# Differences

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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}\}$. | ||

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