Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First, an example is given to show that the class of finitely generated FA-presentable algebras is not closed under forming finitely generated subalgebras, even wi...

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classification of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a difficult pro...

We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type FP∞ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.

We show that every λm-injectivity class (i.e., the class of all the objects injective with respect to some class of λ-presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by a class of λ-presentable morphisms. This was known for small-injectivity classes, and referred to as the “small object argument”. An analogous ...

The concept of a finitely related algebra, as opposed to the ones of finitely presentable and finitely generated ones, is not preserved under categorical equivalences. We propose a categorically well behaved approximation for it in the context of locally presentable categories, which turns out to be a natural counterpart to the (slightly reformulated) categorical definitions of finitely present...

Several researchers have recently established that for every Turing degree c, the real closed field of all c-computable real numbers has spectrum {d : d′ ≥ c′′}. We investigate the spectra of real closed fields further, focusing first on subfields of the field R0 of computable real numbers, then on archimedean real closed fields more generally, and finally on nonarchimedean real closed fields. ...