Universal algebra

Universal algebra studies sets equipped with operations and the identities those operations satisfy. It provides a common framework for comparing structures such as groups, rings, and lattices.

Algebraic signature: A specification of operation symbols and their arities for an algebraic structure. Signatures state which operations a universal-algebra structure is allowed to have.
Term algebra: The algebra of formal terms generated from variables using a fixed algebraic signature. It supplies a free starting point for imposing identities on all structures of a type.
Group theory: The study of groups and their properties, including symmetry and group actions. Groups are a central example of structures specified by operations and identities.
Model theory: The branch of mathematical logic that studies the relationship between formal languages and their interpretations. Model theory allows arbitrary first-order conditions, while universal algebra centers on operations and identities.
Algebraic structure: A set equipped with specified operations and relations satisfying chosen axioms. Universal algebra treats familiar algebraic structures as instances of this general form.
Free algebra: An algebra generated by a set with no relations beyond those required by its defining variety. Free algebras capture the consequences of a signature and identities without extra constraints.
Lattice theory: The study of partially ordered sets in which every pair has a meet and a join. Lattices illustrate how order-theoretic structures fit the same equational framework.
Category theory: A mathematical framework organized around objects, morphisms, and their composition. Category theory emphasizes maps and structural relationships rather than equational presentations of algebras.
Identity (mathematics): An equation asserted to hold for every assignment of its variables in a specified structure. Identities are the axioms universal algebra uses to characterize classes of structures.
Quotient algebra: An algebra formed by identifying elements according to a congruence relation. Identities can be enforced by quotienting term algebras by the congruence they generate.

Linked from 8 pages