Congruence Types

A minimal algebra is a finite nontrivial algebra in which every unary polynomial is either constant or a permutation.

Peter P. Pálfy, Unary polynomials in algebras. I, Algebra Universalis, 18, 1984, 262-273 MRreview shows that if $\mathbf{M}$ is a minimal algebra then $\mathbf{M}$ is polynomially equivalent to one of the following:

  • a unary algebra in which each basic operation is a permutation
  • a vector space
  • the 2-element Boolean algebra
  • the 2-element lattice
  • a 2-element semilattice.

The type of a minimal algebra $\mathbf{M}$ is defined to be permutational (1), abelian (2), Boolean (3), lattice (4), or semilattice (5) accordingly.

The type set of a finite algebra is defined and analyzed extensively in the groundbreaking book now available free online David Hobby and Ralph McKenzie, The structure of finite algebras, Contemporary Mathematics, 76, American Mathematical Society, Providence, RI, 1988, xii+203 MRreview. With each two-element interval $\{\theta,\psi\}$ in the congruence lattice of a finite algebra the authors associate a collection of minimal algebras of one of the 5 types, and this defines the value of $\mbox{typ}(\theta,\psi)$.

For a finite algebra $\mathbf{A}$, $\mbox{typ}(\mathbf{A})$ is the union of the sets $\mbox{typ}(\theta,\psi)$ where $\{\theta,\psi\}$ ranges over all two-element intervals in the congruence lattice of $\mathbf{A}$. For a class $\mathcal{K}$ of algebras, $\mbox{typ}(\mathcal{K}) = \{\mbox{typ}(\mathbf{A}): \mathbf{A} \mbox{ is a finite algebra in }\mathcal{K}\}$.