Minimal algebra is an important concept in tame congruence theory, a theory that has been developed by Ralph McKenzie and David Hobby.[1]
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, -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 falls into one of the following types (P. P. Pálfy) [1][2]
is of type , or unary type, iff , where denotes the universe of , denotes the set of all polynomials of an algebra and is a
subgroup of the
symmetric group over .
is of type , or affine type, iff is polynomially equivalent to a
vector space.
is of type , or Boolean type, iff is polynomially equivalent to a two-element
Boolean algebra.
is of type , or lattice type, iff is polynomially equivalent to a two-element
lattice.
is of type , or semilattice type, iff is polynomially equivalent to a two-element
semilattice.
References
^
abHobby, David; McKenzie, Ralph (1988). The structure of finite algebras. Providence, RI: American Mathematical Society. p. xii+203 pp.
ISBN0-8218-5073-3.