Semirigid spaces

Abstract

Abstract. Semirigid spaces are introduced and used as a means to construct two metrizable spaces with isomorphic monoids of continuous self-maps and nonisomorphic clones; this resolves Problem 1 in [13]. The clone of any free variety of a given type with sufficiently many constants is shown to be isomorphic to the clone of a metrizable (semirigid) space.

Representations of algebraic theories by continuous maps

“…To proceed, we need to recall some notions and facts presented elsewhere. (a) In [9], a topological space X = (P, t) was called C-semirigid (for a set C c P) i f / ( P ) != C for any continuous self map / : X -> X other than a constant or the identity. It is clear that any C-semirigid space X with P\C ^ 0 must be connected.…”

Algebraic theories, clones and their segments

“…The first order language of clones, as used in [7] and then in [8], is, in fact, the first order language of the corresponding algebraic theories viewed as the above o J-sorted universal algebras and the first order language of nsegments of clones is the first order language of the n-sorted reducts of the corresponding w-sorted algebras. Elementary equivalence of clones or clone segments is the elementary equivalence in their respective languages.…”

“…In [9], the general method of [8] was modified to ge t the following result: for every natural number n > 1 there exists a metric Pn on a set P with card P = 2 ~o such that every continuous map in the n-segment of Clo(P, Qn) is uniformly continuous but the (n+ 1)-segment is not elementary equivalent with the category of all uniformly continuous maps of the spaces (P, pn) k, k = 0,..., n. In [9], the general method of [8] was modified to ge t the following result: for every natural number n > 1 there exists a metric Pn on a set P with card P = 2 ~o such that every continuous map in the n-segment of Clo(P, Qn) is uniformly continuous but the (n+ 1)-segment is not elementary equivalent with the category of all uniformly continuous maps of the spaces (P, pn) k, k = 0,..., n.…”

“…In [6], the basic construction of [8] was enriched to obtain, for instance, the following result: for every triple of natural numbers m < n <~ p there exist metrics T, T' on a set P such that the spaces X = (P, T) and Y = (P, T') satisfy simultaneously the following three statements: the/-segment of CIo(X) is equal to the /-segment of Clo(Y) iff i ~< m, isomorphic iff i <~ n and elementarily equivalent iff i ~< p.…”

“…The full version of the construction is contained in [8]. The full version of the construction is contained in [8].…”