# Congruence types

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

[P. P. Pálfy, Unary polynomials in algebras. I, Algebra Universalis 18 (1984) 262--273 MRreview] shows that if M is a minimal algebra then M is polynomially equivalent to one of the following:

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

The type of a minimal algebra 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, Ralph McKenzie, The structure of finite algebras, Contemporary Mathematics 76 American Mathematical Society (1988) xii+203 MRreview]. With each two-element interval {θ,ψ} 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 typ(θ,ψ).

For a finite algebra A, typ(A) is the union of the sets typ(θ,ψ) where {θ,ψ} ranges over all two-element intervals in the congruence lattice of A. For a class K of algebras, typ(K)  = {typ(A) | A is a finite algebra in K}.

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