Engineering↗
Algebras, Types and Signatures
Algebras of arbitrary type: signatures, arities, the significance of nullary operations, and how the choice of type determines subalgebras, homomorphisms and the whole subsequen…
Tag
universal algebra
Every page in the KEVOS library tagged universal algebra. 62 pages.
Engineering↗
Boolean Algebras: Axioms and Structure
Boolean algebras as an equational class, the two-element algebra as the unique subdirectly irreducible member, atoms and atomlessness, and why the variety is arithmetical.
Engineering↗
Boolean Powers
Boolean 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…
Engineering↗
Boolean Products
Boolean products as subdirect products over a Boolean space with a patching condition, the relationship to sheaves, the equaliser condition, and the classes of algebras admittin…
Engineering↗
Boolean Rings and the Boolean Algebra-Ring Correspondence
Boolean 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.
Engineering↗
Class Operators H, S, P and the Definition of a Variety
The 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.
Engineering↗
Closure Operators and Galois Connections
Closure operators and closure systems, the correspondence with complete lattices, the finitary case and algebraic lattices, and Galois connections as the standard source of clos…
Engineering↗
Complete Lattices and Algebraic Lattices
Complete 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.
Engineering↗
Congruences and Quotient Algebras
Congruences as equivalence relations compatible with the operations, the construction of quotient algebras, and why congruences rather than subobjects are the general quotient d…
Engineering↗
Direct Products, Factor Congruences and Direct Indecomposability
Direct products, projection homomorphisms, factor congruences as the internal signature of a decomposition, directly indecomposable algebras and the limits of unique factorisation.
Engineering↗
Directly Representable Varieties
Directly representable varieties, McKenzie's theorem that they are congruence-permutable, the classification of their directly indecomposable members, and the connection to the …
Engineering↗
Discriminator Varieties
The ternary discriminator term, discriminator varieties, the Bulman-Fleming–Keimel–Werner representation theorem, and the exceptional package of structural properties that f…
Engineering↗
Distributive and Modular Lattices: the M5 and N5 Criteria
Distributive 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.
Engineering↗
Elementary Substructures and the Lowenheim-Skolem Theorems
Elementary substructures and extensions, the Tarski–Vaught test, the downward and upward Löwenheim–Skolem theorems, and the Skolem paradox.
Engineering↗
Equational Logic and the Completeness Theorem
The formal system of equational logic, its five inference rules, soundness, and Birkhoff's completeness theorem identifying derivability with semantic consequence.
Engineering↗
Equivalence Relations and the Partition Lattice
The lattice of equivalence relations on a set, its identification with the partition lattice, why joins require alternating composites, and permutability as the condition that m…
Engineering↗
Filters, Ideals and Ultrafilters
Filters and ideals on a Boolean algebra, principal and free filters, ultrafilters and their three equivalent characterisations, and the role of ultrafilters throughout the rest …
Engineering↗
Finite State Acceptors and Recognisable Languages
Finite state acceptors as unary algebras, recognisable languages, the Myhill–Nerode congruence, the minimal automaton as a quotient, and the closure properties that follow.
Engineering↗
First-order Languages and Structures
First-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 i…
Engineering↗
Free Algebras and the Universal Mapping Property
Free 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.
Engineering↗
Fully Invariant Congruences and Equational Theories
Equational theories as fully invariant congruences on the term algebra, the dual isomorphism with the lattice of varieties, and equational bases.
Engineering↗
Functionally Complete Algebras
Functionally complete algebras, the polynomial clone, the characterisation via simplicity and the discriminator as a polynomial, and the contrast with primality.
Engineering↗
Homomorphisms and the Isomorphism Theorems
Homomorphisms, kernels, the first, second and third isomorphism theorems in arbitrary type, the correspondence theorem, and the congruence extension property.
Engineering↗
Identities and Birkhoff's HSP Theorem
Identities, satisfaction, the operators M and Id as a Galois connection, and Birkhoff's HSP theorem identifying varieties with equationally definable classes, with proof.
Engineering↗
Lattice Homomorphisms, Isomorphisms and Sublattices
Lattice homomorphisms, isomorphisms and sublattices: why every lattice homomorphism is monotone but not conversely, how to recognise a genuine sublattice, and the role of embedd…
Engineering↗
Mal'cev Conditions I: Congruence Permutability
Mal'cev conditions in general, the ternary Mal'cev term characterising congruence permutability, and why term conditions are the right way to classify varieties.
