Věra Trnková

Presentation of Set Functors

Accessible set functors can be presented by signatures and equations as quotients of polynomial functors.We determine how preservation of pullbacks and other related properties (often applied in coalgebra) are reflected in the structure of the system of equations.

Symmetries and retracts of quantum logics

We prove that there are arbitrarily many quantum logics, none of which is “similar” to a part of another and each of which has the group of all symmetries isomorphic to a given abstract group. Moreover, each of them contains a given logic with atomic blocks as its sublogic.

Isomorphisms of sums of countable Boolean algebras

In [2], W. Hanf constructed a Boolean algebra H isomorphic to the direct product H X H X H but not to H X H. This result of Hanf was strengthened by J. Ketonen in [4]: there exists even a countable Boolean algebra with this property. In [1], P. R. Halmos examines various problems concerning isomorphism of products of Boolean algebras and asks the corresponding questions for sums in place of products. An analogy of the above result of W. Hanf is proved in [6]: there exists a Boolean algebra B isomorphic to B + B + B but not to B + B. Even, every Boolean algebra is a homomorphic image of a Boolean algebra with this property. All the algebras with this property, constructed in [6], are very large. It is natural to ask whether there exists a countable Boolean algebra B isomorphic to B + B + B but not to B + B. Let us notice that a weaker question about the existence of two nonisomorphic countable Boolean algebras A, B with A + A isomorphic to B + B, was answered affirmatively in [5]. Nevertheless, there exists no countable Boolean algebra B isomorphic to B + B + B but not to B + B. We prove here a stronger result, namely that, for countable Boolean algebras, nB = mB with m > n implies nB -(n + 1)B. This is stated in Theorem 3 of the present paper. Theorems 1 and 2 concern the Schroder-Bernstein property and the pseudoindecomposability of countable Boolean algebras. They form steps in the proof of Theorem 3, but they could be interesting in themselves.

Unexpected properties of locally presentable categories

Locally presentable categories are precisely those complete categories which have a small dense subcategory. Every subcategory of a locally presentable category (i) has a small dense subcategory, (ii) if it is closed under limits, then it is locally presentable and (iii) if it is closed under colimits, then it is coreflective. For density, canonical colimits can be substituted by arbitrary colimits.The above results hold under the assumption of the set-theoretical Vopenkas Principle; in fact, each of them is logically equivalent to that principle.

Relational Automata in a Category and Their Languages

Improved shaped polymeric articles, including ostomy seals, are obtained by radiation crosslinking an acrylamide polymer composition comprising a water-soluble acrylamide polymer plasticized with a quantity of a water-miscible polyol (containing water) that provides a soft, flexible and elastomeric composition.

Continuous and uniformly continuous maps of powers of metric spaces

Abstract For every natural number n > 1, we construct a metric space X such that every continuous map ƒ: X k → X m is uniformly continuous whenever k ⩽ n − 1 but the category of all continuous maps of the spaces X, X2,…, Xn admits no full embedding into the category of all uniformly continuous maps of all powers of X and vice versa.

On elementary equivalence and isomorphism of clone segments

The principal application of a general theorem proved here shows that for any choice 1≤m≤n≤p of integers there exist metric spacesX andY such that the initialk-segments of their clones of continuous maps coincide exactly whenk≤m, are isomorphic exactly whenk≤n, and are elementarily equivalent exactly whenk≤p.

Topological reflections revisited

Two full reflective subcategories of %/i are constructed whose intersection is not reflective.

Nonconstant continuous maps of spaces and of their β-compactifications

Abstract For every cardinal number ɱ and every pair of monoids M 1 ⊆ M 2 there exists a Tychonoff space X such that the collection of all subspaces Y of β X with X ⊆ Y ⊆ βX contains a stiff collection X of the cardinality ɱ (stiff in the sense that if Y , Y ′ ∈ X and ƒ: Y → Y′ is continuous, then either ƒ is constant or Y = Y ′ and ƒ is the identity); moreover, all the nonconstant continuous maps of X into itself form a monoid isomorphic to M 1 and all the nonconstant continuous maps of β X into itself form a monoid isomorphic to M 2 . This assertion is a corollary of the Main Theorem proved here. A more general setting of simultaneous representations of small categories is investigated.

Representation of semigroups by products in a category

Each finite cyclic group has a representation in the category of all Abelian groups (see [2]) and in the category of all Boolean algebras (see [3]). Each semigroup with one generator has a representation in the category of all modules over a suitable ring (see [l], [6]), and in the category of all topological spaces (see [9]). The method of [9] admits a large generalization. It is presented in this paper. We prove that each semigroup with one generator and each Abelian group have representations in each of the following categories: the category of all topological (or uniform or proximity) spaces, of all directed graphs, small categories, unary universal algebras with at least two operations and partial universal algebras of some types. Thus, taking the additive group of all rational numbers as a represented semigroup, we get a space (or a graph or an algebra) which has “negative powers” and “all roots.” It is a pleasure to acknowledge my indebtedness to M. Kat&ov for reading the manuscript and making valuable criticisms, and to my colleague A. Pultr for valuable suggestions.