Next: Universes
Up: Static Algebras and
Previous: Logic Names
A static algebra or (for the sake of brevity) state
S of
vocabulary ![[Upsilon]](upsilon.xbm)
is a nonempty set
X , the superuniverse of S ,
together with interpretations of the function names in on X . An
r-ary function name is interpreted as a function from 
to X , a
basic function of S . The interpretation of an
r-ary relation
name is a function from to {true, false}, a basic
relation of S . The vocabulary is called the vocabulary
of S and denoted Fun(S).
The interpretations of the nullary logic names true, false and
undef are distinct elements of X . The Boolean operations behave in
the usual way on the Boolean values true and false and produce
undef if at least one of the arguments is not Boolean. The equality
sign is interpreted as the characteristic function of the identity relation
on X. If f(![[x]](xbar.xbm)
) evaluates to true in S, we say that f() holds in S; and if f() evaluates to false in S, we say that
f() fails in S.
Formally speaking, basic functions are total. However, we view them as
being partial and define the domain Dom(f)
of an r-ary basic function
f as the set of r-tuples
such that
f() <> undef. Let
us stress though that undef is an ordinary element of the superuniverse.
Often, a basic function produces undef if at least one argument equals
undef, but this is not required and there are exceptions (e.g.
basic relations).