Uniform representations of congruence schemes
Joel D. Berman, George Grätzer · Pacific Journal of Mathematics · 1978
A congruence scheme Σ is a finite sequence of polynomials.A nontrivial equational class K is a representation of Σ iff the principal congruences in K can be described in a natural fashion by Σ.In this paper it is shown that a necessary and sufficient condition for a congruence scheme Σ whose polynomials do not contain constants to have a representation is that each polynomial in the sequence be at least binary.1* Introduction* For an algebra 31 and a, be A, let θ(a, b) denote the smallest congruence relation under which a = 6; such relations are called principal.For instance, in the class D of distributive lattices (see [1]): c = d(θ(a, b)) iff c = Po(b, a, 6, c, d) p o (a, a, b, c, d) = p λ (a f a, b, c, d) p^b, a, b, c, d) = p 2 (δ, a, b, c, d) p 2 (a, a, b, c, d) = p 3 (α, α, 6, c, d) Pz(b, a, b,c,d) = d, where PoThis is one example of a congruence scheme (for a general definition, see §2).The general definition of a congruence scheme permits an arbitrary sequence p 0 , , p n _ x of polynomials and the polynomials may have any number of variables.As the simplest example of a congruence scheme, let K be an equational class and let us assume that for all 9ί 6 K the following holds: c = d(β(a, b)) iff c -a + c and d = b + e for some ee A, where + is a binary operation of K.Congruence schemes have been investigated in [5] under the name 1-good systems and in [1].301