Abstract
AbstractGiven any finite subset A of order n of a distributive lattice and $$k\in \{1,\ldots ,n\}$$
k
∈
{
1
,
…
,
n
}
, there is a natural extension of the median operation to n variables which generalizes the notion of the kth smallest element of A. By applying each of these operations to A, a totally ordered set to(A) is obtained. We refer to to(A) as the total orderization of A. After developing a brief theory of total orderization invariant maps on distributive lattices, it is shown in this paper how these functions generalize and provide new characterizations for symmetric continuous positively homogeneous functions, bounded orthosymmetric multilinear maps, and certain power sum polynomials on vector lattices. These theorems generalize several results by Bernau, Huijsmans, Kusraev, Azouzi, Boulabiar, Buskes, Boyd, Ryan, and Snigireva and in turn reveal novel properties of the various maps studied in this paper.
Publisher
Springer Science and Business Media LLC
Reference17 articles.
1. Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, Orlando (1985)
2. Azzouzi, Y., Boulabiar, K., Buskes, G.: The de Schipper formula and squares of Riesz spaces. Indag. Math. (N.S.) 17(4), 479–496 (2006)
3. Ben Amor, F.: Orthogonally additive homogenous polynomials on vector lattices. Commun. Algebra 43(3), 1118–1134 (2015)
4. Bernau, S.J., Huijsmans, C.B.: Almost $$f$$-algebras and $$d$$-algebras. Math. Proc. Camb. Philos. Soc. 107(2), 287–308 (1990)
5. American Mathematical Society Colloquium Publications;G Birkhoff,1979