Tag
Mathematics
65 resources tagged “Mathematics” across the knowledge library.
65 items · page 2 of 3
Guide6 Aug 2026Lattice Homomorphisms, Isomorphisms and SublatticesLattice homomorphisms, isomorphisms and sublattices: why every lattice homomorphism is monotone but not conversely, how to recognise a genuine sublattice, and the role of embeddings in the M5/N5 characterisation.5 min readRead MoreGuide6 Aug 2026Mal'cev Conditions I: Congruence PermutabilityMal'cev conditions in general, the ternary Mal'cev term characterising congruence permutability, and why term conditions are the right way to classify varieties.5 min readRead MoreGuide6 Aug 2026Mal'cev Conditions II: Congruence Distributivity and Jonsson TermsCongruence-distributive varieties, the Jónsson term characterisation, Jónsson's lemma, and congruence modularity with Day terms.5 min readRead MoreGuide6 Aug 2026Orthogonal Latin Squares and the Refutation of Euler's ConjectureOrthogonal Latin squares, Euler's officer problem, the refutation by Bose, Shrikhande and Parker, and the algebraic reading of orthogonality as a condition on a pair of quasigroup operations.6 min readRead MoreGuide6 Aug 2026Posets and the Two Definitions of a LatticeThe order-theoretic and algebraic definitions of a lattice, the equivalence between them, and why universal algebra needs the algebraic form: only an equationally defined class is a variety.5 min readRead MoreGuide6 Aug 2026Preliminaries: Sets, Classes, Relations and NotationSets, proper classes, relations, kernels and the notational conventions universal algebra shares with model theory: why the collection of all groups cannot be a set, how relation composition is written, and what the symbol ≈ means as against =.5 min readRead MoreGuide6 Aug 2026Preservation Theorems: Horn, Universal and Positive SentencesThe preservation theorems: Łoś–Tarski for substructures, Lyndon for homomorphisms, Keisler–Galvin for reduced products, Chang–Łoś–Suszko for unions of chains, and their relationship to Birkhoff's theorem.5 min readRead MoreGuide6 Aug 2026Primal AlgebrasPrimal algebras, Foster's theorem that they generate varieties equivalent to Boolean algebras, the representation of every member as a Boolean power, and the relationship to functional completeness.5 min readRead MoreGuide6 Aug 2026Principal Congruence FormulasPrincipal congruence formulas, Mal'cev's chain characterisation, when the chains are uniformly bounded, and the application to bounding the size of subdirectly irreducible algebras.5 min readRead MoreGuide6 Aug 2026Quasigroups, Loops and Latin SquaresQuasigroups and loops as equationally defined algebras, the correspondence with Latin squares, why division must be in the type for quasigroups to form a variety, and the congruence properties that follow.6 min readRead MoreGuide6 Aug 2026Quasiprimal Algebras and Pixley's TheoremQuasiprimal algebras, Pixley's characterisation via inner isomorphisms, the relationship to primality, and the varieties quasiprimal algebras generate.5 min readRead MoreGuide6 Aug 2026Recent Developments: the 1981 FrontierThe source's Recent Developments chapter: the commutator programme, the classification of varieties by congruence conditions, decidability questions, Boolean constructions, structure theory, and the applications to computer science and model theory as they stood in 1981.5 min readRead MoreGuide6 Aug 2026Reduced Products and Filtered ProductsReduced products over a filter, the congruence they induce, the direct product and ultraproduct as extreme cases, and the Horn sentence preservation theorem.5 min readRead MoreGuide6 Aug 2026Satisfaction, Truth and Elementary EquivalenceThe satisfaction relation defined by recursion, truth for sentences, theories and models, elementary equivalence, and why elementarily equivalent structures can be non-isomorphic.5 min readRead MoreGuide6 Aug 2026Semantic Embeddings and UndecidabilitySemantic embeddings as a technique for transferring undecidability, the interpretation of graphs and semigroups into varieties, the decidability results that stand in contrast, and the classification the source reports.6 min readRead MoreGuide6 Aug 2026Semisimple VarietiesSemisimple varieties, the relationship to discriminator varieties, residual smallness, and the structure available when subdirect decomposition terminates in simple algebras.4 min readRead MoreGuide6 Aug 2026Skew-free AlgebrasSkew congruences in direct products, skew-free algebras and varieties, the relationship to congruence permutability and distributivity, and the role in Boolean representations.5 min readRead MoreGuide6 Aug 2026Steiner Triple Systems, Squags and SloopsSteiner triple systems recast as algebras: squags (Steiner quasigroups) and sloops (Steiner loops), the equational axioms for each, and what the variety structure delivers that the combinatorial description does not.6 min readRead MoreGuide6 Aug 2026Stone Duality and Boolean SpacesStone duality: the Stone space of ultrafilters, Boolean spaces as compact Hausdorff totally disconnected spaces, the dual equivalence of categories, and how it seeds the Boolean product machinery.6 min readRead MoreGuide6 Aug 2026Subdirect Products and Birkhoff's Subdirect Representation TheoremSubdirect products, subdirect irreducibility, the monolith, and Birkhoff's subdirect representation theorem with its Zorn's lemma proof.4 min readRead MoreGuide6 Aug 2026Subuniverses, Subalgebras and the Generation OperatorSubuniverses, subalgebras, the generation operator Sg(X), and the two equivalent descriptions of generation as intersection from above and as term-closure from below.4 min readRead MoreGuide6 Aug 2026Syntactic Monoids and Kleene's TheoremThe syntactic congruence and syntactic monoid, Kleene's characterisation of recognisable languages as the regular ones, Schützenberger's star-free theorem, and Eilenberg's variety correspondence.6 min readRead MoreGuide6 Aug 2026Tame Congruence TheoryTame congruence theory: minimal sets, the five local types, type sets of varieties, and how the classification refines and largely supersedes the 1981 congruence-condition hierarchy.5 min readRead MoreGuide6 Aug 2026Terms, Term Algebras and Term OperationsTerms as syntactic objects, the term algebra T(X) and its absolute freeness, the distinction between a term and the operation it induces, and why the term algebra is the reference object for equational reasoning.6 min readRead More