Manuscript XXII-F@1 · XXII-F.tex · definition · 67821705

Definition: Faithful generated affine representation

The conditional extraction theorem › Representation and selecting properties · lines 502–517

XXII-F@1#def:representation, XXII-F@1#eq:faithful, XXII-F@1#eq:span, XXII-F@1#eq:generated

For an ordered carrier $(E, ⪯)$ , a representation $(V, ι, K)$ has $ι : E \to V$ injective, $K$ proper, and

p ⪯ q V K ⟺ ι (q) - ι (p) \in K, = span {ι (q) - ι (p) : p, q \in E}, = cone {ι (q) - ι (p) : p ⪯ q, p \neq = q} .

Closure is ordinary finite-dimensional closure.

Displayed formulas in this passage

XXII-F@1#eq:faithful · line 506
$p ⪯ q V K ⟺ ι (q) - ι (p) \in K, = span {ι (q) - ι (p) : p, q \in E}, = cone {ι (q) - ι (p) : p ⪯ q, p \neq = q} .$
XXII-F@1#eq:span · line 506
$p ⪯ q V K ⟺ ι (q) - ι (p) \in K, = span {ι (q) - ι (p) : p, q \in E}, = cone {ι (q) - ι (p) : p ⪯ q, p \neq = q} .$
XXII-F@1#eq:generated · line 506
$p ⪯ q V K ⟺ ι (q) - ι (p) \in K, = span {ι (q) - ι (p) : p, q \in E}, = cone {ι (q) - ι (p) : p ⪯ q, p \neq = q} .$