Engineering

The PMI Learning Curve (Start Here)

Where to begin and the order to climb — all 27 topics on one learning journey.

Engineering Mathematics7 Aug 2026 · 2 min read

The Second and Third Isomorphism Theorems

The remaining isomorphism theorems in their general algebraic form, and the hypotheses each requires — including the one that fails without congruence permutability.

Quotient Algebras and the Natural Map

The construction of the quotient algebra modulo a congruence, the natural surjection onto it, and the universal property that makes quotients the right notion.

Engineering Mathematics9 Aug 2026 · 27 min read

Locally Nilpotent Derivations | KEVOS® Mathematics

A derivation is locally nilpotent when some power of it kills every element. In characteristic zero such derivations exponentiate to automorphisms, and a slice makes them partia…

Engineering Mathematics9 Aug 2026 · 25 min read

Commutation Relations in the Weyl Algebra | KEVOS® Mathematics

The relations between the generators of the Weyl algebra, how they follow from the product rule, and the identities and structural consequences they generate.

The Weyl Algebra and the Birth of Quantum Mechanics | KEVOS® Mathematics

How the Heisenberg commutation relation of 1925 became the Weyl algebra, why it admits no finite matrices, and how Dirac's quantum algebra survives in modern notation.

Homomorphisms, Kernels and the First Isomorphism Theorem

Homomorphisms as structure-preserving maps, kernels as congruences, and the first isomorphism theorem in the generality where it belongs.

Business Analysis & Its 6 Domains

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The Stone Representation Theorem

Every Boolean algebra is isomorphic to a field of sets. The theorem, its proof from the prime ideal theorem, and what it does and does not deliver.

Models, Methods & Artifacts — Overview

What models, methods and artifacts are, how Figure 4-1 connects them through tailoring, and how to avoid waste.

Filters and Ideals in Boolean Algebras

Filters as upward-closed meet-closed subsets, ideals as their duals, and their correspondence with congruences.

Engineering Mathematics9 Aug 2026 · 23 min read

The Bernstein-Sato Polynomial | KEVOS® Mathematics

The Bernstein-Sato polynomial of p is the monic generator of the ideal of all b(s) admitting an operator D(s) with b(s) p^s = D(s) p^(s+1).

Engineering Mathematics12 Aug 2026 · 1 min read

Artifacts — Strategy, Logs & Plans

Strategy artifacts, the logs-and-registers set, and the full catalog of project plans.

Engineering Mathematics9 Aug 2026 · 22 min read

Holomorphic Functions as a Module over the Weyl Algebra | KEVOS® Mathematics

The holomorphic functions on an open subset of the complex plane form a Weyl module that is neither simple nor torsion, as the function exp(exp z) shows.

Engineering Mathematics7 Aug 2026 · 2 min read

The Commutator and the Center: Modern Developments

The commutator theory for congruence-modular varieties: the generalisation of the group commutator that the source's centre section anticipates.

Semigroups, Monoids and Quasigroups as Algebras

The one-operation structures and the divisibility structures, presented as algebras. Quasigroups in particular require a careful choice of type, and that choice illustrates a ge…

Engineering Mathematics7 Aug 2026 · 4 min read

Notation and Conventions

A consolidated reference for the notation used throughout the subject, drawn from the source's own special-notation tables and organised by what the symbol is for rather than by…

Bibliography and Further Reading Guide

A guide to the source's bibliography and to the standard references for the subject, organised by what each is for.

Partial Orders, Posets and Bounds

Partial orders, the posets they generate, and the bound notions — upper and lower bounds, suprema and infima — that make the order-theoretic definition of a lattice possible.

Lattices as Algebras: the Equational Definition

A lattice can be defined purely equationally, as a set with two binary operations satisfying four pairs of identities. This is the definition that makes lattices algebras in the…

Proving the Canonical Basis Is a Basis | KEVOS® Mathematics

How the monomials x-alpha d-beta are proved to be a K-basis of the Weyl algebra: straightening gives spanning, and a test polynomial gives independence.

Engineering Mathematics9 Aug 2026 · 26 min read

Derivations of a Commutative Ring | KEVOS® Mathematics

Derivations of a commutative K-algebra: the Leibniz rule, the module and Lie algebra structure, and why they are the differential operators of order one.

Engineering Mathematics12 Aug 2026 · 1 min read

PMBOK 7 — The 8 Performance Domains

The eight interacting domains of project delivery from the PMBOK Guide, 7th Edition.

