Combinatorial and Automata Applications

Quasigroups, Loops and Latin Squares

A Latin square is the multiplication table of a quasigroup. That single observation turns a combinatorial object into an algebra and makes the whole apparatus available.

Engineering · Mathematics5 min readKV-MATH-0228

Learning objectives

Define a quasigroup by the unique-solvability condition and equationally.
Explain why the equational definition requires two division operations.
Translate between a Latin square and a quasigroup in both directions.
Distinguish loops from quasigroups and from groups.
State the congruence properties of the variety of quasigroups.
Recognise isotopy as a coarser equivalence than isomorphism.

01Two definitions of a quasigroup

The classical definition: a set with a binary operation such that for all a and b, each of a·x = b and y·a = b has a unique solution. Equivalently, left and right translations are bijections.

CautionThe classical definition does not give a variety

Unique solvability is an existence-and-uniqueness statement, not an identity. A subalgebra of a quasigroup in type (2) — a subset closed under the product alone — need not be a quasigroup, because solutions may escape the subset. So the class is not closed under S and is not a variety in that type.

The repair is to place the two division operations in the type, making them primitive rather than derived. With type (2, 2, 2) the class becomes equationally definable and every general theorem applies.

02The equational definition

Type (2, 2, 2) with ·, \ (left division), / (right division):
x \ (x · y) ≈ y x · (x \ y) ≈ y
(y · x) / x ≈ y (y / x) · x ≈ y

Four identities. Each division operation is both a left and right inverse to multiplication in the appropriate slot.

Type choice and its consequences
Type	Class	Variety?	Subalgebras
(2)	quasigroups by unique solvability	No	sub-groupoids, may not be quasigroups
(2, 2, 2)	quasigroups equationally	Yes	subquasigroups
(2, 2, 2, 0)	loops	Yes	subloops, all containing the identity
(2, 1, 0)	groups	Yes	subgroups

This is the clearest illustration in the source of the principle stated in the Core stream: the type is part of the data. The same objects, described in two types, give a variety in one case and not the other.

03Latin squares

A Latin square of order n is an n × n array on n symbols in which each symbol occurs exactly once in each row and each column. The multiplication table of a finite quasigroup is a Latin square, and conversely.

ProcedureTranslating between Latin squares and quasigroups

in: Latin square of order n → out: quasigroup of order n, and back

square → quasigroup:
index rows and columns by the symbol set Q
define a · b := the entry in row a, column b
each row is a permutation ⟹ left translations are bijections
each column is a permutation ⟹ right translations are bijections
define a \ b and a / b as the unique solutions
quasigroup → square:
tabulate the operation; the Latin condition is exactly unique solvability
the correspondence is a bijection between Latin squares on Q
and quasigroup operations on Q

Correctness: the Latin row condition is precisely bijectivity of left translations, and likewise for columns. Caveat: the correspondence is with labelled squares — two Latin squares differing by a relabelling give isomorphic quasigroups, so counting squares and counting quasigroups are different problems.

The bijection means every question about Latin squares is a question about quasigroups. Orthogonality of Latin squares, treated on the next page, becomes a statement about a pair of quasigroup operations on the same set.

04Loops and the road to groups

A loop is a quasigroup with a two-sided identity element, placed in the type as a nullary operation. A group is an associative loop.

Quasigroup
Unique solvability only. No identity, no associativity. The Latin square is arbitrary.
Loop
Add an identity element. The Latin square has a row and column matching the border in order.
Moufang loop, Bol loop and friends
Add weakened associativity laws. Each is equational, so each is a variety, and they form a hierarchy between loops and groups.
Group
Full associativity. The Latin square is a Cayley table, and the whole of group theory becomes available.

Key resultAssociativity is the expensive axiom

Everything below associativity — quasigroups, loops, Moufang loops — is combinatorially abundant, with the number of objects growing very fast in the order. Groups are comparatively rare. Associativity is what collapses the combinatorial explosion, which is why Latin squares are a much larger world than Cayley tables.

05Congruence properties

Quasigroups are congruence-permutable, which is what makes their quotient theory behave like that of groups.

Mal'cev term for quasigroups: p(x, y, z) = (x / (y \ y)) · (y \ z)

Constructed from the two divisions. Verifying p(x,y,y) ≈ x and p(x,x,y) ≈ y is a direct computation with the four defining identities.

Quasigroups in the classification
Property	Holds?	Consequence
Congruence-permutable	Yes	Joins of congruences are single composites.
Congruence-modular	Yes	Commutator theory available.
Congruence-distributive	No	Jónsson's lemma unavailable.
Congruence extension property	Yes	Congruences on subquasigroups extend.
Locally finite	No	Free quasigroups on finitely many generators are infinite.

Permutability means normal subloops do coordinatise congruences in the loop case, recovering something close to the group picture. For quasigroups proper, without an identity, the congruence itself remains the object of study.

06Isotopy: a coarser equivalence

Two quasigroups are isotopic when one can be obtained from the other by relabelling rows, columns and symbols independently. Isotopy is strictly coarser than isomorphism, and it is the natural equivalence for Latin squares.

Isomorphism

One relabelling

A single bijection applied to arguments and values alike. The algebraic notion, preserved by all the universal-algebraic machinery.

Isotopy

Three independent relabellings

Separate bijections for rows, columns and symbols. The combinatorial notion. Every quasigroup is isotopic to a loop, so isotopy classes are coarser than isomorphism classes.

CautionUniversal algebra does not see isotopy

Isotopy is not preserved by the class operators and has no equational characterisation. Results proved algebraically hold up to isomorphism, not up to isotopy, and combinatorial literature counting Latin squares up to isotopy is answering a different question from algebraic literature counting quasigroups up to isomorphism. Compare figures only after checking which equivalence is in use.

Frequently asked

Why does every quasigroup have an isotopic loop?

Pick any element and use its left and right translations to relabel rows and columns. The chosen element becomes an identity in the relabelled table. This principal isotopy construction shows loops are isotopy-representatives of quasigroups, which is why loop theory carries most of the structural content.

Are the four quasigroup identities independent?

Yes — each of the four is needed. Dropping one gives a strictly larger class. This contrasts with the lattice axioms, where idempotency is derivable from the others, and is worth checking rather than assuming when comparing axiom lists across texts.

How many Latin squares are there of a given order?

The counts grow extremely fast and are known only for small orders, having been extended by successive computer searches. They are catalogue data: consult a current combinatorics handbook or the relevant online sequence rather than a printed table, and check whether the figure counts labelled squares, reduced squares, isotopy classes or main classes — the four differ by large factors.

Sources and further reading

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.

Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.