Structure
Let 
be a language of the first-order logic. Assume that the language 
has the following sets of nonlogical symbols:
1. 
is the set of constant symbols of 
. (These are nullary function symbols.)
2. 
is the set of predicate symbols of 
, and for each 
, 
is the arity of 
. The symbols in 
are also called relation symbols of the language 
.
3. 
is the set of function symbols of 
, and for each 
, 
is the arity of 
. The symbols in 
are also called operation symbols of the language 
.
4. 
is the universal quantifier symbol of 
.
A structure for 
is a tuple 
, 
, 
where 
is a set (called the underlying set of 
), and the following hold:
1. For each 
, 
,
2. For each 
, 
,
3. For each 
, 
.
REFERENCES
Bell, J. L. and Slomson, A. B. Models and Ultraproducts: an Introduction. Amsterdam, Netherlands: North-Holland, 1971.
Enderton, H. B. A Mathematical Introduction to Logic. New York: Academic Press, 1972.
