Manuscript XIII-E@1 · XIII-E.tex · section · 74eaba48

E3: overlap equations and ordered values

E3: overlap equations and ordered values · lines 1015–1036

XIII-E@1#eq:overlap

The isolated primitive extensions are inherited constructions. The remaining problem is compatible extension of the complete graphs, not another extension of the same primitive.

Suppose, conditionally, that local extensions $u_{i}$ have been constructed on an open cover ${U_{i}}$ of a graph domain. A permitted adjustment $c_{i}$ must be a well-defined contact of the frozen class. Agreement on overlaps requires

(u_{i} + c_{i}) ∣_{U_{i} \cap U_{j}} = (u_{j} + c_{j}) ∣_{U_{i} \cap U_{j}},

or equivalently

c_{j} - c_{i} = u_{i} - u_{j} on U_{i} \cap U_{j} .

Displayed formulas in this passage

XIII-E@1#eq:overlap · line 1026
$(u_{i} + c_{i}) ∣_{U_{i} \cap U_{j}} = (u_{j} + c_{j}) ∣_{U_{i} \cap U_{j}},$
unlabeled · line 1032
$c_{j} - c_{i} = u_{i} - u_{j} on U_{i} \cap U_{j} .$

Nearby in this section

lemma: Lemma: Conditional gluing
section: E3: overlap equations and ordered values
scope: E3 status