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Knowledge articles across every category of the library.
887 items · page 2 of 37
Article7 Aug 2026The Definition of an Algebra and its TypeThe central definition of the subject: an algebra is a set with a family of finitary operations indexed by a type. Everything that follows is an elaboration of this one idea.3 min readRead MoreArticle7 Aug 2026The First Two Finite Basis TheoremsTwo results giving conditions under which a variety has a finite equational basis, and the general shape of finite basis arguments.3 min readRead MoreArticle7 Aug 2026Subdirectly Irreducible AlgebrasThe algebras that admit no non-trivial subdirect decomposition. They are characterised by a single lattice-theoretic condition and serve as the atoms of the structure theory.3 min readRead MoreArticle7 Aug 2026Sublattices and Lattice IsomorphismSubsets of a lattice that are lattices in their own right under the inherited operations, the difference between a sublattice and a sub-poset that happens to be a lattice, and the classification of lattices up to isomorphism.3 min readRead MoreArticle7 Aug 2026Subuniverses and the Generation Operator SgThe operator Sg that produces the smallest subuniverse containing a given set, its two equivalent descriptions, and the finitary character that makes it an algebraic closure operator.3 min readRead MoreArticle7 Aug 2026Term Operations and Polynomial OperationsThe functions terms induce on an algebra, the polynomials obtained by allowing parameters, and the clones these families form.3 min readRead MoreArticle7 Aug 2026Terms and the Term Algebra T(X)Terms as formal expressions built from variables and operation symbols, the term algebra they form, and its absolute freeness.3 min readRead MoreArticle7 Aug 2026The Boolean Algebra / Boolean Ring CorrespondenceThe term-equivalence between Boolean algebras and Boolean rings: each structure's operations are term operations of the other, so the two varieties are the same variety in different notation.3 min readRead MoreArticle7 Aug 2026The Center of an AlgebraThe centre of a general algebra, defined by a term condition generalising the group centre, and its use in characterising modules up to polynomial equivalence.3 min readRead MoreArticle7 Aug 2026The Classification of Varieties and Tame Congruence TheoryThe programme of classifying locally finite varieties by the local structure of their finite members, developed after the source text.3 min readRead MoreArticle7 Aug 2026Steiner Triple Systems as AlgebrasSteiner triple systems recast as algebras, so that combinatorial questions about them become questions about varieties and congruences.3 min readRead MoreArticle7 Aug 2026Stone Duality for Boolean AlgebrasThe full categorical duality between Boolean algebras and Boolean spaces: objects correspond, morphisms correspond with reversed direction, and every construction on one side has a counterpart on the other.2 min readRead MoreArticle7 Aug 2026Structure Theory and Finite Basis DevelopmentsAdvances in the structure theory of varieties and in the finite basis problem after the source's period.3 min readRead MoreArticle7 Aug 2026Subalgebras and Algebra IsomorphismSubalgebras as subsets closed under the operations, the notion of embedding, and isomorphism as the equivalence under which algebras are classified.3 min readRead MoreArticle7 Aug 2026Subdirect Products and Subdirect EmbeddingsSubalgebras of a direct product that project onto every factor. The construction is weaker than a direct product but available everywhere, and it is the decomposition the subject actually uses.3 min readRead MoreArticle7 Aug 2026Semigroups, Monoids and Quasigroups as AlgebrasThe one-operation structures and the divisibility structures, presented as algebras. Quasigroups in particular require a careful choice of type, and that choice illustrates a general principle.3 min readRead MoreArticle7 Aug 2026Semisimple and Directly Representable VarietiesVarieties whose subdirectly irreducible members are all simple, and those whose finite members are all direct products from a fixed finite list.3 min readRead MoreArticle7 Aug 2026Set-Theoretic PreliminariesThe set-theoretic apparatus the subject actually uses: classes as well as sets, indexed families, direct products and powers, and the specific conventions that differ from ordinary practice.3 min readRead MoreArticle7 Aug 2026Simple Algebras and Congruence SimplicityAlgebras whose only congruences are the two trivial ones. Simplicity is the strongest indecomposability condition and appears throughout the structure theory of varieties.3 min readRead MoreArticle7 Aug 2026Sizes of Subdirectly Irreducible AlgebrasBounding how large the subdirectly irreducible members of a variety can be, and the compactness arguments that produce the bounds.3 min readRead MoreArticle7 Aug 2026Skew-Free Algebras and IndependenceAlgebras whose congruences on a product decompose into products of congruences on the factors, and the independence conditions that force this.3 min readRead MoreArticle7 Aug 2026Squags and SloopsThe two varieties of algebras associated with Steiner triple systems — one idempotent, one with a distinguished element — and the relationship between them.3 min readRead MoreArticle7 Aug 2026Principal and Generated CongruencesThe congruence generated by a set of pairs, the principal congruences generated by a single pair, and the reason principal congruences are the compact building blocks of Con(A).3 min readRead MoreArticle7 Aug 2026Quasigroups, Loops and Latin SquaresThe identification of finite quasigroups with Latin squares, and the algebraic vocabulary this supplies for a purely combinatorial object.2 min readRead More