On the algebraic dimension of Riesz spaces

We prove that the algebraic dimension of an infinite dimensional $C$-$\sigma$-complete Riesz space (in particular, of a Dedekind $\sigma$-complete and a laterally $\sigma$-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known.

Inverse complements and strongly unit regular elements

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

Sheaf representations and locality of Riesz spaces with order unit

We present an algebraic study of Riesz spaces (=real vector lattices) with a (strong) order unit. We exploit a categorical equivalence between those structures and a variety of algebras called RMV-algebras. We prove two different sheaf representations for Riesz spaces with order unit: the first represents them as sheaves of linearly ordered Riesz spaces over a spectral space, the second represent them as sheaves of "local" Riesz spaces over a compact Hausdorff space. Motivated by the latter representation we study the class of local RMV-algebras. We study the algebraic properties of local RMV-algebra and provide a characterisation of them as special retracts of the real interval [0,1]. Finally, we prove that the category of local RMV-algebras is equivalent to the category of all Riesz spaces.

Unital topology on a unital l-group

In this paper we investigate some fundamental properties of unital topology on a lattice ordered group with order unit. We show that some essential properties of order unit norm on a vector lattice with order unit, are valid for unital l-groups. For instance we show that for an Archimedean Riesz space G with order unit u, the unital topology and the strong link topology are the same.

Ideal structure and factorization properties of the regular kernel operators

We consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on $$L^p$$ L p -spaces. We show, in particular, that for any $$L^p(\mu )$$ L p ( μ ) space, with $$\mu $$ μ $$\sigma $$ σ -finite and $$1<p<\infty $$ 1 < p < ∞ , the norm-closure of the ideal of finite-rank operators on $$L^p(\mu )$$ L p ( μ ) , is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the $$L^p$$ L p setting is the fact that every regular kernel operator on an $$L^p(\mu )$$ L p ( μ ) space ($$\mu $$ μ and p as before) factors with regular factors through $$\ell _p$$ ℓ p . We show that a similar but weaker factorization property, where $$\ell _p$$ ℓ p is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit.

The Unbounded Fuzzy Order Convergence in Fuzzy Riesz Spaces

The fuzzy order convergence in fuzzy Riesz spaces is defined only for fuzzy order bounded nets. The aim of this paper is to define and study unbounded fuzzy order convergence and some of its applications. Furthermore, some theoretical concepts like the fuzzy weak order unit and fuzzy ideals are studied in relation to unbounded fuzzy order convergence.