Selected Topics and Applications
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.
Learning objectives
- Define squags and sloops by their identities
- Describe the correspondence between the two varieties
- Explain why the choice of type matters
The two varieties
An algebra ⟨S, ·⟩ of type ⟨2⟩ satisfying idempotence x·x ≈ x, commutativity, and the Steiner law x·(x·y) ≈ y.
An algebra ⟨S, ·, 1⟩ of type ⟨2, 0⟩ satisfying commutativity, x·1 ≈ x, x·x ≈ 1, and x·(x·y) ≈ y.
| Squag | Sloop | |
|---|---|---|
| Type | ⟨2⟩ | ⟨2, 0⟩ |
| x · x | x | 1 |
| Order of the algebra | ≡ 1 or 3 (mod 6) | ≡ 2 or 4 (mod 6) |
| Corresponds to | Steiner triple system on n points | Steiner triple system on n − 1 points, plus the identity |
| Idempotent | Yes | No |
The correspondence
The two varieties encode the same combinatorial data with different conventions.
The sloop presentation has an identity element, which makes it a genuine loop and connects to group theory. The squag presentation is idempotent, which makes every element a one-element subalgebra and simplifies the subalgebra lattice. Different questions are easier in different presentations.
Congruences and simplicity
The two varieties have different congruence behaviour, which is the main practical consequence of the type difference.
- In a sloop, congruences are determined by the class of the identity element 1, since sloops are congruence-permutable loops. The situation resembles group theory.
- In a squag, there is no distinguished element and congruences carry more information. Every element is a subalgebra, so Sub(A) has all singletons as atoms.
- Both varieties are congruence-permutable, since both are quasigroup-like and admit Mal'cev terms.
Many Steiner triple systems give simple algebras, meaning they admit no non-trivial quotient system. The projective and affine systems — the Fano plane among them — are the classical examples of systems with rich subsystem structure but few congruences.
The type-choice lesson again
Squags and sloops repeat a pattern already seen with groups, monoids and quasigroups: the same underlying mathematics admits several types, and the choice determines what the algebraic machinery sees.
| Choice | Consequence |
|---|---|
| Include a constant (sloop) | Subalgebras must contain it; congruences reduce to one class |
| Omit the constant (squag) | Singletons are subalgebras; congruence structure is richer |
| Include division operations | The class becomes a variety rather than a quasivariety |
None of these choices is more correct than another. What matters is stating which one is in force, since theorems about subalgebras and congruences are sensitive to it.
Frequently asked questions
Are squags associative?
No, and they cannot be. An associative idempotent commutative quasigroup would be trivial. Non-associativity is essential to the Steiner structure.
Which variety do combinatorialists prefer?
Usually neither explicitly — they work with the systems directly. The algebraic presentations are the tool that lets universal-algebraic methods be applied, which is the point of Chapter III.
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 III.1, book pages 111-113.
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.