We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation subclasses whose axiomatisations are recursively enumerable in our second-order fragment can also be recursively axiomatised in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. As applications of these results, by showing that certain classes can be straightforwardly defined as separation subclasses, we obtain first-order axiomatisability results for these classes. In particular we apply this technique to graph colourings and a class of partial algebras arising from separation logic.

Given a partial action $\unicode[STIX]{x1D703}$ of a group on a set with an algebraic structure, we construct a reflector of $\unicode[STIX]{x1D703}$ in the corresponding subcategory of global actions and study the question when this reflector is a globalization. In particular, if $\unicode[STIX]{x1D703}$ is a partial action on an algebra from a variety $\mathsf{V}$, then we show that the problem reduces to the embeddability of a certain generalized amalgam of $\mathsf{V}$-algebras associated with $\unicode[STIX]{x1D703}$. As an application, we describe globalizable partial actions on semigroups, whose domains are ideals.

Let K be a category of structures with all its homomorphisms, Uα K → Set the α-th power of its forgetful functor U An α-ary implicit partial operation (O.P.I.) in K is a diagram of natural transformations and functors. We first study various properties which O P I's can have, as maximality, definability and closure under products or equalizers. Revisiting various concepts and results of Isbell, Linton, Bacsich and Herrera, we note, among other things, that the dominion (resp. the stable dominion) of a subset of a structure K is its closure under O P I's (resp. under equalizer-closed O P I's), and we show that in a variety, all epis are surjective (resp. all monos are regular) iff all limit-closed (resp. product-closed) O P I's are restrictions of total implicit operations.