Structure

A synonym for algebraic system. Similarly, substructure is a synonym for "subsystem" , and elementary substructure is a synonym for "elementary subsystem" (cf. Model theory). If is a first-order language of signature and is an algebraic system of signature , then is called an -structure.

A first-order language is uniquely determined by its signature (cf. Algebraic system). Therefore, in a more informal way of writing, many authors describe by just giving the set of function symbols (also called "functionals" ) and relation symbols (also called "predicates" ). Here, one omits the mapping , which assigns to every symbol its arity, that is, the number of variables of the function or relation which interprets the symbol in a structure. This is justified when the arity can be read off from the symbol by convention. It is even common to write . For example, is called the language of groups (additively written), is the language of ordered groups, is the language of rings with , and is the language of ordered rings. Here, "+" and "" are binary function symbols, "-" is a unary function symbol (for additive inversion), "<" is a binary relation symbol, and "0" and "1" are constant symbols. A constant symbol, belonging to , is a -ary function symbol, but often constant symbols are considered separately. The field axioms can be formulated in the language of rings, but one can also use the language of fields obtained from by adjoining a unary function symbol for multiplicative inversion.

If the language is given by a set of relation symbols, function symbols and constant symbols, then an -structure is a set , called the universe of the structure , together with relations and functions on and elements of (constants) which interpret the corresponding symbols of . Often, is identified with ; for example, may denote the set of real numbers as well as the field of real numbers. In this spirit, an element of is also called an element of , and the cardinality of a structure is defined to be the cardinality of its universe.

If is a language containing (as sets of symbols) and is an -structure such that and the interpretations of the relation, function and constant symbols of are the same in as in , then is called an expansion of , and is called a reduct of . For example, a field is an expansion of the underlying additive group, and this group is a reduct of the field.

If , then to add a name for every element in to the language, one passes to the expansion of , which has signature (cf. Model theory; Elimination of quantifiers).