Ideals and slices over the initial algebra

Let V be a variety and c a 0-ary term. I will discuss a simple universal-algebraic construction that extracts from V the structures that behave like c-ideals. The construction is based on the terms theta such that theta(c,...,c) = c, and on the corresponding quasivariety V_c of subreducts of algebras in V. This point of view gives familiar examples: rngs from unital rings, generalized Boolean algebras from Boolean algebras, and Wajsberg hoops from MV-algebras. I will explain how V_c can be compared with V/0, the category of homomorphisms from algebras in V to the initial algebra 0. This comparison is always a coreflection of V_c into V/0, and I will describe a Mal’tsev-like condition characterizing when it is an equivalence. Informally, the question is: when is an algebra over the initial algebra completely determined by one of its fibres? This is joint work in progress with Giuseppe Metere and Serafina Lapenta.