arXiv subjects

Jani Jokela

Publications and source records attributed to Jani Jokela.

6 recordsLinked to original sources

arXiv2025

A note on disjointness and discrete elements in partially ordered vector spaces

The notions of disjointness and discrete elements play a prominent role in the classical theory of vector lattices. There are at least three different generalizations of the notion of disjointness to a larger class of partially ordered vector spaces. In recent years, one of these generalizations has been widely studied in the context of pre-Riesz spaces. The notion of $D$-disjointness is the most general of the three disjointness concepts. In this paper we study $D$-disjointness and the related concept of a $D$-discrete element. We establish some basic properties of $D$-discrete elements in Archimedean partially ordered spaces, and we investigate their relationship to discrete elements in the theory of pre-Riesz spaces. We then apply our results to establish the equivalence of pervasiveness and weak pervasiveness in finite-dimensional Archimedean pre-Riesz spaces.

math.FA↗

arXiv2025

Partially ordered semigroups and groups with two mixed partial orderings

A mixed lattice is a partially ordered set with two mixed partial orderings that are linked by asymmetric upper and lower envelopes. These notions generalize the join and meet operations of a lattice. In the present paper, we study different types of partially ordered semigroups with two mixed orderings, and investigate their relationship to subsemigroups of mixed lattice groups, which are partially ordered groups with a similar order structure. We also consider Archimedean orderings, and we show that elements of finite order cannot exist in a rather general class of Archimedean mixed lattice groups. Moreover, we give an example of a non-Archimedean mixed lattice group that contains an element of finite order.

math.GR↗

arXiv2024

General mixed lattices

A mixed lattice is a lattice-type structure consisting of a set with two partial orderings, and generalizing the notion of a lattice. Mixed lattice theory has previously been studied in various algebraic structures, such as groups and semigroups, while the more general notion of a mixed lattice remains unexplored. In this paper, we study the fundamental properties of mixed lattices and the relationships between the various properties, such as the one-sided associative, distributive and modular laws. We also give an alternative definition of mixed lattices and mixed lattice groups as non-commutative and non-associative algebras satisfying a certain set of postulates. The algebraic and the order-theoretic definitions are then shown to be equivalent.

math.CO↗

arXiv2023

Ideals, bands and direct sum decompositions in mixed lattice vector spaces

A mixed lattice vector space is a partially ordered vector space with two partial orderings and certain lattice-type properties. In this paper we first give some fundamental results in mixed lattice groups, and then we investigate the structure theory of mixed lattice vector spaces, which can be viewed as a generalization of the theory of Riesz spaces. More specifically, we study the properties of ideals and bands in mixed lattice spaces, and the related idea of representing a mixed lattice space as a direct sum of disjoint bands. Under certain conditions, these decompositions can also be given in terms of order projections.

math.FA↗

arXiv2022

Mixed lattice structures and cone projections

Problems related to projections on closed convex cones are frequently encountered in optimization theory and related fields. To study these problems, various unifying ideas have been introduced, including asymmetric vector-valued norms and certain generalized lattice-like operations. We propose a new perspective on these studies by describing how the problem of cone projection can be formulated using an order-theoretic formalism developed in this paper. The underlying mathematical structure is a partially ordered vector space that generalizes the notion of a vector lattice by using two partial orderings and having certain lattice-type properties with respect to these orderings. In this note we introduce a generalization of these so-called mixed lattice spaces, and we show how such structures arise quite naturally in some of the applications mentioned above.

math.OC↗

arXiv2022

Compatible topologies on mixed lattice vector spaces

A mixed lattice vector space is a partially ordered vector space with two partial orderings, generalizing the notion of a Riesz space. Whereas the algebraic theory of mixed lattice structures dates back to the 1970s, the topological theory of mixed lattice spaces remains largely unexplored. The purpose of this paper is to develop the basic topological theory of mixed lattice spaces. A vector topology is said to be compatible with the mixed lattice structure if the mixed lattice operations are continuous. We obtain a characterization of compatible mixed lattice topologies, which is similar to the well known Roberts-Namioka characterization of locally solid Riesz spaces. Moreover, we study locally convex topologies and the associated seminorms, as well as connections between mixed lattice topologies and locally solid topologies on Riesz spaces. We also briefly discuss asymmetric norms and cone norms on mixed lattice spaces.

math.FA↗