scholarly journals WHEN LATTICE HOMOMORPHISMS OF ARCHIMEDEAN VECTOR LATTICES ARE RIESZ HOMOMORPHISMS

2009 ◽  
Vol 87 (2) ◽  
pp. 263-273 ◽  
Author(s):  
MOHAMED ALI TOUMI
Keyword(s):  
London Math ◽  
Lattice Homomorphism ◽  
Vector Lattices ◽  
Lattice Homomorphisms ◽  
Orthogonal Systems

AbstractLet A, B be Archimedean vector lattices and let (ui)i∈I, (vi)i∈I be maximal orthogonal systems of A and B, respectively. In this paper, we prove that if T is a lattice homomorphism from A into B such that $T\left ( \lambda u_{i}\right ) =\lambda v_{i}$ for each λ∈ℝ+ and i∈I, then T is linear. This generalizes earlier results of Ercan and Wickstead (Math. Nachr279 (9–10) (2006), 1024–1027), Lochan and Strauss (J. London Math. Soc. (2) 25 (1982), 379–384), Mena and Roth (Proc. Amer. Math. Soc.71 (1978), 11–12) and Thanh (Ann. Univ. Sci. Budapest. Eotvos Sect. Math.34 (1992), 167–171).

‘POSITIVELY HOMOGENEOUS LATTICE HOMOMORPHISMS BETWEEN RIESZ SPACES NEED NOT BE LINEAR’

2016 ◽  
Vol 102 (3) ◽  
pp. 444-445
Author(s):  
FETHI BEN AMOR
Keyword(s):  
Main Result Theorem ◽  
Riesz Spaces ◽  
Vector Lattices ◽  
Homogeneous Lattice ◽  
Result Theorem ◽  
Lattice Homomorphisms

This note furnishes an example showing that the main result (Theorem 4) in Toumi [‘When lattice homomorphisms of Archimedean vector lattices are Riesz homomorphisms’, J. Aust. Math. Soc. 87 (2009), 263–273] is false.

scholarly journals Free and projective Banach lattices

2015 ◽  
Vol 145 (1) ◽  
pp. 105-143 ◽  
Author(s):  
Ben de Pagter ◽  
Anthony W. Wickstead
Keyword(s):  
Finite Number ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Closed Ideal ◽  
Lattice Homomorphism ◽  
Linear Lattice ◽  
Fundamental Properties ◽  
Number Of Generators ◽  
Lattice Homomorphisms

We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms, and establish some of their fundamental properties. We give much more detailed results about their structure in the case when there are only a finite number of generators, and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice P to be projective if, whenever X is a Banach lattice, J is a closed ideal in X, Q : X → X/J is the quotient map, T : P → X/J is a linear lattice homomorphism and ε > 0, there exists a linear lattice homomorphism : P → X such that T = Q º and ∥∥ ≤ (1 + ε)∥T∥. We establish the connection between projective Banach lattices and free Banach lattices, describe several families of Banach lattices that are projective and prove that some are not.

Some Homological Pathology in Vector Lattices

1965 ◽  
Vol 17 ◽  
pp. 411-428 ◽  
Author(s):  
David M. Topping
Keyword(s):  
Chain Condition ◽  
Boolean Algebras ◽  
Real Vector ◽  
Linear Lattice ◽  
Free Objects ◽  
Vector Lattices ◽  
Countable Chain Condition ◽  
Projective Vector ◽  
Countable Chain ◽  
Lattice Homomorphisms

The purpose of this paper is to point out a number of curious phenomena in the category of (real) vector lattices and linear lattice homomorphisms. Birkhoff (3, p. 221, Ex. 2 and Problem 96) called attention to the question of constructing models of the free objects with more than one generator in this category, a problem recently solved by E. C. Weinberg (9). In §6 we construct a more manageable class of (non-free) projective vector lattices. Here, however, there is a countability restriction which suggests strong connections with free and projective Boolean algebras (in the category of Boolean algebras and their homomorphisms, such algebras must satisfy the countable chain condition (6)).

scholarly journals Homomorphisms Between Lattices of Zero-Sets

1978 ◽  
Vol 21 (1) ◽  
pp. 1-5 ◽  
Author(s):  
S. Broverman
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Lattice Homomorphism ◽  
Completely Regular ◽  
Zero Sets ◽  
Hausdorff Topological Space ◽  
Continuous Map ◽  
Lattice Homomorphisms

AbstractFor a completely regular Hausdorff topological space X, let Z(X) denote the lattice of zero-sets of X. If T is a continuous map from X to Y, then there is a lattice homomorphism T” from Z(Y) to Z(X) induced by T which is defined by τ‘(A) = τ←(A). A characterization is given of those lattice homomorphisms from Z(Y) to Z(X) which are induced in the above way by a continuous function from X to Y.

