Definition·D087
Assignment into a structure
A function naming, for every variable, an element of a structure's domain.
For an L-structure
, an assignment into
is a function
In words
An assignment into a structure is a function from the variables to the structure's domain.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Recall
(D078), so an assignment names, once and for all, a domain element
for every one of the infinitely many variables
; only the values at the (finitely many) variables actually occurring in a given term or formula will ever matter, once satisfaction is defined. For
and
, write
for the assignment agreeing with
everywhere except sending
to
(a function again, by the same argument as the extension trick used for finite sets); this is how a quantifier will update the assignment when its bound variable is stripped off.