Algebraic Structures

In abstract algebra, an algebraic structure on a set 𝐴 (called carrier set or underlying set) is a collection of finitary operations on 𝐴. The set 𝐴 with this structure is also called an algebra.

Algebraic Structures - 1 Operator Types

Algebraic Structure Type	Binary Operation Properties					Description
Algebraic Structure Type	Closed	Associativity	Identity	Invertibility	Commutativity	Description
Partial Magma	Unneeded	Unneeded	Unneeded	Unneeded	Unneeded
Semigroupoid	Unneeded	Required	Unneeded	Unneeded	Unneeded
Small Category	Unneeded	Required	Required	Unneeded	Unneeded
Groupoid	Unneeded	Required	Required	Required	Unneeded	Click here to expand... groupoid can be seen as a: Group with a partial function replacing the binary operation; Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation, called inverse by analogy with group theory. Notice that a groupoid where there is only one object is a usual group
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Commutative Magma	Required	Unneeded	Unneeded	Unneeded	Required
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded	is a magma whose elements are invertible
	Required	Unneeded	Unneeded	Required	Required
Unital Magma	Required	Unneeded	Required	Unneeded	Unneeded
	Required	Unneeded	Required	Unneeded	Required
Loop	Required	Unneeded	Required	Required	Unneeded	is a quasigroup with an identity element
	Required	Unneeded	Required	Required	Required
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded	is a magma whose binary operation is associative
Semilattice	Required	Required	Unneeded	Unneeded	Required	is a semigroup whose binary operation is commutative and idempotent
Inverse Semigroup Associative Quasigroup	Required	Required	Unneeded	Required	Unneeded	is a semigroup whose elements are invertible
	Required	Required	Unneeded	Required	Required
Monoid	Required	Required	Required	Unneeded	Unneeded	is a semigroup with an identity element
Commutative Monoid	Required	Required	Required	Unneeded	Required	is a monoid whose binary operation is also commutative
Group	Required	Required	Required	Required	Unneeded	is a monoid whose elements are invertible is a loop whose binary operation is associative is an inverse group with an identity element
Abelian Group	Required	Required	Required	Required	Required	is a group where the binary operation is commutative

Algebraic Structures - 2 Operator Types

Algebraic Structure Type	Binary Operation Properties						Description
Algebraic Structure Type	Closed	Associativity	Identity	Invertibility	Commutativity	Distributivity	Description
Semiring	Required	Required	Required	Unneeded	Required		is similar to a ring, but without the requirement that each element must have an additive inverse while a ring is (algebra) an algebraic structure as above, but only required to be a semigroup under the multiplicative operation, that is, there need not be a multiplicative identity element.
Semiring	Required	Required	Unneeded	Unneeded	Unneeded
Ring	Required	Required	Required	Required	Required		is an abelian group (under addition, say) that happens to have a second closed, associative, binary operation as well. And these two operations satisfy a distribution law. (You may or may not require rings to have an identity with the second operation)
Ring	Required	Required	Unneeded	Unneeded	Unneeded
Unitary Ring	Required	Required	Required	Required	Required		TODO
Unitary Ring	Required	Required	Required	?	?		TODO
Field	Required	Required	Required	Required	Required		is a ring where both operations are commutative, where every element has both an additive inverse (i.e. the first operation) and a multiplicative inverse (i.e. the second operation) (and thus there is a multiplicative identity), and the extra requirement that if xy=0 for some x≠0, then we must have y=0 (we call this having no zero-divisors)
Field	Required	Required	Required	Required	Required

Algebraic Structures - Complex Types

Algebraic Structures - Examples

see: Algebraic Structures - Examples

Algebraic Structures

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Algebraic Structures - 1 Operator Types

Algebraic Structures - 2 Operator Types

Algebraic Structures - Complex Types

Algebraic Structures - Examples

Algebraic Structures