On Two Concepts of Ultrafilter Extensions of First-Order Models and Their Generalizations

Abstract

There exist two known concepts of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them [1] comes from modal logic and universal algebra, and in fact goes back to [2]. Another one [3,4] comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups [5] as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification. The main result of [3,4], which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in [6]. Results of such type are referred to as extension theorems.After a brief introduction to this area, we offer a uniform approach to both types of extensions. It is based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order models in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and an appropriate semantic for generalized models of this form. We provide two specific operations which turn generalized models into ordinary ones, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and provide a topological characterization of generalized models. Defining a natural concept of homomorphisms between generalized models, we generalize a restricted version of the extension theorem to generalized models. To formulate the full version, we provide even a wider concept of ultrafilter interpretations together with their semantic based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a partial case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn generalized models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which generalized models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we define homomorphisms between generalized models in the wide sense, and establish for them three full versions of the extension theorem.The results of first three sections of this paper were partially announced in [7].

On ultrafilter extensions of first-order models and ultrafilter interpretations

“…A part of the results mentioned in Sects. 1-3 was announced in [25]; here we provide complete proofs of all our results. 3…”

“…In[25], it was erroneously stated that the set of right continuous maps forms a compact Hausdorff space w.r.t. the pointwise convergence topology; actually, the intended topology was a restricted pointwise convergence topology, as explained in details below.…”

More about divisibility in βN

“…The reason for including (ii) as a separate condition in the theorem above is that it shows that our relation is exactly one of the two extensions that are canonical in sense of [15]. So, using [20, Theorem 4] we get that is the topological closure of (seen as a subset of ) in .…”

“…Lately extensions of first‐order models were considered in several papers, cf. [15] for references. In particular, [15] describes two canonical ways to construct ultrafilter extensions of relations.…”

“…[15] for references. In particular, [15] describes two canonical ways to construct ultrafilter extensions of relations. The first of them, described explicitly by Goldblatt in [4] and further developed by Goranko in [6], coincides with the relation considered in this paper, as we will show in Theorem 3.4.…”