Minimal algebra

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. 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, -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

  1. ^ a b Hobby, David; McKenzie, Ralph (1988). The structure of finite algebras. Providence, RI: American Mathematical Society. p. xii+203 pp. ISBN 0-8218-5073-3.
  2. ^ Pálfy, P. P. (1984). "Unary polynomials in algebras. I". Algebra Universalis. 18 (3): 262–273. doi:10.1007/BF01203365. S2CID 15991530.

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