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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 where it first appears.

Category Engineering / MathematicsSource Special NotationPages xv-xviReading 2 minReviewed 2026-08-07

Learning objectives

Decode the standard symbol set without consulting the source's index
Distinguish the visually similar notations that carry different meanings
Recognise which symbols are near-universal and which are source-specific

Structural notation

A = &langle;A, F&rangle;: an algebra with universe A and operation set F
fA: the interpretation in A of the operation symbol f
Sg(X): the subuniverse generated by X
Sub(A) / Sub(A): the set / the lattice of subuniverses of A
Con A / Con A: the set / the lattice of congruences on A
Θ(a1, a2): the principal congruence generated by the pair
Θ(X): the congruence generated by a set of pairs
A/θ, A/θ: the quotient set / quotient algebra modulo θ
ker(α): the kernel of a homomorphism
Δ, ∇: the least and greatest congruences (identity, all-pairs)

Lattice and order notation

&or;, &and;: join and meet
≤: the partial order — also used for subalgebra in some contexts
≺: covering relation: a is covered by b
l.u.b., sup / g.l.b., inf: least upper bound / greatest lower bound
M5, N5: the two five-element lattices in the forbidden-sublattice theorems
Eq(A): the lattice of equivalence relations on A
Π(A): the set of partitions of A
a/θ: the equivalence class of a modulo θ
LC: the lattice of closed sets of a closure operator

Class operators, varieties and terms

I, S, H, P, PS: isomorphic images, subalgebras, homomorphic images, products, subdirect products
V: the variety operator — V(K) = HSP(K)
T(X) / T(X): the set / algebra of terms over X
pA: the term operation on A induced by term p
FK(X): the free algebra over X in the class K
ΦK(X), θK(X): the congruences used in constructing free algebras
p ≈ q: an identity (equation)
⊧: satisfaction — A ⊧ p ≈ q
IdK(X): the identities holding in K
M(Σ): the class of models of a set of identities Σ

Boolean and model-theoretic notation

2: the two-element Boolean algebra
I(X), F(X): the ideal / filter generated by X
B*: the Stone space (dual) of a Boolean algebra
A[B]*: the Boolean power of A by B
Spec A: the spectrum of an algebra
∏i∈I Ai/U: the ultraproduct modulo an ultrafilter U
PU(K), PR(K): ultraproducts / reduced products of members of K
Th(K), Th∀: the theory of a class; its universal consequences
A ≺ B: A is an elementary substructure of B
Z(A): the centre of an algebra

Traps worth memorising

Four collisions to watch

Su(A) vs Sub(A) — power set of a set versus subuniverses of an algebra.
Con A vs Con A — the set versus the lattice; roman versus bold.
≤ — partial order in Chapter I, substructure relation in Chapter V.
⊧ — equational satisfaction in Chapter II, first-order satisfaction in Chapter V. The second generalises the first, but the pages are far apart.

Frequently asked questions

Is this notation standard across the literature?

Mostly. H, S, P for the class operators, Con and Sub for the lattices, and V for the variety operator are near-universal. The bold-face convention for algebras versus universes is common but not invariable — some authors use A for both and rely on context.

What does the double-turnstile mean exactly?

Satisfaction. Written between a structure and a sentence or identity, it asserts that the structure makes the statement true. In Chapter II it applies to identities; in Chapter V it is extended to arbitrary first-order formulas.

Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section Special Notation, book pages xv-xvi.

This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.