Simultaneous representations in categories of algebras

Abstract

Hedrlin and Pultr proved that every small category is isomorphic to a full subcategory of the category Alg (Δ) of all algebras of type Δ whenever the sum ∑Δ of their arities satisfies ∑Δ>2. This article deals with simultaneous representation in categories of algebras, a generalization of the related question: given a subcategory k′ of a small category k, when does there exist an extension Δ′ of a type Δ with ∑Δ≽2 such that k′ is a full subcategory of Alg(Δ′) while the Alg (Δ)-redacts of algebras representing k′ determine a category isomorphic to k? We characterize simultaneous representability by algebras and their reducts completely, and show that it is closely related to Isbell's dominion. A consequence of the main result states that algebraically representable pairs (k′, k) of one-object categories k′ k are exactly those for which k′ coincides with its dominion in k, and provides an alternative characterization of the dominion. Simultaneous representability by partial algebras is not subject to any such restriction.