Structure (mathematical logic)

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The concept of structure (English ( first order ) structures) is a fundamental concept of mathematical branches of model theory and universal algebra. A structure is a set, called the universe of the structure, provided with operations on this set. A variety of mathematical structures ( as an informal term ) can be regarded as such a structure, in particular, any algebraic structure and each order structure. An example of a structure provided with the natural numbers of addition, multiplication and comparison. In the model theory structures are sometimes also called models.

• 6.1 Redukte and expansions
• 6.2 substructures
• 6.3 Products, groups and quotients

Definition

A structure (called the universe, the basic region or carrier of ) a lot equipped with

• Functions for an arbitrary natural number, to each of an index set
• And - place relations for an arbitrary natural number, to each of an index set,

Can therefore be defined as triple. A zero- ary function is of a constant. A zero- ary relation is either or and can be interpreted as the truth, verum or falsum

The respective functions and relations can be represented by the symbols of a suitable set of symbols and signatures. The similarity type or type of the structure is given by a function which clearly assigns each character in the arity of the associated functions, and relations. However, the type can also be easily identified by the family of all arities. A structure with the signature is called short structure. If a structure contains no relations, so it is called algebraic structure, it does not contain any functions, whereas relational structure.

Variants

Sometimes the definition is modified in the following ways:

• It should be explicitly counted constants.
• Zero digit relations are excluded or explicitly added counted.
• The index sets must therefore be well-ordered ordinal.

Relation to logic

The model theory examines the relationship between logical formulas and structures for which such formulas in a certain sense to be defined apply. The structures presented here are examined in particular in relation to the first order predicate logic. Predicate logic formulas are interpreted as elements of an elementary language, which permits the use of certain function and relation symbols of fixed arity in the formulas. This information is called the signature of the language. Does this match the signature of a structure, so the structure can be regarded as interpretation of the formula. Under this interpretation, the formula is replaced by certain rules a truth value ( the respective functions or relations for the function and relation symbols are spoken informally used ). Is this the verum, this means the interpretation model of the formula.

Special cases

In many cases limited to relational structures is possible. Each binary function can be regarded as - place relation. The same is true for partial functions. Heterogeneous algebras can be understood as relational structures: Each basic set is interpreted as a unary relation on the union of the basic quantities. However, it change the homomorphism and Substrukturbegriffe. However, the respective characteristics (function, partial function, etc.) are definable in the first order predicate logic. Thus, considerations can be about respect axiomatization, restrict elementary equivalence, satisfiability and decidability relational structures. The notion of elementary substructure does not change. Algebraic structures in contrast, form an important special case, which is examined in particular in the universal algebra. Using defined by equation logic classes of algebraic structures here can be far-reaching statements make than in the general model theory of first-order predicate logic. If a structure contains only zero binary relations, it is called propositional interpretation. Such structures allow for a model-theoretic view of propositional logic.

Examples

Consider a signature consisting of an index set and an index set. and may the arity possess, and against the arity. have the arity.

The structure of the natural numbers consists of the set of natural numbers, wherein the index or symbol is assigned to the natural numbers, the addition, the multiplication of the integers, the constant, the constant and the comparison.

Analog can be defined in the same signature as the structures of the integers or rational numbers with their known links.

Homomorphisms

Construction of derived structures

Redukte and expansions

Omission of relations or functions can be derived from a structure, a new structure form: Is a structure with signature and so there exists a unique structure with signature with the same universe as that matches up with, called reduct of. Conversely, let structures grow by additional relations or functions. Is a reduct of so called expansion. A common special case in the model theory is the expansion constants.

Substructures

A sub- structure or substructure with universe of a structure with universe is a structure with the same signature as, so that the relations and functions in yield by limiting the relations and functions in the universe. For relational structures exist for every subset a unique induced subtree with this universe. For general structures this is not necessarily the case, since not every subset of the universe must be closed under the functions of the structure. The substructures of a structure form an algebraic envelope system.

In the model theory play as a special case of elementary substructures a central role.

Products, groups and quotients

From a family of structures is the direct product ( Cartesian product ) can ( briefly ) form as a structure over the Cartesian product of the universes as the universe, so for relation symbols apply: (functions are interpreted as relations, this product is but in this case again a function ). This construction yields a product in the sense of category theory in the category of structures over the given signature with homomorphisms as morphisms arbitrary.

For relational structures can be a disjoint union of a family defined by forming the set-theoretic disjoint unions of the universes and of the respective relations, which is the disjoint union of relations is identified for any obvious way with a relation on the disjoint union of the universes. This provides a coproduct in the above category.

Nor can a quotient of a relational structure with respect to an equivalence relation form. Universe form the equivalence classes of which relations are defined by. The canonical surjection provides a homomorphism. Conversely, any homomorphism as the core of an equivalence relation. Claims to the homomorphism such that there is a strong homomorphism, can be translated into claims on the corresponding equivalence relation. Compare also the increased demand for a congruence in the algebraic case.

As a special quotients of direct products, Ultra products result.