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 {\displaystyle 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 {\displaystyle \mathbb {M} } falls into one of the following types (P. P. Pálfy)
M {\displaystyle \mathbb {M} } is of type 1 {\displaystyle {\bf {1}}} , or unary type, iff P o l M = P o l ⟨ M , G ⟩ {\displaystyle {\rm {Pol}}~\mathbb {M} ={\rm {Pol}}\langle M,G\rangle } , where M {\displaystyle M} denotes the universe of M {\displaystyle \mathbb {M} } , P o l A {\displaystyle {\rm {Pol~\mathbb {A} }}} denotes the set of all polynomials of an algebra A {\displaystyle \mathbb {A} } and G {\displaystyle G} is a subgroup of the symmetric group over M {\displaystyle M} .
M {\displaystyle \mathbb {M} } is of type 2 {\displaystyle {\bf {2}}} , or affine type, iff M {\displaystyle \mathbb {M} } is polynomially equivalent to a vector space.
M {\displaystyle \mathbb {M} } is of type 3 {\displaystyle {\bf {3}}} , or Boolean type, iff M {\displaystyle \mathbb {M} } is polynomially equivalent to a two-element Boolean algebra.
M {\displaystyle \mathbb {M} } is of type 4 {\displaystyle {\bf {4}}} , or lattice type, iff M {\displaystyle \mathbb {M} } is polynomially equivalent to a two-element lattice.
M {\displaystyle \mathbb {M} } is of type 5 {\displaystyle {\bf {5}}} , or semilattice type, iff M {\displaystyle \mathbb {M} } is polynomially equivalent to a two-element semilattice.
References
