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In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets and is a set of ordered pairs consisting of elements from and from . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element is related to an element , if and only if the pair belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case of an -ary relation over sets , which is a subset of the Cartesian product
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
In mathematics, the concept of an inverse element generalises the concepts of opposite and reciprocal of numbers.
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading.
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed.
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements.
In abstract algebra, a semiring is an algebraic structure. It is a generalization of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices.
In mathematics, an aperiodic semigroup is a semigroup S such that every element is aperiodic, that is, for each x in S there exists a positive integer n such that xn = xn+1. An aperiodic monoid is an aperiodic semigroup which is a monoid.
In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup is associated with the composite of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are acting as transformations of the set. From an algebraic perspective, a semigroup action is a generalization of the notion of a group action in group theory. From the computer science point of view, semigroup actions are closely related to automata: the set models the state of the automaton and the action models transformations of that state in response to inputs.
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.
In algebra, a presentation of a monoid is a description of a monoid in terms of a set Σ of generators and a set of relations on the free monoid Σ∗ generated by Σ. The monoid is then presented as the quotient of the free monoid by these relations. This is an analogue of a group presentation in group theory.
In mathematics, a regular semigroup is a semigroup S in which every element is regular, i.e., for each element a in S there exists an element x in S such that axa = a. Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations.
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation R; S from two given binary relations R and S. In the calculus of relations, the composition of relations is called relative multiplication, and its result is called a relative product. Function composition is the special case of composition of relations where all relations involved are functions.
In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed.
In mathematics, a near-semiring, also called a seminearring, is an algebraic structure more general than a near-ring or a semiring. Near-semirings arise naturally from functions on monoids.
In algebra, a transformation semigroup is a collection of transformations that is closed under function composition. If it includes the identity function, it is a monoid, called a transformationmonoid. This is the semigroup analogue of a permutation group.
In mathematics, a null semigroup is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero. If every element of a semigroup is a left zero then the semigroup is called a left zero semigroup; a right zero semigroup is defined analogously.
References
- Howie, John M. (1995). Fundamentals of Semigroup Theory. Clarendon Press. ISBN 0-19-851194-9.
- Kilp, M.; Knauer, U.; Mikhalev, A.V. (2000), "Monoids, Acts and Categories with Applications to Wreath Products and Graphs", De Gruyter Expositions in Mathematics, 29, Walter de Gruyter, ISBN 3-11-015248-7
- Golan, Jonathan S. (1999). Semirings and Their Applications. Springer. ISBN 0-7923-5786-8.