where is the satisfaction relation. When , we says that satisfies , or that is satisfied by .
More generally, we say that a -structure is a model of a theory over , if for every . When is a model of , we say that satisfies , or that is satisfied by , and is written
Then it is easy to see that any model of is a semigroup, and vice versa.
Next, let , where is a constant symbol, and
Then is a model of iff is a group. Clearly any group is a model of . To see the converse, let be a model of and let be the interpretation of and be the interpretation of . Let us write for the product . For any , let such that and such that . Then , so that . This shows that is the identity of with respect to . In particular, , which implies , or that is a inverse of with respect to .
Remark. Let be a theory. A class of -structures is said to be axiomatized by if it is the class of all models of . is said to be the set of axioms for this class. This class is necessarily unique, and is denoted by . When consists of a single sentence , we write .
|Date of creation||2013-03-22 13:00:14|
|Last modified on||2013-03-22 13:00:14|
|Last modified by||CWoo (3771)|