Algebraic Operations & Abstract Systems¶

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Abstractions about algebraic systems defined by operations, identities, inverses, homomorphisms, order, and representation. They include rings, fields, magmas, operator algebras, operads, lattices, formal series, and generalized associative or commutative structures.

32 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

Absolute value (algebra) — A nonnegative multiplicative or submultiplicative magnitude function on a field or integral domain that separates zero and satisfies the triangle inequality.
Absorbing element — An element that returns itself whenever combined with any element under a specified binary operation.
Action algebra — An algebraic-logic structure combining Kleene algebra's composition and iteration with residuated semilattice implication operations.
Additive inverse — For an element in an additive algebraic structure, another element whose sum with it is the additive identity.
Anticommutative property — A binary-operation property in which exchanging the two arguments yields the additive inverse of the original result.
Calkin algebra — The C*-algebra obtained by quotienting all bounded operators on an infinite-dimensional separable Hilbert space by the ideal of compact operators.
Cancellation property — An algebraic property allowing a common left or right factor to be removed from an equality, even when no inverse element is available.
Commutative magma — A set with a closed binary operation satisfying commutativity but not necessarily associativity, identity or inverses.
Congruence lattice problem — The representation problem asking which distributive algebraic lattices occur as congruence lattices of lattices.
Dirichlet algebra — A closed unital subalgebra of continuous functions on a compact Hausdorff space whose real parts are uniformly dense in all continuous real-valued functions.
E∞-operad — An operad whose spaces of n-ary operations are contractible with suitably free symmetric-group actions, encoding multiplication associative and commutative up to coherent higher homotopies.
Filtered algebra — An algebra equipped with a nested exhaustive sequence of subspaces whose multiplication sends filtration levels p and q into level p+q.
Finite lattice representation problem — The question whether every finite lattice is isomorphic to the congruence lattice of some finite algebra.
Formal power series — An infinite coefficient sequence manipulated as an algebraic series in an indeterminate, without any requirement that numerical substitution converge.
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.
Homomorphism — A map between algebraic structures of the same signature that preserves each distinguished operation and constant.
Hurwitz problem — The problem of determining when sums-of-squares quadratic forms admit bilinear multiplicative composition formulas.
Jordan operator algebra — A normed Jordan algebra modeled on self-adjoint operators with the symmetrized product a circle b equal to one half of ab plus ba.
Lindenbaum–Tarski algebra — The quotient algebra of formulas or sentences of a logical theory by provable equivalence, with logical connectives inducing well-defined algebraic operations on equivalence classes.
Locally nilpotent — A local finiteness condition under which every finitely generated subobject is nilpotent, or an ideal becomes nilpotent after localization at a specified prime.
Necklace ring — A ring on infinite sequences over a commutative ring whose multiplication combines indices by least common multiple and weights products by greatest common divisor.
Nuclear C*-algebra — A C-algebra whose algebraic tensor product with every C-algebra has a unique C*-norm, equivalently whose identity approximately factors through matrix algebras by completely positive maps.
Ordered field — A field with a total order preserved by addition and multiplication by positive elements.
Partial groupoid — A set equipped with a binary operation that is defined only for a specified subset of ordered pairs.
Quasi-Hopf algebra — A quasi-bialgebra equipped with an antipode and distinguished elements whose identities replace the strict Hopf antipode laws in the presence of a nontrivial associator.
Quasifield — A nonassociative division-like algebra whose additive structure is a group and whose multiplication supports division while satisfying only selected distributive laws.
Quasitrace — A positive tracial functional on a C-star algebra that is homogeneous and additive on commuting positive elements but need not be globally additive.
Quasivariety — A class of algebraic structures of a fixed signature axiomatizable by quasi-identities, equivalently closed under isomorphisms, subalgebras, direct products, and ultraproduct or reduced-product conditions under standard formulations.
Subfield of an algebra — An F-subalgebra of an F-algebra that is itself a field, with maximal and strictly maximal variants recording containment and dimension conditions.
Term algebra — The freely generated algebra of formal terms built from variables and operation symbols in a signature, initial among algebras receiving those generators.
Tomita–Takesaki theory — The modular theory deriving a one-parameter automorphism group and commutant duality from a von Neumann algebra with a cyclic separating vector or faithful normal weight.
Total algebra — An algebra of all coefficient functions on a suitably finite-factorization monoid, with convolution multiplication extending the finite-support monoid algebra to infinite formal sums.