term algebra

Term algebras can be shown decidable using quantifier elimination.

Consider, for simplicity, a term algebra, that is, a collection of free variables, constants, and operators which may be freely combined.

In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature.

The signature (Sort and Ops in the example below) which gives the valid constants and operations of the term algebra.

Given a signature for the function symbols, the set of all possible terms that can be freely generated from the constants, variables and functions form a term algebra.

Those tokens are elements of a term algebra built upon the signature of the AADT(in the example, terms that represent either a philosopher or a fork).

An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely free algebra") divided by the equivalence relations generated by a set of identities.

Because of the ubiquity of associative algebras, and because many textbooks teach more associative algebra than nonassociative algebra, it is common for authors to use the term algebra to mean associative algebra.

Loosely speaking, they can be understood as a kind of memoization for free variables; that is, as an optimization technique for rapidly locating the free variables in a term algebra or in a lambda expression.

Although the above definition is formulated in terms of a term algebra, the general concept applies more generally, and can be defined both for combinatory algebras and for lambda calculus proper, specifically, within the framework of explicit substitution.

In mathematics, the Persian mathematician Muhammad ibn Musa al-Khwarizmi gave his name to the concept of the algorithm, while the term algebra is derived from al-jabr, the beginning of the title of one of his publications.

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, anarchic algebra.

From a category theory perspective, a term algebra is the initial object for the category of all 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.

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.

In contrast to string rewriting systems, whose objects are flat sequences of symbols, the objects of a term rewriting system form a term algebra, which can be visualized as a tree of symbols, the structure of the tree fixed by the signature used to define the terms.