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congruence_types [2010/08/20 20:47] (current)
jipsen created
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 +=====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 [[http://www.ams.org/mathscinet-getitem?mr=86h:08001a|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
 +[[http://www.ams.org/online_bks/conm76/|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 [[http://www.ams.org/mathscinet-getitem?mr=89m:08001|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, $\mbox{typ}(\mathcal{K}) = \{\mbox{typ}(\mathbf{A}): \mathbf{A}
 +\mbox{ is a finite algebra in }\mathcal{K}\}$.