On elementary equivalence and isomorphism of clone segments

Abstract

The principal application of a general theorem proved here shows that for any. choice 1 5 m 5 n 5 p of integers, there exist metric spaces X and Y such that the initial k-segments of their clones of continuous maps coincide exactly when k 5 m, are isomorphic exactly when k 5 n, and are elementarily equivalent exactly when k IP.

Representations of algebraic theories by continuous maps

“…These three sections thus form the 'topological' part of the paper. Moreover, the spaces F(a { ) and F(a 2 ) constructed in Sections 4 and 5 are metrizable, and this provides a joint strengthening of those results of [8,9] and [6] which concern clone segment equality and isomorphism.…”

“…To prove that (P, i) = ((A, u), (£, Q)) determines a representation of in Top, we need certain specific properties of the metric Q on P. The proof follows, in part, the reasoning used in [6] to deal with metric spaces. Here, however, the insertion of a space (A, u) into the space (P, t) determining the representation of ^( E , Q.)…”

“…The result of [12] describing their internal structure is recalled in the section below, and used extensively afterwards. J. Sichler and V. Trnkova [6] ( REMARK. Theorem 1.3 can be applied to the category MTop of metrizable topological spaces and their continuous maps because, clearly, MTop is well-pointed and, as shown implicitly in [6] and stated explicitly in [12], it is also comprehensive.…”

“…J. Sichler and V. Trnkova [6] ( REMARK. Theorem 1.3 can be applied to the category MTop of metrizable topological spaces and their continuous maps because, clearly, MTop is well-pointed and, as shown implicitly in [6] and stated explicitly in [12], it is also comprehensive. Hence any two r-ultrametrics <p < (p can be represented by the equality and the isomorphism of clone segments of metrizable topological spaces.…”

Representability and local representability of algebraic theories

“…, X n − 1 . Papers [20,[24][25][26] present some variants and improvements of this result (and also investigate categories other than the category Top of continuous maps of topological spaces). In these constructions, representations of certain algebraic theories are implicitly constructed within the proof showing that the internal structure of these algebraic theories implies the required properties of the spaces X and Y determining their representation in Top.…”