Study of algebraic structures from the signature-and-equations level of abstraction — independent of the particular operations (groups, rings, lattices, boolean algebras etc.) and instead characterising varieties (classes closed under H,…
universal-algebra
Birkhoff HSP theorem: variety = class closed under H, S, P
Birkhoff 1935 (Proc. Camb. Phil. Soc. 31:433). A class V of algebras of a fixed signature is a variety (equational class) iff it is closed…
Con(A) is a complete algebraic lattice; Mal'tsev conditions
For any algebra A, the set Con(A) of congruences (equivalence relations compatible with every fundamental operation) forms a complete…
Free algebra F_V(X) and its universal mapping property
The free algebra F_V(X) over a set X in a variety V is the algebra of 'terms over X' modulo the identities of V. Universal property: for…
Mal'tsev term t(x,y,z) = xy⁻¹z: residuals 0 in Z and S_3
Exact symbolic verification that the group-theoretic Mal'tsev term t(x,y,z) = xy⁻¹z satisfies both Mal'tsev identities t(x,y,y) = x and…
Idempotent variety {x∘x = x} closed under P: product residual 0
Concrete witness of the product closure (the P in HSP) for the simplest non-trivial equational variety: the idempotent variety defined by…
Free monoid on 1 generator: f̂(aʲ · aᵏ) − (f̂(aʲ) + f̂(aᵏ)) ≡ 0
Free monoid on 1 generator: F_{Mon}({a}) ≅ (N, +, 0) via aᵏ ↔ k. Universal property with target (Z, +): for any n ∈ Z, the map f: {a} → Z,…