scholarly journals Vector Lattices Admitting a Positively Homogeneous Continuous Function Calculus

2020 ◽  
Vol 71 (1) ◽  
pp. 281-294
Author(s):  
Niels Jakob Laustsen ◽  
Vladimir G Troitsky
Keyword(s):  
Continuous Function ◽  
Vector Space ◽  
Vector Lattice ◽  
Positive Element ◽  
Order Ideal ◽  
Lattice Homomorphism ◽  
Vector Lattices ◽  
Coordinate Projection ◽  
Function Calculus

Abstract We characterize the Archimedean vector lattices that admit a positively homogeneous continuous function calculus by showing that the following two conditions are equivalent for each $n$-tuple $\boldsymbol{x} = (x_1,\ldots ,x_n)\in X^n$, where $X$ is an Archimedean vector lattice and $n\in{\mathbb{N}}$: • there is a vector lattice homomorphism $\Phi _{\boldsymbol{x}}\colon H_n\to X$ such that $$\begin{equation*}\Phi_{\boldsymbol{x}}(\pi_i^{(n)}) = x_i\qquad (i\in\{1,\ldots,n\}),\end{equation*}$$where $H_n$ denotes the vector lattice of positively homogeneous, continuous, real-valued functions defined on ${\mathbb{R}}^n$ and $\pi _i^{(n)}\colon{\mathbb{R}}^n\to{\mathbb{R}}$ is the $i^{\text{}}$th coordinate projection;• there is a positive element $e\in X$ such that $e\geqslant \lvert x_1\rvert \vee \cdots \vee \lvert x_n\rvert$ and the norm$$\begin{equation*}\lVert x\rVert_e = \inf\bigl\{ \lambda\in[0,\infty)\:\colon\:\lvert x\rvert{\leqslant}\lambda e\bigr\},\end{equation*}$$defined for each $x$ in the order ideal $I_e$ of $X$ generated by $e$, is complete when restricted to the closed sublattice of $I_e$ generated by $x_1,\ldots ,x_n$. Moreover, we show that a vector space which admits a ‘sufficiently strong’ $H_n$-function calculus for each $n\in{\mathbb{N}}$ is automatically a vector lattice, and we explore the situation in the non-Archimedean case by showing that some non-Archimedean vector lattices admit a positively homogeneous continuous function calculus, while others do not.

scholarly journals On maps between Stone-Cech compactifications induced by lattice homomorphisms

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2465-2474 ◽  
Author(s):  
Themba Dube
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lattice Homomorphism ◽  
Completely Regular ◽  
Zero Sets ◽  
Continuous Map ◽  
Necessary And Sufficient ◽  
Lattice Homomorphisms ◽  
Tychonoff Spaces

Broverman has shown that if X and Y are Tychonoff spaces and t:Z(Y)?Z(X) is a lattice homomorphism between the lattices of their zero-sets, then there is a continuous map ?: ?X ? ?Y induced by t. In this note we expound this idea and supplement Broverman?s results by first showing that this phenomenon holds in the category of completely regular frames. Among results we obtain, which were not considered by Broverman, are necessary and sufficient conditions (in terms of properties of the map t) for the induced map ? to be (i) the inclusion of a subspace, (ii) surjective, and (iii) irreducible. We show that if X and Y are pseudocompact then t pulls back z-ultrafilters to z-ultrafilters if and only if cl?X t(Z) = ?? [cl?YZ] for every Z ? Z(Y) if and only if t is ?-homomorphism.

Algorithm for the synthesis of orthogonal systems of signals and their properties

1996 ◽  
Vol 2 (5-6) ◽  
pp. 69-73
Author(s):  
Yu.V. Stasev ◽  
◽  
N.V. Pastukhov ◽  
Keyword(s):  
Orthogonal Systems

XVI.—On Triple Systems of Surfaces and Non-Orthogonal Curvilinear Coordinates

1927 ◽  
Vol 46 ◽  
pp. 194-205 ◽  
Author(s):  
C. E. Weatherburn
Keyword(s):  
Curvilinear Coordinates ◽  
Triple Systems ◽  
Orthogonal Curvilinear Coordinates ◽  
Orthogonal Systems ◽  
Considerable Detail

The properties of “triply orthogonal” systems of surfaces have been examined by various writers and in considerable detail; but those of triple systems generally have not hitherto received the same attention. It is the purpose of this paper to discuss non-orthogonal systems, and to investigate formulæ in terms of the “oblique” curvilinear coordinates u, v, w which such a system determines.

Theorems on decomposition of operators in L 1 and their generalization to vector lattices

2006 ◽  
Vol 58 (1) ◽  
pp. 30-41 ◽  
Author(s):  
O. V. Maslyuchenko ◽  
V. V. Mykhailyuk ◽  
M. M. Popov
Keyword(s):  
Vector Lattices
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