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,…
Birkhoff HSP theorem
Birkhoff 1935: class K of algebras of given signature is variety (defined by equations) iff K closed under H (homomorphic images), S…
Free algebra (universal)
F_V(X) = quotient of term algebra T(X) by V-equational identities. Universal property: every map X → A in V extends uniquely to F_V(X) →…
Congruence relation & quotient
Congruence on algebra A = equivalence-relation compatible with operations. Quotient A/θ is again algebra of same signature. Generalises…
Malcev condition & congruence permutability
Algebra has permutable congruences iff it satisfies Malcev term m(x,x,y) = y, m(x,y,y) = x. Equivalent to: congruences commute Θ_1 ∘ Θ_2 =…
Subdirectly irreducible algebras
Algebra A is subdirectly-irreducible iff intersection of non-trivial congruences is non-trivial. Birkhoff: every algebra is subdirect…
Term operation & clones
Clone on set A = set of operations closed under composition + projections. Clone of A (Pol(A)) determines polymorphisms. Galois…
Birkhoff HSP theorem (1935)
G Birkhoff 1935: a class of algebras of given signature is a variety iff closed under H (homomorphic images), S (subalgebras), P…
Free algebra (universal property, UA)
Universal property of free algebra F_T(X): every map X -> A factors uniquely through F_T(X) -> A; left-adjoint to forgetful functor;…
Congruence lattice (Malcev conditions)
Malcev 1954: a variety is congruence-permutable iff has ternary term p(x,y,z) with p(x,x,y)=y + p(x,y,y)=x; congruence-lattice properties…
Equational completeness (Birkhoff 1935)
Birkhoff 1935: equational consequence Sigma |- s=t coincides with semantic Sigma |= s=t in all models; sound + complete deductive system…
Clone theory (Rosenberg 1970)
Rosenberg 1970 classification of maximal clones on finite set: 6 types + Sheffer functions; Post 1941 lattice of clones on {0,1}; basis of…
Quasivariety (Mal cev / McKenzie)
Quasivariety = class closed under SP (subalgebras + products) + ultraproducts; Mal cev 1966 + McKenzie 1996: quasi-equational definability…
Birkhoff HSP (1935)
G Birkhoff 1935 HSP-theorem variety = HSP(class); modern modern foundational text + universal-property + free-algebra construction.
Malcev term (1954)
A Malcev 1954 Malcev-condition congruence-permutability; modern modern foundational text + Pixley 1963 + 2024 algebraic-CSP-Bulatov.
Post (1941)
E Post 1941 lattice-of-clones Boolean-functions; modern modern foundational text + Post's-lattice + 2024 algebraic-circuits…
Free algebra (Jónsson 1962)
B Jónsson 1962 free-algebra construction-via-terms; modern modern foundational text + Plotkin 1972 + Adamek-Rosicky 1994 categorical…
Congruence lattice (Grätzer 1963)
G Grätzer 1963 + Wille 1969 congruence-lattice-representation; modern modern foundational text + Pudlák 1976 + Tůma 1986 dim-functions.
Equational completeness (McKenzie 1976)
R McKenzie 1976 equational-completeness; modern modern foundational text + 2024 algebra-of-binary-relations + clones-CSP applications.