Mathematical logic | Unification (computer science) | Universal algebra | Free algebraic structures

Term algebra

In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. For example, in a signature consisting of a single binary operation, the term algebra over a set X of variables is exactly the free magma generated by X. Other synonyms for the notion include absolutely free algebra and anarchic algebra. From a category theory perspective, a term algebra is the initial object for the category of all X-generated algebras of the same signature, and this object, unique up to isomorphism, is called an initial algebra; it generates by homomorphic projection all algebras in the category. A similar notion is that of a Herbrand universe in logic, usually used under this name in logic programming, which is (absolutely freely) defined starting from the set of constants and function symbols in a set of clauses. That is, the Herbrand universe consists of all ground terms: terms that have no variables in them. An atomic formula or atom is commonly defined as a predicate applied to a tuple of terms; a ground atom is then a predicate in which only ground terms appear. The Herbrand base is the set of all ground atoms that can be formed from predicate symbols in the original set of clauses and terms in its Herbrand universe. These two concepts are named after Jacques Herbrand. Term algebras also play a role in the semantics of abstract data types, where an abstract data type declaration provides the signature of a multi-sorted algebraic structure and the term algebra is a concrete model of the abstract declaration. (Wikipedia).

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Related pages

Initial algebra | Quantifier elimination | Algebraic structure | Predicate (mathematical logic) | Logic programming | Isomorphism | Jacques Herbrand | Clause (logic) | Clone (algebra) | Universal algebra | Arity | Atomic formula | Abstract data type | Term (logic) | Universe (mathematics) | F-algebra | Infix notation | Category theory | Signature (logic) | Domain of discourse | Mathematical logic | Binary operation | String (computer science)