Abstract Algebra

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14 domain-specific abstractions whose origin domain is Abstract Algebra.

• Additive inverse — For an element in an additive algebraic structure, another element whose sum with it is the additive identity.
• Automorphism Group — The group obtained by collecting every structure-preserving self-isomorphism of a fixed mathematical object and using composition as the group operation.
• Bialgebra — A vector space carrying compatible unital associative algebra and counital coassociative coalgebra structures, so multiplication and unit are coalgebra maps equivalently comultiplication and counit are algebra maps.
• Bicomplex Number — Extend complex arithmetic with a second commuting imaginary unit, producing a four-real-dimensional commutative algebra whose idempotent decomposition reveals two coupled complex components and zero divisors.
• Commutative magma — A set with a closed binary operation satisfying commutativity but not necessarily associativity, identity or inverses.
• Direct sum of groups — A group assembled from mutually commuting normal subgroups with trivial intersections so every element decomposes uniquely into component elements, with finite support in infinite families.
• Free monoid — The monoid of all finite words over an alphabet under concatenation with the empty word as identity, characterized by unique extension of every generator map to a monoid homomorphism.
• Loop (Algebra) — A quasigroup with a two-sided identity: multiplication has uniquely solvable left and right division without requiring associativity.
• Module (Algebra) — An additive abelian group equipped with a compatible left or right action by a ring, generalizing vector spaces from field scalars to ring scalars.
• Partial groupoid — A set equipped with a binary operation that is defined only for a specified subset of ordered pairs.
• Polynomial Ring — Adjoin one or more algebraically free commuting indeterminates to a coefficient ring, with finite coefficient support and the universal substitution property.
• Semisimple module — A module that is a direct sum of simple submodules, equivalently one in which every submodule has a complementary submodule.
• Splitting field — Form the field extension generated by all roots of a polynomial so that it factors completely into linear terms, with minimality and uniqueness understood relative to the base field.
• Subquotient — Obtain an algebraic object by first selecting a subobject and then quotienting it by a compatible normal subobject or congruence.