ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
▼ Theory
▼ Zermelo-Fraenkel set theory
▼ Set
▼ Binary cartesian set product
▼ Binary relation
▼ Map
▼ Operation
▼ N-operation
▼ Binary operation
▼ Enclosed binary operation
▼ Groupoid
▼ Ringoid
▼ Semiring
▼ Ring
▼ Left ring action
▼ Module
▼ Linear combination
▼ Linear map
▼ Affine map
Definition D5121
Subaffine map
Formulation 0
Let $R$ be an D5119: Ordered division ring such that
(i) $1_R$ is a D577: Multiplicative identity in $R$
Let $V$ be a D29: Vector space over $R$.
Let $W$ be an D1963: Ordered vector space over $R$ such that
(i) $\preceq$ is the D378: Ordering relation on $W$
A D18: Map $f : V \to W$ is subaffine from $V$ to $W$ over $R$ if and only if \begin{equation} \forall \, N \in 1, 2, 3, \ldots : x \in V^N : r \in R^N \left[ \sum_{n = 1}^N r_n = 1_R \quad \implies \quad f \left( \sum_{n = 1}^N r_n x_n \right) \preceq \sum_{n = 1}^N r_n f(x_n) \right] \end{equation}