homomorphism, we mean that the peripherical spectrum of the operator is identified with the spectrum of the induced isometric lattice homomorphism. A simple proof of a "zero-two" law follows easily from our representation technique.

Sums and extensions of vector lattice homomorphisms

Sums and extensions of vector lattice homomorphisms

α-Dedekind complete archimedean vector lattices versus α-quasi- F spaces

α denotes an uncountable cardinal number, and W (α) denotes the category of archimedean vector lattices with distinguished waek unit and unit preserving vector lattice homomorphisms. In this paper we show that in W (α), the full subcategory of α-Dedekind complete objects is epireflective. We explain how the Yosida functor connects the algebraic notions of α-Dedekind complete, α-dense, and α-jam-dense with the topological notions of α-quasi- F, α-irreducible, and α-SpFi morphism. We then go on to show that the α-quasi- F cover of a compact space X, denoted (QF α X, q α) is the Yosida space of the α-Dedekind complete epireflection of C( X) in W (α). Finally we show that in the topological category α-SpFi, the full subcategory of α-quasi- F spaces is monocoreflective, and for each compact space X,(QF α X, q α) is the α-quasi- F monocoreflection of X.