Core Structure Theory
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 general principle.
Learning objectives
- Present semigroups and monoids as algebras and identify the varieties
- Explain why quasigroups need three binary operations rather than one
- Connect quasigroups to Latin squares
Semigroups and monoids
An algebra ⟨S, ·⟩ of type ⟨2⟩ in which multiplication is associative.
Adding a nullary operation for the identity gives a monoid, of type ⟨2, 0⟩. Both classes are varieties.
A semigroup may happen to possess an identity element without that element being part of its type. The distinction matters for subalgebras: a subsemigroup of a monoid need not contain the identity, whereas a submonoid must. Changing the type changes the subalgebra lattice.
Commutative semigroups, bands (idempotent semigroups) and semilattices (commutative bands) are all subvarieties obtained by adding identities.
Quasigroups: the type problem
A set with a binary operation such that for all a, b the equations a · x = b and y · a = b have unique solutions.
This definition is not equational — it asserts existence and uniqueness of solutions rather than stating identities. With this type, quasigroups are not a variety: a subalgebra of a quasigroup under multiplication alone need not be a quasigroup.
Introduce the solutions as operations. A quasigroup is an algebra ⟨Q, ·, /, \⟩ of type ⟨2, 2, 2⟩ satisfying:
- x \ (x · y) ≈ y and x · (x \ y) ≈ y
- (y · x) / x ≈ y and (y / x) · x ≈ y
With this type the class of quasigroups is a variety. This is a recurring move in the subject: an existential condition is converted into an operational one by naming the witness, which brings the class inside the equational framework.
Loops and the connection to Latin squares
A quasigroup with a two-sided identity element, included in the type as a nullary operation.
Groups are exactly the associative loops. The chain of specialisation is:
- Quasigroup — unique solvability
- Loop — plus identity
- Group — plus associativity
- Abelian group — plus commutativity
- Group — plus associativity
- Loop — plus identity
The multiplication table of a finite quasigroup is precisely a Latin square: an n × n array in which every symbol occurs exactly once in each row and each column. Unique solvability of a · x = b is exactly the row condition; the column condition comes from the other equation.
This identification is what allows the refutation of Euler's conjecture on orthogonal Latin squares to be carried out algebraically, which is the subject of Chapter III §3.
Why the type choice is a general lesson
Three examples of the same phenomenon have now appeared:
| Structure | Naive type | Repaired type | Gained |
|---|---|---|---|
| Group | ⟨2⟩ | ⟨2, 1, 0⟩ | Closure under subalgebras |
| Monoid | ⟨2⟩ | ⟨2, 0⟩ | Identity preserved by subalgebras and homomorphisms |
| Quasigroup | ⟨2⟩ | ⟨2, 2, 2⟩ | Equational definability |
If a class is closed under H, S and P but its natural axioms are not identities, the remedy is usually to enlarge the type by naming witnesses. If the class is not closed under all three, no enlargement of the type will make it a variety — the obstruction is structural.
Frequently asked questions
Are quasigroups congruence-permutable?
Yes. The term p(x,y,z) = (x / (y \ y)) · (y \ z) can be used to build a Mal'cev term, so quasigroups — and hence groups and loops — are congruence-permutable.
Is every Latin square the table of a group?
No, far from it. Every group of order n gives a Latin square, but the number of Latin squares of order n grows far faster than the number of groups. Most Latin squares correspond to quasigroups that are not associative.
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 II.1, book pages 26-30.
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.