Partial Magma

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Semigroupoid

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Small Category

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Groupoid

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• 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

groupoid can be seen as a:

Magma

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Commutative Magma

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Quasigroup

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is a magma whose elements are invertible

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Unital Magma

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Loop

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is a quasigroup with an identity element

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Semigroup

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is a magma whose binary operation is associative

Semilattice

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is a semigroup whose binary operation is commutative and idempotent

Inverse Semigroup
Associative Quasigroup

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is a semigroup whose elements are invertible

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Monoid

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is a semigroup with an identity element

Commutative Monoid

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is a monoid whose binary operation is also commutative

Group

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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

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is a group where the binary operation is commutative

Semiring

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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 ✅ to be a semigroup under the multiplicative operation, that is, there need not be a multiplicative identity element.

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Ring

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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)

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Unitary Ring

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TODO

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Field

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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)

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