The Correspondence Theorem for Algebras

The bijection between congruences above a fixed congruence and congruences on the quotient — a lattice isomorphism that makes Con of a quotient an interval in Con of the origi…

Lattices as Posets and the Equivalence Theorem

The order-theoretic definition of a lattice and the theorem establishing that it agrees exactly with the equational definition. Both directions of the construction are given, to…

Congruences and the Substitution Property

Congruences: the equivalence relations compatible with the operations. The substitution property, why it is the right condition, and the failure modes when it is absent.

The Congruence Extension Property

The condition that congruences on a subalgebra extend to the whole algebra, the varieties that satisfy it, and its role in transferring structural results.

Satisfaction and the Tarski Truth Definition

The recursive definition of truth in a structure, and the reason a recursion over formulas requires assignments rather than sentences alone.

Distributive Lattices and their Characterisation

The distributive law for lattices, its self-dual character, and the concrete examples that make distributivity the most important special condition in lattice theory.

Engineering Mathematics7 Aug 2026 · 2 min read

The Compactness Theorem via Ultraproducts

The theorem that a finitely satisfiable theory has a model, proved algebraically by an ultraproduct construction, and its consequences.

Why the Polynomial Ring Is a Simple Weyl Module | KEVOS® Mathematics

Over a field of characteristic zero every non-zero polynomial generates the whole polynomial ring as a module over the Weyl algebra, so the module is simple.

Engineering Mathematics12 Aug 2026 · 1 min read

Methods — Data Gathering & Analysis

The full data-gathering and analysis method catalog, including the business-justification methods.

Boolean Products: Definition and Motivation

A representation of an algebra as continuous sections over a Boolean space, replacing sheaf theory with a simpler and equally powerful formulation.

Modules and R-Modules as Algebras

Modules over a fixed ring as algebras with one unary operation per scalar, and the sense in which modules are the model case for the whole structure theory of congruence-modular…

Engineering Mathematics9 Aug 2026 · 24 min read

Order of a Differential Operator and the Order Filtration | KEVOS® Mathematics

The order of a differential operator, the increasing filtration it defines, and why composition adds orders while commutators lose one.

The 8 Performance Domains

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Squags and Sloops

The two varieties of algebras associated with Steiner triple systems — one idempotent, one with a distinguished element — and the relationship between them.

Subdirectly Irreducible Algebras

The 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.

Why the Weyl Algebra Is a Domain | KEVOS® Mathematics

The Weyl algebra has no zero divisors. The proof is one line from additivity of degree, and it fails in positive characteristic for an instructive reason.

Reading Paths: the Short Course and the Research Track

The source text divides into a short introductory course and a research-oriented remainder. This page sets out both routes explicitly, so that a reader can take the material at …

Engineering Mathematics9 Aug 2026 · 24 min read

Holonomic Modules: Definition and First Examples | KEVOS® Mathematics

A finitely generated module over the n-th Weyl algebra is holonomic when it is zero or has dimension exactly n, the least value Bernstein's inequality allows.

Engineering Mathematics9 Aug 2026 · 24 min read

Automorphisms of the Weyl Algebra | KEVOS® Mathematics

Every endomorphism of the Weyl algebra is injective because the algebra is simple, so the Dixmier conjecture is a surjectivity question, and it implies the Jacobian conjecture.

Atoms and Finite Boolean Algebras

Atoms as the minimal non-zero elements, the classification of finite Boolean algebras as power sets, and the failure of that classification in the infinite case.

Mechanical12 Aug 2026 · 13 min read

General Principles of Technical Drawing

A practitioner's guide to the general principles of technical drawing under AS 1100.101: drawing sheets, line conventions, lettering, scales, projection, sectioning, dimensionin…

Simple Algebras and Congruence Simplicity

Algebras whose only congruences are the two trivial ones. Simplicity is the strongest indecomposability condition and appears throughout the structure theory of varieties.

Engineering Mathematics9 Aug 2026 · 22 min read

The Hilbert Function and the Hilbert Polynomial | KEVOS® Mathematics

The Hilbert function of a finitely generated graded module counts dimensions degree by degree, and for large arguments it agrees exactly with a polynomial.

Algebraic Lattices and Compact Elements

Compact elements, algebraic lattices, and the theorem that the subuniverse and congruence lattices of any algebra are algebraic — the structural fact that finitary arity buys.

Elicitation, Analysis & Requirements

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