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33 relations: Abstract rewriting system, Algebra, Algorithm, Binary relation, Buchberger's algorithm, Closure (mathematics), Composition of relations, Computational group theory, Confluence (abstract rewriting), Converse relation, Coset, Critical pair (logic), Decidability (logic), Donald Knuth, E theorem prover, Encompassment ordering, Equation, Free abelian group, Generating set of a group, Gröbner basis, Group (mathematics), Jean-Pierre Jouannaud, Newman's lemma, Normal form (abstract rewriting), Polynomial ring, Presentation of a group, Presentation of a monoid, Rewrite order, Rewriting, Shortlex order, Term (logic), Well-order, Word problem (mathematics).
Abstract rewriting system
In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviation ARS) is a formalism that captures the quintessential notion and properties of rewriting systems.
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Algebra
Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.
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Algorithm
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.
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Binary relation
In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.
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Buchberger's algorithm
In computational algebraic geometry and computational commutative algebra, Buchberger's algorithm is a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order.
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Closure (mathematics)
A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.
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Composition of relations
In the mathematics of binary relations, the composition relations is a concept of forming a new relation from two given relations R and S. The composition of relations is called relative multiplication in the calculus of relations.
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Computational group theory
In mathematics, computational group theory is the study of groups by means of computers.
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Confluence (abstract rewriting)
In computer science, confluence is a property of rewriting systems, describing which terms in such a system can be rewritten in more than one way, to yield the same result.
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Converse relation
In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation.
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Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.
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Critical pair (logic)
In mathematical logic, a critical pair arises in term rewriting systems where rewrite rules overlap to yield two different terms.
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Decidability (logic)
In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas, or, more precisely, an algorithm that can and will return a boolean true or false value that is correct (instead of looping indefinitely, crashing, returning "don't know" or returning a wrong answer).
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Donald Knuth
Donald Ervin Knuth (born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University.
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E theorem prover
E is a high performance theorem prover for full first-order logic with equality.
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Encompassment ordering
In theoretical computer science, in particular in automated theorem proving and term rewriting, the containment, or encompassment, preorder (≤) on the set of terms, is defined by It is used e.g. in the Knuth–Bendix completion algorithm.
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Equation
In mathematics, an equation is a statement of an equality containing one or more variables.
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Free abelian group
In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis.
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Generating set of a group
In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.
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Gröbner basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Jean-Pierre Jouannaud
Jean-Pierre Jouannaud is a French computer scientist, known for his work in the area of term rewriting.
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Newman's lemma
In mathematics, in the theory of rewriting systems, Newman's lemma, also commonly called the diamond lemma, states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is, one in which there are no infinite reduction sequences, is confluent if it is locally confluent.
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Normal form (abstract rewriting)
In abstract rewriting, an object is in normal form if it cannot be rewritten any further.
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Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.
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Presentation of a group
In mathematics, one method of defining a group is by a presentation.
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Presentation of a monoid
In algebra, a presentation of a monoid (respectively, a presentation of a semigroup) is a description of a monoid (respectively, a semigroup) in terms of a set of generators and a set of relations on the free monoid (respectively, the free semigroup) generated by.
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Rewrite order
In theoretical computer science, in particular in automated theorem proving and term rewriting, a binary relation (→) on the set of terms is called a rewrite relation if it is closed under contextual embedding and under instantiation; formally: if l→r implies uσp→up for all terms l, r, u, each path p of u, and each substitution σ. If (→) is also irreflexive and transitive, then it is called a rewrite ordering, or rewrite preorder.
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Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of (potentially non-deterministic) methods of replacing subterms of a formula with other terms.
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Shortlex order
In mathematics, and particularly in the theory of formal languages, shortlex is a total ordering for finite sequences of objects that can themselves be totally ordered.
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Term (logic)
In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact.
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Well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.
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Word problem (mathematics)
In mathematics and computer science, a word problem for a set S with respect to a system of finite encodings of its elements is the algorithmic problem of deciding whether two given representatives represent the same element of the set.
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References
[1] https://en.wikipedia.org/wiki/Knuth–Bendix_completion_algorithm
