Homogeneous orthogonally additive polynomials on vector lattices

It is proved that an orthogonally additive order bounded homogeneous polynomial acting between uniformly complete vector lattices admits a representation in the form of the composition of a linear order bounded operator and a special homogeneous polynomial playing the role of a power-law function, which is absent in the vector lattice. This result helps to establish a criterion for the integral representability of an orthogonally additive homogeneous polynomial.     

Representation of orthogonally additive polynomials

We prove that each bounded orthogonally additive homogeneous polynomial acting from an Archimedean vector lattice into a separated convex bornological space, under the additional assumption that the bornological space is complete or the vector lattice is uniformly complete, can be represented as the composite of a bounded linear operator and a special homogeneous polynomial which plays the role of the exponentiation absent in the vector lattice. The approach suggested is based on the notions of convex bornology and vector lattice power.     

On compact domination of homogeneous orthogonally additive polynomials

We prove an analog of the Dodds–Fremlin–Wickstead Theorem on compact domination for homogeneous orthogonally additive polynomials in Banach lattices. The proof is based on linearization of the polynomials which was established earlier by the author.     

Canonical extensions of bounded archimedean vector lattices

Canonical extensions of Boolean algebras with operators were introduced in the seminal paper of Jónsson and Tarski. The two defining properties of canonical extensions are the density and compactness axioms. While the density axiom can be extended to the setting of vector lattices of continuous real-valued functions, the compactness axiom requires appropriate weakening. This provides a motivation for defining the concept of canonical extension in the category \(\varvec{ bav }\) of bounded archimedean vector lattices. We prove existence and uniqueness theorems for canonical extensions in \(\varvec{ bav }\). We show that the underlying vector lattice of the canonical extension of \(A\in \varvec{ bav }\) is isomorphic to the vector lattice of all bounded real-valued functions on the Yosida space of A, and give an intrinsic characterization of those \(B \in \varvec{ bav }\) that arise as the canonical extension of some \(A \in \varvec{ bav }\).     

On the set of orthogonally additive functions with orthogonally additive second iterate

Let E be a real inner product space of dimension at least 2. We show that both the set of all orthogonally additive functions mapping E into E having orthogonally additive second iterate and its complement are dense in the space of all orthogonally additive functions from E into E with the Tychonoff topology.     

Narrow orthogonally additive operators

We generalize the notion of narrow operators to nonlinear maps on vector lattices. The main objects are orthogonally additive operators and, in particular, abstract Uryson operators. Most of the results extend known theorems obtained by O. Maslyuchenko, V. Mykhaylyuk and the second named author published in Positivity 13:459–495 (2009) for linear operators. For instance, we prove that every orthogonally additive laterally-to-norm continuous C-compact operator from an atomless Dedekind complete vector lattice to a Banach space is narrow. Another result asserts that the set \({\mathcal U}_{on}^{lc}(E,F)\) of all order narrow laterally continuous abstract Uryson operators is a band in the vector lattice of all laterally continuous abstract Uryson operators from an atomless vector lattice \(E\) with the principal projection property to a Dedekind complete vector lattice \(F\) . The band generated by the disjointness preserving laterally continuous abstract Uryson operators is the orthogonal complement to \({\mathcal U}_n^{lc}(E,F)\) .     

Transversally bounded lattices

A problem stemming from a boundedness question for torsion modules and its translation into ideal lattices is explored in the setting of abstract lattices. Call a complete lattice L transversally bounded (resp., uniformly transversally bounded) if for all families (X _i)_iIof nonempty subsets of L with the property that {x _iiI}<1 for all choices of x _iX _i, almost all of the sets X _ihave join smaller than 1 (resp., _jJ X _jhas join smaller than 1 for some cofinite subset J of I). It is shown that the lattices which are transversally bounded, but not uniformly so, correspond to certain ultrafilters with peculiar boundedness properties similar to those studied by Ramsey. The prototypical candidates of the two types of lattices which one is led to construct from ultrafilters (in particular the lattices arising from what will be called Ramsey systems) appear to be of interest beyond the questions at stake.     

