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, [math]\displaystyle{ 0 }[/math]-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 [math]\displaystyle{ \mathbb M }[/math] falls into one of the following types (P. P. Pálfy)

• [math]\displaystyle{ \mathbb M }[/math] is of type [math]\displaystyle{ \bf 1 }[/math], or unary type, iff [math]\displaystyle{ {\rm Pol} ~\mathbb M={\rm Pol} \langle M,G\rangle }[/math], where [math]\displaystyle{ M }[/math] denotes the universe of [math]\displaystyle{ \mathbb M }[/math], [math]\displaystyle{ \rm Pol~ \mathbb A }[/math] denotes the set of all polynomials of an algebra [math]\displaystyle{ \mathbb A }[/math] and [math]\displaystyle{ G }[/math] is a subgroup of the symmetric group over [math]\displaystyle{ M }[/math].

• [math]\displaystyle{ \mathbb M }[/math] is of type [math]\displaystyle{ \bf 2 }[/math], or affine type, iff [math]\displaystyle{ \mathbb M }[/math] is polynomially equivalent to a vector space.

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

• [math]\displaystyle{ \mathbb M }[/math] is of type [math]\displaystyle{ \bf 4 }[/math], or lattice type, iff [math]\displaystyle{ \mathbb M }[/math] is polynomially equivalent to a two-element lattice.

• [math]\displaystyle{ \mathbb M }[/math] is of type [math]\displaystyle{ \bf 5 }[/math], or semilattice type, iff [math]\displaystyle{ \mathbb M }[/math] is polynomially equivalent to a two-element semilattice.