Minimal algebra

Minimal algebra is an important concept in tame congruence theory, a theory that has been developed by Ralph McKenzie and David Hobby.

Definition

A minimal algebra is a finite algebra with more than one element, in which every non-constant unary polynomial is a permutation on its domain.

Classification

A polynomial of an algebra is a composition of its basic operations, $0$-ary operations and the projections. Two algebras are called polynomially equivalent if they have the same universe and precisely the same polynomial operations. A minimal algebra $\mathbb{𝕄}$ falls into one of the following types (P. P. Pálfy)

• $\mathbb{𝕄}$ is of type $\mathbf{𝟏}$, or unary type, iff $\mathrm{P}\mathrm{o}\mathrm{l}\mathbb{𝕄}=\mathrm{P}\mathrm{o}\mathrm{l}⟨M,G⟩$, where $M$ denotes the universe of $\mathbb{𝕄}$, $\mathrm{P}\mathrm{o}\mathrm{l}\mathbb{𝔸}$ denotes the set of all polynomials of an algebra $\mathbb{𝔸}$ and $G$ is a subgroup of the symmetric group over $M$.

• $\mathbb{𝕄}$ is of type $\mathbf{𝟐}$, or affine type, iff $\mathbb{𝕄}$ is polynomially equivalent to a vector space.

• $\mathbb{𝕄}$ is of type $\mathbf{𝟑}$, or Boolean type, iff $\mathbb{𝕄}$ is polynomially equivalent to a two-element Boolean algebra.

• $\mathbb{𝕄}$ is of type $\mathbf{𝟒}$, or lattice type, iff $\mathbb{𝕄}$ is polynomially equivalent to a two-element lattice.

• $\mathbb{𝕄}$ is of type $\mathbf{𝟓}$, or semilattice type, iff $\mathbb{𝕄}$ is polynomially equivalent to a two-element semilattice.