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Multiplication in Vector Lattices
Canadian Journal of Mathematics, 1968B. Z. Vulih has shown (13) how an essentially unique intrinsic multiplication can be defined in a Dedekind complete vector lattice L having a weak order unit. Since this work is available only in Russian, a brief outline is given in § 2 (cf. also the review by E. Hewitt (4), and for details, consult (13) or (11)).
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Canadian Journal of Mathematics, 1980
Let G be a lattice-ordered group (l-group). If X ⊆ G, then letThen X’ is a convex l-subgroup of G called a polar. The set P(G) of all polars of G is a complete Boolean algebra with ‘ as complementation and set-theoretic intersection as meet. An l-subgroup H of G is large in G (G is an essential extension of H) if each non-zero convex l-subgroup of G ...
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Let G be a lattice-ordered group (l-group). If X ⊆ G, then letThen X’ is a convex l-subgroup of G called a polar. The set P(G) of all polars of G is a complete Boolean algebra with ‘ as complementation and set-theoretic intersection as meet. An l-subgroup H of G is large in G (G is an essential extension of H) if each non-zero convex l-subgroup of G ...
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2018
The first of two results is a 1-1-correspondence between isomorphism classes of finite-dimensional vector lattices and finite rooted unlabelled trees. Thus the problem of counting isomorphism classes of finite-dimensional vector lattices reduces to the well-known combinatorial problem of counting these trees.
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The first of two results is a 1-1-correspondence between isomorphism classes of finite-dimensional vector lattices and finite rooted unlabelled trees. Thus the problem of counting isomorphism classes of finite-dimensional vector lattices reduces to the well-known combinatorial problem of counting these trees.
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Contact Vectors of Point Lattices
Mathematical Notes, 2023The contact vectors of a lattice \(L\) are vectors \(l\) which are minimal in the \(l^{2}\)-norm in their parity class. In this paper, the author shows that, in the space of all symmetric matrices, the set of all contact vectors of the lattice \(L\) defines the subspace \(M(L)\) containing the Gram matrix \(A\) of the lattice \(L\).
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Vector Lattices Associated with Ordered Vector Spaces
Mediterranean Journal of Mathematics, 2010The vector lattice generated by a real Archimedean vector space \(V\) is considered. It is proved that the cone \(C\) of all positive linear forms on \(V\) separates elements of \(V\). Then the positive linear forms are determined in terms of conical measures on the cone \(C\).
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Prime Ideals in Vector Lattices
Canadian Journal of Mathematics, 1962Projectors, spectral functions, carriers, and collections of these objects are some of the tools which have been used to study vector lattices. One of our objectives in this paper is to show that these various approaches are not essentially different. We do this by proving that each of the above-mentioned objects can be identified with a collection of ...
Johnson, D. G., Kist, J. E.
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Girsanov’s theorem in vector lattices
Positivity, 2019The authors formulate and prove Girsanov's theorem in vector lattices. The methodology involves the theory of cross-variation processes, specifically the Kunita-Watanabe inequality, exponential processes, Itô's rule for multi-dimensional processes, and the integration by parts formula for martingales.
Grobler, Jacobus J. +1 more
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Sampling short lattice vectors and the closest lattice vector problem
Proceedings 17th IEEE Annual Conference on Computational Complexity, 2003We present a 2/sup O(n)/ time Turing reduction from the closest lattice vector problem to the shortest lattice vector problem. Our reduction assumes access to a subroutine that solves SVP exactly and a subroutine to sample short vectors from a lattice, and computes a (1+/spl epsi/)-approximation to CVP As a consequence, using the SVP algorithm from ...
M. Ajtai, R. Kumar, D. Sivakumar
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Extension of Vector Lattice Homomorphisms
Journal of the London Mathematical Society, 1986Suppose B is a complete Boolean algebra, D a distributive lattice and \(\phi\) a lattice homomorphism from a sublattice, \(D_ 0\), of D, into B, then \(\phi\) can be extended to a lattice homomorphism of D into B. This generalizes Sikorski's extension theorem for Boolean algebras. It also leads to a new proof that if N is a complete vector lattice, L a
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On Vector Lattice-Valued Measures
Canadian Mathematical Bulletin, 1965E. Hewitt [1] used the Daniell approach to define a real-valued measure function on a σ-algebra of the real line. He began by defining an arbitrary non-negative linear functional I on L∞ ∞(R), (the space of all complex-valued continuous functions on the real line R which vanish off some compact subset of R).
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