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. 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 $\mathbb {M}$ falls into one of the following types (P. P. Pálfy)

• $\mathbb {M}$ is of type ${\bf {1}}$, or unary type, iff ${\rm {Pol}}~\mathbb {M} ={\rm {Pol}}\langle M,G\rangle$, where $M$ denotes the universe of $\mathbb {M}$, ${\rm {Pol~\mathbb {A} }}$ 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 ${\bf {2}}$, or affine type, iff $\mathbb {M}$ is polynomially equivalent to a vector space.

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

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

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