Reflections on lower bounded lattices

Lattices in the variety of lower bounded lattices of rank k are characterized. A sufficient condition for a lattice to be lower bounded is given, and used to produce a new example of a non-finitely-generated lower bounded lattice. Lattices that are subdirect products of finite lower bounded lattices are characterized.In memory of Ivan RivalReceived September 18, 2003; accepted in final form October 5, 2004.This revised version was published online in August 2005 with a corrected cover date.     

On orthogonally additive mappings,IV

Summary The conditional Cauchy functional equationF: (X, +, ) (Y, +), F(x + y) = F(x) + F(y) x, y X, x y, has first been studied under regularity (mainly continuity and boundedness) conditions and by referring to the inner product and the Birkhoff—James orthogonalities (A. Pinsker 1938, K. Sundaresan 1972, S. Gudder and D. Strawther 1975). The latter authors proposed an axiomatic framework for the space (X, +, ), and it then became possible to modify their axioms so that it could be proved without any regularity condition that the odd solutions of (*) are additive and the even ones are quadratic (cf., e.g., ([8], [12]). The results obtained included the classical case of the inner product orthogonality as well as the three following generalizations thereof: (i) Birkhoff—James orthogonality on a normed space, (ii) orthogonality induced by a non-isotropic sesquilinear functional, (iii) semi-inner product orthogonality.Making a further step in the modifications of the axioms for the space (X, +, ), the additive/quadratic representation of the solutions of (*) now can be proved in a much more general situation which includes also the case of the orthogonality induced by an isotropic symmetric bilinear functional.     

Orthogonality spaces and orthogonally additive mappings

Summary An open problem on orthogonality spaces posed by Jürg R?tz in [10] (also cf. [14] is completely solved in this paper, so that orthogonality spaces admitting nonzero even orthogonally additive mappings are completely described. As a by-product, a characterization of real inner product spaces is also given.     

States on bounded commutative residuated lattices

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X,

1. If s is a state, then X/ker(s) is an MV-algebra.
2. If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.

Moreover we show that for a state s on X, the following statements are equivalent:

1. s is a state-morphism on X.
2. ker(s) is a maximal filter of X.
3. s is extremal on X.

     

Narrow orthogonally additive operators in lattice-normed spaces

We consider a new class of narrow orthogonally additive operators in lattice-normed spaces and prove the narrowness of every C-compact norm-laterally-continuous orthogonally additive operator from a Banach–Kantorovich space V into a Banach space Y. Furthermore, every dominated Urysohn operator from V into a Banach sequence lattice Y is also narrow. We establish that the order narrowness of a dominated Urysohn operator from a Banach–Kantorovich space V into a Banach space with mixed norm W implies the order narrowness of the least dominant of the operator.     

Domination problem for narrow orthogonally additive operators

The “Up-and-down” theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result in operator theory. We prove an analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result is used to prove a theorem of domination for order narrow positive abstract Uryson operators from a vector lattice E to a Banach lattice F with an order continuous norm.     

Q-universal varieties of bounded lattices

A quasivariety K of algebraic systems of finite type is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.? It is known that, for every variety K of (0, 1)-lattices, if K contains a finite nondistributive simple (0, 1)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, 1)-lattices. This paper shows that in general the opposite implication is not true. A family (A _i : i < 2ω) of locally finite varieties of (0, 1)-lattices is exhibited each of which contains no simple non-distributive (0, 1)-lattice and each of which is Q-universal. Received July 19, 2001; accepted in final form July 11, 2002.     

Orthogonally additive polynomials on Fourier algebras

We show that an n-homogeneous polynomial P   on the Fourier algebra A(G)$A(G)$ of a locally compact group G   can be represented in the form P(f)=〈T,fⁿ〉$P(f)=〈T,f^n〉$(f∈A(G))$(f\in A(G))$ for some T   in the group von Neumann algebra VN(G)$\mathit{VN}(G)$ of G if and only if it is orthogonally additive and completely bounded.