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 $$ \bf 1$$, 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.