Engineering

Ultrafilters and the Boolean Prime Ideal Theorem

Maximal proper filters, their characterisation by the decision property, and the existence theorem that underwrites Stone duality and the ultraproduct construction.

Engineering Mathematics9 Aug 2026 · 23 min read

What Algebraic D-modules Are | KEVOS® Mathematics

A D-module is a module over a ring of differential operators. This page explains the dictionary between differential equations and modules over the Weyl algebra.

Engineering Mathematics9 Aug 2026 · 27 min read

Computing a Direct Limit: Worked Constructions | KEVOS® Mathematics

How to identify a direct limit in practice: a recognition criterion, then germs of holomorphic functions, localisation and the microfunction system computed in full.

Boolean Algebras: Axioms and First Examples

Boolean algebras as a variety: the axioms, the two-element algebra that generates everything, and the examples that motivate the theory.

Terms 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.

The Prerequisite Dependency Graph

The source carries an explicit diagram of prerequisites showing which sections depend on which. This page renders that dependency structure as navigable text and draws out the c…

The Boolean Algebra / Boolean Ring Correspondence

The 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 diffe…

Theories, Models and Axiomatisability

Theories as sets of sentences, model classes, and the question of which classes of algebras are first-order axiomatisable.

Identities, Satisfaction and Equational Classes

Identities as pairs of terms, what it means for an algebra to satisfy one, and the Galois connection between classes of algebras and sets of identities.

Models — Complexity, Team & Other

Cynefin, Stacey, Tuckman, Drexler/Sibbet, plus conflict, negotiation, planning, process groups and salience.

Boolean Constructions: Recent Work

Developments in Boolean product representations and discriminator varieties following the period the source describes.

Engineering Mathematics9 Aug 2026 · 20 min read

Why the Weyl Algebra Has No Finite-dimensional Representations | KEVOS® Mathematics

No non-zero finite-dimensional vector space carries an action of the Weyl algebra in characteristic zero. Three independent proofs, and exactly where each one uses the character…

Engineering Mathematics12 Aug 2026 · 42 min read

Complete Wall — All 27 (single print)

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Engineering Mathematics9 Aug 2026 · 22 min read

Systems of Differential Equations and Presentation Matrices | KEVOS® Mathematics

Several unknown functions give a matrix of operators, a quotient of a free module over the Weyl algebra, and a solution space that is again a space of homomorphisms.

Term Operations and Polynomial Operations

The functions terms induce on an algebra, the polynomials obtained by allowing parameters, and the clones these families form.

Methods — Estimating, Meetings & Other

The nine estimating methods, the meetings-and-events catalog, and other methods (impact mapping, NPS, timebox).

Equivalence Relations and the Partition Lattice Eq(A)

Equivalence relations on a set, their equivalent description as partitions, and the complete lattice Eq(A) they form — the ambient lattice inside which every congruence lattice …

Engineering Mathematics9 Aug 2026 · 26 min read

The Dixmier Conjecture and Its Link to the Jacobian Conjecture | KEVOS® Mathematics

Dixmier asked whether every endomorphism of the Weyl algebra is an automorphism. A yes would prove the Jacobian conjecture; the two are now known equivalent.

Engineering Mathematics12 Aug 2026 · 2 min read

Needs Assessment

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Ultraproducts in Universal Algebra

The ultraproduct construction from the algebraic side: reduced products modulo an ultrafilter, and the properties they inherit.

Directly Indecomposable Algebras

Algebras that admit no non-trivial direct decomposition, the congruence-lattice condition characterising them, and the limits of unique factorisation.

Discriminator Varieties and their Structure

Varieties generated by classes of algebras sharing a discriminator term, and the representation theory that makes them, in the source's words, remarkably well-behaved.

Birkhoff's Subdirect Representation Theorem

Every algebra is a subdirect product of subdirectly irreducible algebras. The theorem holds with no hypotheses whatever and is the foundation of the structure theory.

Engineering Mathematics9 Aug 2026 · 20 min read

Simplicity of the Weyl Algebra | KEVOS® Mathematics

The Weyl algebra has no two-sided ideals except zero and itself. The proof brackets a minimal-degree element down to a non-zero constant.

Models — Motivation & Change

Herzberg, intrinsic/extrinsic, McClelland, Theory X/Y/Z, plus ADKAR, Kotter, Satir, Bridges change models.

