Varieties in which the Pierce stalks are directly indecomposable. (English) Zbl 0868.08003

If the sublattice of the factor congruences of the congruences of an algebra \(A\) is a Boolean algebra with the Stone space \(X\) of maximal ideals, then the natural map \(A\to\prod\{A/\bigcup m:m\in X\}\) is an isomorphism between \(A\) and the so-called algebra of global sections of the Pierce sheaf. This construction is possible, for example, for the varieties with distributive congruence lattices or for the variety of rings with unit. In this paper, the author characterizes varieties in which the stalks of the Pierce sheaves are directly indecomposable. A variety of algebras is called a Pierce variety if there exists a term \(u= u(x,y,x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n)\) such that the following two equalities \(u(x,y,w,\vec 0,\vec 1)=x\) and \(u(x,y,w,\vec 1,\vec 0)= y\) hold for some \(n\)-ary terms \(\vec 0\), \(\vec 1\). The following is proved: the stalks of the Pierce sheaf of every element of a variety \({\mathbf V}\) are directly indecomposable iff \({\mathbf V}\) is a Pierce variety such that every subalgebra of any subdirectly irreducible member is directly indecomposable and every ultraproduct of any family of subdirectly irreducible members is directly indecomposable.


08B05 Equational logic, Mal’tsev conditions
08A30 Subalgebras, congruence relations
08B26 Subdirect products and subdirect irreducibility
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