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.