Engineering↗
Mal'cev Conditions II: Congruence Distributivity and Jonsson Terms
Congruence-distributive varieties, the Jónsson term characterisation, Jónsson's lemma, and congruence modularity with Day terms.
Engineering↗
Orthogonal Latin Squares and the Refutation of Euler's Conjecture
Orthogonal 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 quasigro…
Engineering↗
Posets and the Two Definitions of a Lattice
The 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 …
Engineering↗
Preliminaries: Sets, Classes, Relations and Notation
Sets, 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 relatio…
Engineering↗
Preservation Theorems: Horn, Universal and Positive Sentences
The preservation theorems: Łoś–Tarski for substructures, Lyndon for homomorphisms, Keisler–Galvin for reduced products, Chang–Łoś–Suszko for unions of chains, and their relation…
Engineering↗
Primal Algebras
Primal 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 fun…
Engineering↗
Principal Congruence Formulas
Principal congruence formulas, Mal'cev's chain characterisation, when the chains are uniformly bounded, and the application to bounding the size of subdirectly irreducible algeb…
Engineering↗
Quasigroups, Loops and Latin Squares
Quasigroups 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 congru…
Engineering↗
Quasiprimal Algebras and Pixley's Theorem
Quasiprimal algebras, Pixley's characterisation via inner isomorphisms, the relationship to primality, and the varieties quasiprimal algebras generate.
Engineering↗
Recent Developments: the 1981 Frontier
The source's Recent Developments chapter: the commutator programme, the classification of varieties by congruence conditions, decidability questions, Boolean constructions, stru…
Engineering↗
Reduced Products and Filtered Products
Reduced products over a filter, the congruence they induce, the direct product and ultraproduct as extreme cases, and the Horn sentence preservation theorem.
Engineering↗
Satisfaction, Truth and Elementary Equivalence
The satisfaction relation defined by recursion, truth for sentences, theories and models, elementary equivalence, and why elementarily equivalent structures can be non-isomorphic.
Engineering↗
Semantic Embeddings and Undecidability
Semantic embeddings as a technique for transferring undecidability, the interpretation of graphs and semigroups into varieties, the decidability results that stand in contrast, …
Engineering↗
Semisimple Varieties
Semisimple varieties, the relationship to discriminator varieties, residual smallness, and the structure available when subdirect decomposition terminates in simple algebras.
Engineering↗
Skew-free Algebras
Skew congruences in direct products, skew-free algebras and varieties, the relationship to congruence permutability and distributivity, and the role in Boolean representations.
Engineering↗
Steiner Triple Systems, Squags and Sloops
Steiner triple systems recast as algebras: squags (Steiner quasigroups) and sloops (Steiner loops), the equational axioms for each, and what the variety structure delivers that …
Engineering↗
Stone Duality and Boolean Spaces
Stone 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…
Engineering↗
Subdirect Products and Birkhoff's Subdirect Representation Theorem
Subdirect products, subdirect irreducibility, the monolith, and Birkhoff's subdirect representation theorem with its Zorn's lemma proof.
Engineering↗
Subuniverses, Subalgebras and the Generation Operator
Subuniverses, subalgebras, the generation operator Sg(X), and the two equivalent descriptions of generation as intersection from above and as term-closure from below.
Engineering↗
Syntactic Monoids and Kleene's Theorem
The syntactic congruence and syntactic monoid, Kleene's characterisation of recognisable languages as the regular ones, Schützenberger's star-free theorem, and Eilenberg's varie…
Engineering↗
Tame Congruence Theory
Tame congruence theory: minimal sets, the five local types, type sets of varieties, and how the classification refines and largely supersedes the 1981 congruence-condition hiera…
Engineering↗
Terms, Term Algebras and Term Operations
Terms 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 referen…
Related
Other tags in the library.
mathematics 387
abstract algebra 252
Machine Design 184
business 179
project management 158
Computational Number Theory 123
algebra handbook 85
graduate mathematics 85
handbook 79
templates and examples 78
History of Engineering 77
Manufacturing 76
algebra and trigonometry 74
algebra 60
project delivery 60
principles of project management 58
advanced algebra 53
Fasteners and Joints 53
Civil Engineering 48
CAD 42
Context and scope 42
Machine Elements 40
Threading and Gaging 37
Mechanical Engineering 35
Power Transmission 34
template 34
Operational Excellence 32
example 31
project leadership 31
project teams 31
strategy 31
Business Expansion 29
Metrology 29
project management research 29
Sales & Marketing 29
personal development 27
behaviour change 26
habits 26
systems 26
Electrical Engineering 25