Engineering Mathematics12 Aug 2026 · 2 min read

PMI Ecosystem Master Map

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Engineering Mathematics9 Aug 2026 · 25 min read

The Weyl Algebra Defined as a Ring of Operators | KEVOS® Mathematics

The n-th Weyl algebra is the algebra of operators on a polynomial ring generated by multiplication by each variable and differentiation with respect to it.

Engineering Mathematics9 Aug 2026 · 23 min read

When Are Two Twisted Modules Isomorphic? | KEVOS® Mathematics

Twisting the polynomial module by automorphisms of the Weyl algebra yields simple modules that are isomorphic only when the twisting automorphisms agree.

Sublattices and Lattice Isomorphism

Subsets 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 t…

Engineering Mathematics9 Aug 2026 · 23 min read

Noetherian Modules and the Ascending Chain Condition | KEVOS® Mathematics

A module is Noetherian when every submodule is finitely generated, a condition equivalent to the ascending chain condition and the maximal condition.

Closure Operators and Algebraic Closure

Closure operators, the complete lattices of closed sets they generate, and the correspondence that makes them the organising device behind subuniverses, congruences and generate…

Engineering Mathematics9 Aug 2026 · 24 min read

Canonical Form of an Element of the Weyl Algebra | KEVOS® Mathematics

Every element of the Weyl algebra is uniquely a finite sum of scalar multiples of monomials with all variables written to the left of all derivatives.

The Seventeen Open Problems: a Status Register

A status register for the open problems stated in the source's closing chapter, reporting what is known rather than asserting resolutions.

The Classification of Varieties and Tame Congruence Theory

The programme of classifying locally finite varieties by the local structure of their finite members, developed after the source text.

Models, Methods & Artifacts — Applied Across Domains

A synthesis of which models, methods and artifacts are most useful in each of the eight performance domains.

Engineering Mathematics9 Aug 2026 · 24 min read

The Polynomial Ring as a Module over the Weyl Algebra | KEVOS® Mathematics

The polynomial ring carries a canonical action of the Weyl algebra by multiplication and differentiation, and is isomorphic to the quotient by the left ideal of the partials.

Irredundant Bases and the Irredundant Basis Theorem

Minimal generating sets that admit no proper generating subset, and the theorem constraining the possible sizes of such bases in an algebra with an algebraic closure operator.

Class Operators H, S, P and their Composition

The operators taking homomorphic images, subalgebras and products of a class, the inclusions among their composites, and the identity HSP that computes the generated variety.

Elementary Equivalence and Elementary Substructures

Structures indistinguishable by first-order sentences, and substructures that agree with the ambient structure on every formula.

A Catalogue of Algebras: Groups, Rings, Lattices

The standard algebraic structures presented uniformly as algebras, showing how each familiar definition translates into a type plus a set of identities.

The Primal Algebra Characterisation Theorem

The theorem identifying primal algebras by intrinsic conditions, and the representation of the generated variety by Boolean powers.

Engineering Mathematics9 Aug 2026 · 22 min read

The Weyl Algebra in Positive Characteristic | KEVOS® Mathematics

Over a field of characteristic p the two definitions of the Weyl algebra give different rings. One has nilpotents, the other is not simple, and both have finite-dimensional modu…

Engineering Mathematics9 Aug 2026 · 24 min read

The Jacobian Conjecture | KEVOS® Mathematics

The conjecture that a polynomial self-map of affine space with Jacobian determinant one has a polynomial inverse, what is proved and what is open.

Structure Theory and Finite Basis Developments

Advances in the structure theory of varieties and in the finite basis problem after the source's period.

Engineering Mathematics9 Aug 2026 · 25 min read

Differential Operators on an Affine Variety | KEVOS® Mathematics

What the ring of differential operators looks like on an affine variety, why smoothness makes it well behaved, and how the cusp breaks the naive picture.

Engineering Mathematics9 Aug 2026 · 25 min read

The Associated Graded Algebra | KEVOS® Mathematics

How a filtration of a ring produces a graded algebra of symbols, and why the Weyl algebra's Bernstein filtration yields a polynomial ring in 2n variables.

Engineering Mathematics12 Aug 2026 · 2 min read

Models, Methods & Artifacts

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The Ternary Discriminator Function

The operation that tests equality and branches, and the reason it is the single most consequential term operation in the subject.