Minimal algebra

Template:Orphan Template:More references 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. In simpler terms, it’s an algebraic structure where unary operations (those involving a single input) behave like permutations (bijective mappings). These algebras provide intriguing connections between mathematical concepts and are classified into different types, including affine, Boolean, lattice, and semilattice types.

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 $𝕄$ falls into one of the following types (P. P. Pálfy)

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

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

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

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

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