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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Outline of algebraic structures | 2/3 | https://en.wikipedia.org/wiki/Outline_of_algebraic_structures | reference | science, encyclopedia | 2026-05-05T08:12:44.195968+00:00 | kb-cron |
Semiring: a ringoid such that S is a monoid under each operation. Addition is typically assumed to be commutative and associative, and the monoid product is assumed to distribute over the addition on both sides, and the additive identity 0 is an absorbing element in the sense that 0 x = 0 for all x. Near-ring: a semiring whose additive monoid is a (not necessarily abelian) group. Ring: a semiring whose additive monoid is an abelian group. Commutative ring: a ring in which the multiplication operation is commutative. Division ring: a nontrivial ring in which division by nonzero elements is defined. Integral domain: A nontrivial commutative ring in which the product of any two nonzero elements is nonzero. Field: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element). Nonassociative rings: These are like rings, but the multiplication operation need not be associative. Lie ring: a ringoid whose additive monoid is an abelian group, but whose multiplicative operation satisfies the Jacobi identity rather than associativity. Jordan ring: a commutative nonassociative ring that respects the Jordan identity Boolean ring: a commutative ring with idempotent multiplication operation. Kleene algebras: a semiring with idempotent addition and a unary operation, the Kleene star, satisfying additional properties. *-algebra or -ring: a ring with an additional unary operation () known as an involution, satisfying additional properties. Arithmetic: addition and multiplication on an infinite set, with an additional pointed unary structure. The unary operation is injective successor, and has distinguished element 0. Robinson arithmetic. Addition and multiplication are recursively defined by means of successor. 0 is the identity element for addition, and annihilates multiplication. Robinson arithmetic is listed here even though it is a variety, because of its closeness to Peano arithmetic. Peano arithmetic. Robinson arithmetic with an axiom schema of induction. Most ring and field axioms bearing on the properties of addition and multiplication are theorems of Peano arithmetic or of proper extensions thereof. Lattice-like structures have two binary operations called meet and join, connected by the absorption law.
Latticoid: meet and join commute but need not associate. Skew lattice: meet and join associate but need not commute. Lattice: meet and join associate and commute. Complete lattice: a lattice in which arbitrary meet and joins exist. Bounded lattice: a lattice with a greatest element and least element. Complemented lattice: a bounded lattice with a unary operation, complementation, denoted by postfix ⊥. The join of an element with its complement is the greatest element, and the meet of the two elements is the least element. Modular lattice: a lattice whose elements satisfy the additional modular identity. Distributive lattice: a lattice in which each of meet and join distributes over the other. Distributive lattices are modular, but the converse does not hold. Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation. This can be shown to be equivalent with the ring-like structure of the same name above. Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by the infix operator →, and governed by the axioms: x → x = 1 x (x → y) = x y y (x → y) = y x → (y z) = (x → y) (x → z)
=== Module-like structures on two sets === The following module-like structures have the common feature of having two sets, A and B, so that there is a binary operation from A×A into A and another operation from A×B into A. Modules, counting the ring operations, have at least three binary operations.
Group with operators: a group G with a set Ω and a binary operation Ω × G → G satisfying certain axioms. Module: an abelian group M and a ring R acting as operators on M. Usually M is defined as "over R". The members of R are sometimes called scalars, and the binary operation of scalar multiplication is a function R × M → M, which satisfies several axioms. Vector spaces: A module where the ring R is a division ring or a field. Graded vector spaces: Vector spaces which are equipped with a direct sum decomposition into subspaces or "grades". Quadratic space: a vector space V over a field F with a quadratic form on V taking values in F. Other special types of modules, including free modules, projective modules, injective modules and flat modules are studied in abstract algebra.
=== Algebra-like structures on two sets === These structures are defined over two sets, a ring R and an R-module M equipped with an operation called multiplication. This can be viewed as a system with five binary operations: two operations on R, two on M and one involving both R and M. Many of these structures are hybrid structures of the previously mentioned ones.