Congruence Types

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

Peter P. Pálfy, \emph{Unary polynomials in algebras. I}, Algebra Universalis, \textbf{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 \emph{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, \emph{The structure of finite algebras}, Contemporary Mathematics, \textbf{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, $\mbox{typ}(\mathcal{K}) = \{\mbox{typ}(\mathbf{A}): \mathbf{A} \mbox{ is a finite algebra in }\mathcal{K}\}$.

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