Tag
abstract algebra
62 resources tagged “abstract algebra” across the knowledge library.
62 items · page 1 of 3
Guide6 Aug 2026Algebras, Types and SignaturesAlgebras of arbitrary type: signatures, arities, the significance of nullary operations, and how the choice of type determines subalgebras, homomorphisms and the whole subsequent theory.4 min readRead MoreGuide6 Aug 2026Boolean Algebras: Axioms and StructureBoolean algebras as an equational class, the two-element algebra as the unique subdirectly irreducible member, atoms and atomlessness, and why the variety is arithmetical.6 min readRead MoreGuide6 Aug 2026Boolean PowersBoolean powers A[B]*, their construction as locally constant functions on a Stone space, the identities they preserve, and filtered Boolean powers as the refinement that carries the decidability results.6 min readRead MoreGuide6 Aug 2026Boolean ProductsBoolean products as subdirect products over a Boolean space with a patching condition, the relationship to sheaves, the equaliser condition, and the classes of algebras admitting such representations.5 min readRead MoreGuide6 Aug 2026Boolean Rings and the Boolean Algebra-Ring CorrespondenceBoolean rings, the mutual translation with Boolean algebras, term equivalence as the precise relationship, and why the ring picture makes ideals and the prime spectrum available.6 min readRead MoreGuide6 Aug 2026Class Operators H, S, P and the Definition of a VarietyThe class operators I, S, H, P and P_S, the inclusions and idempotency relations among their composites, and the identification of HSP as the variety-generating closure operator.5 min readRead MoreGuide6 Aug 2026Closure Operators and Galois ConnectionsClosure operators and closure systems, the correspondence with complete lattices, the finitary case and algebraic lattices, and Galois connections as the standard source of closure operators in algebra and logic.5 min readRead MoreGuide6 Aug 2026Complete Lattices and Algebraic LatticesComplete lattices, algebraic lattices, compact elements, and the theorem that a lattice is algebraic exactly when it is the lattice of closed sets of a finitary closure operator.5 min readRead MoreGuide6 Aug 2026Congruences and Quotient AlgebrasCongruences as equivalence relations compatible with the operations, the construction of quotient algebras, and why congruences rather than subobjects are the general quotient device.5 min readRead MoreGuide6 Aug 2026Direct Products, Factor Congruences and Direct IndecomposabilityDirect products, projection homomorphisms, factor congruences as the internal signature of a decomposition, directly indecomposable algebras and the limits of unique factorisation.4 min readRead MoreGuide6 Aug 2026Directly Representable VarietiesDirectly representable varieties, McKenzie's theorem that they are congruence-permutable, the classification of their directly indecomposable members, and the connection to the (discriminator) ⊗ (modular Abelian) decomposition.5 min readRead MoreGuide6 Aug 2026Discriminator VarietiesThe ternary discriminator term, discriminator varieties, the Bulman-Fleming–Keimel–Werner representation theorem, and the exceptional package of structural properties that follows.5 min readRead MoreGuide6 Aug 2026Distributive and Modular Lattices: the M5 and N5 CriteriaDistributive and modular lattices, the self-duality of both conditions, and the forbidden-sublattice theorems that characterise them by the non-embeddability of M5 and N5.4 min readRead MoreGuide6 Aug 2026Elementary Substructures and the Lowenheim-Skolem TheoremsElementary substructures and extensions, the Tarski–Vaught test, the downward and upward Löwenheim–Skolem theorems, and the Skolem paradox.6 min readRead MoreGuide6 Aug 2026Equational Logic and the Completeness TheoremThe formal system of equational logic, its five inference rules, soundness, and Birkhoff's completeness theorem identifying derivability with semantic consequence.5 min readRead MoreGuide6 Aug 2026Equivalence Relations and the Partition LatticeThe lattice of equivalence relations on a set, its identification with the partition lattice, why joins require alternating composites, and permutability as the condition that makes joins easy.5 min readRead MoreGuide6 Aug 2026Filters, Ideals and UltrafiltersFilters and ideals on a Boolean algebra, principal and free filters, ultrafilters and their three equivalent characterisations, and the role of ultrafilters throughout the rest of the subject.6 min readRead MoreGuide6 Aug 2026Finite State Acceptors and Recognisable LanguagesFinite state acceptors as unary algebras, recognisable languages, the Myhill–Nerode congruence, the minimal automaton as a quotient, and the closure properties that follow.6 min readRead MoreGuide6 Aug 2026First-order Languages and StructuresFirst-order languages with relation and operation symbols, structures as the semantic objects, the syntax of terms and formulas, free and bound variables, and how algebras sit inside the wider class of structures.5 min readRead MoreGuide6 Aug 2026Free Algebras and the Universal Mapping PropertyFree algebras over a class, their construction as quotients of the term algebra, the universal mapping property, and the fact that F_K(X) lies in the variety generated by K.6 min readRead MoreGuide6 Aug 2026Fully Invariant Congruences and Equational TheoriesEquational theories as fully invariant congruences on the term algebra, the dual isomorphism with the lattice of varieties, and equational bases.5 min readRead MoreGuide6 Aug 2026Functionally Complete AlgebrasFunctionally complete algebras, the polynomial clone, the characterisation via simplicity and the discriminator as a polynomial, and the contrast with primality.5 min readRead MoreGuide6 Aug 2026Homomorphisms and the Isomorphism TheoremsHomomorphisms, kernels, the first, second and third isomorphism theorems in arbitrary type, the correspondence theorem, and the congruence extension property.4 min readRead MoreGuide6 Aug 2026Identities and Birkhoff's HSP TheoremIdentities, satisfaction, the operators M and Id as a Galois connection, and Birkhoff's HSP theorem identifying varieties with equationally definable classes, with proof.5 min readRead More