Remarks on representations of universal algebras by sheaves of quotient algebras

Nontrivial sheaf representation theorems are known for the class of algebras whose congruence lattices are distributive and the class whose congruence lattices are normal unital quantales, which include Keimel's representations of F-rings and Cornish's representations of normal lattices. Unfortunately such results do not include the sheaf representation theorems for rings with identity. In this paper, a sheaf representation for a class of universal algebras with congruence lattice containing a particular type of subframe is presented. As a consequence a representation of non commutative rings is given. This solves a problem posed by A. Wolf [in Recent advances in the representation theory of rings and C^∗-algebras by continuous sections (New Orleans, LA, 1973), 87–93, Mem. Amer. Math. Soc., 148, Amer. Math. Soc., Providence, RI, 1974; MR0369223], but it does not yield Grothendieck representations, nor some of the representations of J. Lambek [Canad. Math. Bull. 14 (1971), 359–368; MR0313324].