The shells of Vaggione: Some comments and questions (Q1866827)

The concept of a shell was introduced by the author in 1972 as an algebra of type \((2,2,0,0)\) and generalized by \textit{D. Vaggione} [Algebra Universalis 36, 483-487 (1996; Zbl 0901.08010)] to algebras with \(2n\) \((2n+1)\)-ary operations and \(2n\) unary operations. The author proves that every variety of these generalized shells is categorically equivalent to a variety of original shells and polynomially equivalent to that one where the unary polynomials are constant. The concepts of factor element and ideal are introduced, and the correspondence to factor congruences is established. Hence, all three of these form isomorphic Boolean algebras. Thus any shell is isomorphic to the shell of all global sections of the sheaf over the Boolean space of all prime ideals of factor objects. Several open problems are listed in the paper.