Stuart Vella Bonello - Academia.edu (original) (raw)

Papers by Stuart Vella Bonello

A poset is a set P endowed with a partial order, i.e. a relation ≤ on P such that, if a, b, c ∈ P... more A poset is a set P endowed with a partial order, i.e. a relation ≤ on P such that, if a, b, c ∈ P , then (1) a ≤ a (reflexivity), (2) a ≤ b and b ≤ a imply a = b (anti-symmetry), (3) a ≤ b and b ≤ c imply a ≤ c (transitivity).

A partially ordered set that contains the supremum and infimum of each pair of elements is called... more A partially ordered set that contains the supremum and infimum of each pair of elements is called a lattice. The theory of lattices is particularly rich for vector spaces that are algebraically related to the ordering. For such structures (referred to as Riesz spaces) a number of characterisations hold for concepts such as Dedekind completeness and order separability, as well as identities and relations associated with the absolute of a vector.

For nets in a partially ordered set there are a number of non-equivalent ways in which convergence may be defined in terms of the ordering. We consider one type [1] which in turn will be used to define the order and sequential order topologies. Whilst these topologies are both T_1, it turns out [2] that they may fail to be Hausdorff even for Boolean algebras i.e. even for complemented distributive lattices.

In the setting of a measure space the quotient space L0 of cosets of measurable functions (identifying two functions if they are equal almost everywhere with respect to the measure) may be turned into a Riesz space based on a pointwise ordering. The function spaces Lp of pth-power integrable measurable functions (where 1<p<infinity) and L∞ of essentially bounded functions are contained in L0 and hence inherit its partial ordering. Characterisations of order properties of these spaces may be given in terms of types of measure spaces. Besides being able to consider the above order topologies, these spaces are also complete with respect to a particular norm.

L0 may be endowed with a linear topology (the topology of convergence in measure) which can be restricted to the other functions spaces mentioned above. For semi-finite measure spaces the existence of isometric embeddings of L∞ in B(Lp) and (L1)* allow for the consideration of the strong-operator and the weak* topologies on it.

References

[1] Birkhoff, G. Lattice Theory, New York, 1948.

[2] Floyd, E. E. Boolean algebras with pathological order topologies, Pacific J. Math. 5 (1955), 687-689.

A poset is a set P endowed with a partial order, i.e. a relation ≤ on P such that, if a, b, c ∈ P... more A poset is a set P endowed with a partial order, i.e. a relation ≤ on P such that, if a, b, c ∈ P , then (1) a ≤ a (reflexivity), (2) a ≤ b and b ≤ a imply a = b (anti-symmetry), (3) a ≤ b and b ≤ c imply a ≤ c (transitivity).

A partially ordered set that contains the supremum and infimum of each pair of elements is called... more A partially ordered set that contains the supremum and infimum of each pair of elements is called a lattice. The theory of lattices is particularly rich for vector spaces that are algebraically related to the ordering. For such structures (referred to as Riesz spaces) a number of characterisations hold for concepts such as Dedekind completeness and order separability, as well as identities and relations associated with the absolute of a vector.

For nets in a partially ordered set there are a number of non-equivalent ways in which convergence may be defined in terms of the ordering. We consider one type [1] which in turn will be used to define the order and sequential order topologies. Whilst these topologies are both T_1, it turns out [2] that they may fail to be Hausdorff even for Boolean algebras i.e. even for complemented distributive lattices.

In the setting of a measure space the quotient space L0 of cosets of measurable functions (identifying two functions if they are equal almost everywhere with respect to the measure) may be turned into a Riesz space based on a pointwise ordering. The function spaces Lp of pth-power integrable measurable functions (where 1<p<infinity) and L∞ of essentially bounded functions are contained in L0 and hence inherit its partial ordering. Characterisations of order properties of these spaces may be given in terms of types of measure spaces. Besides being able to consider the above order topologies, these spaces are also complete with respect to a particular norm.

L0 may be endowed with a linear topology (the topology of convergence in measure) which can be restricted to the other functions spaces mentioned above. For semi-finite measure spaces the existence of isometric embeddings of L∞ in B(Lp) and (L1)* allow for the consideration of the strong-operator and the weak* topologies on it.

References

[1] Birkhoff, G. Lattice Theory, New York, 1948.

[2] Floyd, E. E. Boolean algebras with pathological order topologies, Pacific J. Math. 5 (1955), 687-689.