The groupoid of main classes of Latin squares
Content created by Fredrik Bakke, Jonathan Prieto-Cubides and Egbert Rijke.
Created on 2022-05-01.
Last modified on 2025-08-30.
module univalent-combinatorics.main-classes-of-latin-squares where
Imports
open import elementary-number-theory.natural-numbers open import foundation.1-types open import foundation.mere-equivalences open import foundation.set-truncations open import foundation.universe-levels open import univalent-combinatorics.main-classes-of-latin-hypercubes open import univalent-combinatorics.standard-finite-types open import univalent-combinatorics.truncated-pi-finite-types open import univalent-combinatorics.untruncated-pi-finite-types
Idea
The groupoid of main classes of latin squares¶ consists of unordered triples of inhabited types equipped with a ternary 1-1 correspondence.
Definition
Main classes of general latin squares
Main-Class-Latin-Squares : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) Main-Class-Latin-Squares l1 l2 = Main-Class-Latin-Hypercube l1 l2 2
Main classes of latin squares of fixed finite order
Main-Class-Latin-Square-of-Order : (m : ℕ) → UU (lsuc lzero) Main-Class-Latin-Square-of-Order m = Main-Class-Latin-Hypercube-of-Order 2 m
Properties
The groupoid of main classes of latin squares of fixed order is a groupoid
is-1-type-Main-Class-Latin-Square-of-Order : (m : ℕ) → is-1-type (Main-Class-Latin-Square-of-Order m) is-1-type-Main-Class-Latin-Square-of-Order = is-1-type-Main-Class-Latin-Hypercube-of-Order 2
The groupoid of main classes of latin squares of fixed order is π₁-finite
is-untruncated-π-finite-Main-Class-Latin-Square-of-Order : (k m : ℕ) → is-untruncated-π-finite k (Main-Class-Latin-Square-of-Order m) is-untruncated-π-finite-Main-Class-Latin-Square-of-Order k = is-untruncated-π-finite-Main-Class-Latin-Hypercube-of-Order k 2 is-truncated-π-finite-Main-Class-Latin-Square-of-Order : (m : ℕ) → is-truncated-π-finite 1 (Main-Class-Latin-Square-of-Order m) is-truncated-π-finite-Main-Class-Latin-Square-of-Order = is-truncated-π-finite-Main-Class-Latin-Hypercube-of-Order 2
The sequence of the number of main classes of latin squares of finite order
The following sequence defines A003090 in the OEIS.
number-of-main-classes-of-Latin-squares-of-order : ℕ → ℕ number-of-main-classes-of-Latin-squares-of-order = number-of-main-classes-of-Latin-hypercubes-of-order 2 mere-equiv-number-of-main-classes-of-Latin-squares-of-order : (m : ℕ) → mere-equiv ( Fin (number-of-main-classes-of-Latin-squares-of-order m)) ( type-trunc-Set (Main-Class-Latin-Square-of-Order m)) mere-equiv-number-of-main-classes-of-Latin-squares-of-order = mere-equiv-number-of-main-classes-of-Latin-hypercubes-of-order 2
Recent changes
- 2025-08-30. Fredrik Bakke. Closure properties of π-finite types (#1311).
- 2025-01-06. Fredrik Bakke and Egbert Rijke. Fix terminology for π-finite types (#1234).
- 2023-03-13. Jonathan Prieto-Cubides. More maintenance (#506).
- 2023-03-10. Fredrik Bakke. Additions to
fix-import
(#497). - 2023-03-09. Jonathan Prieto-Cubides. Add hooks (#495).