Algebraic structures

algebraic set with an invertible, associative internal operation admitting a neutral element

algebraic structure that has compatible structures of an abelian group and a monoid, in particular having multiplicative identity

commutative ring in which every nonzero element is inversible

set equipped with one or more finitary operations defined on it

In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the natural numbers with addition form a monoid, the identity element being .

additive subgroup of a ring closed under multiplcation by arbitrary ring element

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. In mathematical analysis, the term also appears in the theory of one-parameter operator semigroups: see C0-semigroup.

generalization of vector space, with scalars in a ring instead of a field

branch of abstract algebra in mathematics

algebraic structure of objects and morphisms between objects, which can be associatively composed if the (co)domains agree

algebraic structure

partially ordered set that admits greatest lower and least upper bounds of any two elements

algebraic structure with a binary operation

branch of mathematics that studies representations of abstract algebraic structures as linear transformations on vector spaces

lattice that models the classical propositional logic

In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices.

mathematical element

A commutative group where every element is the sum of elements from one finite subset

algebraic structure similar to ring but not necessarily having a multiplicative identity

Abelian group constructed universally from a commutative monoid

unital ring with no zero divisors other than 0; noncommutative generalization of integral domains

In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.

group of which the group operation is to be thought of as addition

ring of all matrices of fixed size

several notions

a set equipped with a choice of a specific element

algebra of the operations on a logical theory up to equivalence

idempotent semiring endowed with a closure operator

In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.

In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a small category, except possibly for the requirement that there be an identity at each object. Semigroupoids generalise semigroups in the same way that small categories generalise monoids and groupoids generalise groups. Semigroupoids have applications in the structural theory of semigroups.

regular semigroup in which every element has a unique inverse

semigroup such that, for every element x, there exists another element y such that xyx=x

check digit algorithm presented by H. Michael Damm