Volume 65, Issue 2

The relational complexity, introduced by G. Cherlin, G. Martin, and D. Saracino, is a measure of ultrahomogeneity of a relational structure. It provides an information on minimal arity of additional invariant relations needed to turn given structure into an ultrahomogeneous one. The original motivation was group theory. This work focuses more on structures and provides an alternative approach. Our study is motivated by related concept of lift complexity studied by Hubička and Nešetřil.

Products of locales (generalized spaces) are coproducts of frames. Because of the algebraic nature of the latter they are often viewed as algebraic objects without much topological connotation. In this paper we first analyze the frame construction emphasizing its tensor product carrier. Then we show how it can be viewed topologically, that is, in the sum-of-the-open-rectangles perspective. The main aim is to present the product from different points of view, as an algebraic and a geometric object, and persuade the reader that both of them are fairly transparent.

This article considers free objects and free extensions over posets in the category of frames. Its primary goal is to present novel representations for these objects as subdirect products of certain chains. Constructions for the corresponding objects in the category of bounded distributive lattices are also presented.

We revisit uniformly paracompact uniform frames and show that, in analogy with their spatial counterparts, they have a characterisation in terms of a “completeness property”. Namely, they are precisely those in which every weakly Cauchy filter clusters. We also give another characterisation in terms of the Čech-Stone compactification of the underlying frame. By tweaking the definition of uniformly paracompact frames, we define uniformly para-Lindelöf frames (analogously to same-named uniform spaces) and characterise them in terms of the Lindelöf coreflection of the underlying frame. This latter characterisation has no spatial counterpart.

This paper establishes that the familiar rôle of nonmeasurable cardinals in classical topology extends to pointfree topology, that is, the setting of frames. For this, it considers the frames which are the pointfree form of the extremally disconnected P-spaces, namely the extremally disconnected 0-dimensional frames in which any countable join of complemented elements is complemented, and shows that they (1) have discrete spectrum and (2) are realcompact whenever they have nonmeasurable cardinal. An important tool obtained for this purpose is the result that, for a Boolean frame L, any σ-frame homomorphism L → 2 preserves the joins of all subsets of nonmeasurable cardinal.

The cozero part of a sigma-frame is considered here for the first time. The fundamental notion of a trail in a frame is adapted for sigma-frames via the notion of a witness and, as a consequence, one obtains characterisations for the cozero elements, and of pseudocompactness, of sigma-frames. In the presence of complete regularity, pseudocompactness is seen to be equivalent to (countable) compactness which, in this setting, and unlike its spatial and frame counterparts, gives rise to a coreflection which is the pseudocompactification.

The notion of torsion radical of cyclically ordered groups is defined analogously as in the case of lattice ordered groups. We denote by T the collection of all torsion radicals of cyclically ordered groups. For τ1, τ2 ∈ T, we put τ1 ∈ τ2 if τ1(G) □ τ2(G) for each cyclically ordered group G. We show that T is a proper class; nevertheless, we apply for T the usual terminology of the theory of partially ordered sets. We prove that T is a complete completely distributive lattice. The analogous result fails to be valid for torsion radicals of lattice ordered groups. Further, we deal with products of torsion classes of cyclically ordered groups.

In this paper we explore generalizations of Neumann’s theorem proving that weak commutativity in ordered groups actually implies the group is abelian. We show that a natural generalization of Neumann’s weak commutativity holds for certain Scrimger ℓ-groups.

In the category Arch of archimedean l-groups, the r.u. completion of the divisible hull, rdA, is the maximum essential reflection and the maximum majorizing reflection (Ball-Hager, 1999). In the weak-unital subcategory W, the reflections c 3 A and mA (Aron-Hager, 1981) are respectively maximum essential, and maximum majorizing (Ball-Hager, 1993), and rdA ≤ mA always. These situations are reviewed here, and further, it is shown that: W-epic A ≤ B is Arch-epic if the unit of B is a near unit; rdA = mA if and only if A ≤ mA is Arch-epic, and this obtains when the unit of A is a near unit. (If A ∈ W has a compatible f-ring multiplication, then the unit (the identity) is a near unit.) A point here is that mA has a concrete and understandable description as realvalued functions on the Yosida space of A, perforce, when rdA = mA so does rdA.

An n-ary local polymorphism of a given monounary algebra A is a homomorphism from a finitely generated subalgebra of A n to A. A is n-polymorphism-homogeneous if each n-ary local polymorphism can be extended to a global polymorphism. Then A is called polymorphism-homogeneous, if it is n-polymorphism-homogeneous for each positive integer n. In this paper we describe all polymorphism-homogeneous monounary algebras.

Let α denote an infinite cardinal or ∞ which is used to signify no cardinal constraint. This work introduces the concept of an αcc-Baer ring and demonstrates that a commutative semiprime ring A with identity is αcc-Baer if and only if Spec(A) is αcc-disconnected. Moreover, we prove that for each commutative semprime ring A with identity there exists a minimum αcc-Baer ring of quotients, which we call the αcc-Baer hull of A. In addition, we investigate a variety of classical α-Baer ring results within the contexts of αcc-Baer rings and apply our results to produce alternative proofs of some classical results such as A is α-Baer if and only if Spec(A) is α-disconnected. Lastly, we apply our results within the contexts of archimedean f-rings.

With modest standing assumptions on a category C, it is shown that a Galois connection exists between subclasses of C-objects (on the one hand) and classes of epimorphisms of C (on the other). In this connection the following classes are in a one-to-one correspondence, which reverses inclusion: the epireflective classes of C-objects with the classes ε of epimorphisms which are pushout invariant, in the sense that, for each pushout diagram in which e ∈ ε, then it follows that n ∈ ε. The paper then examines some of the consequences of this result, and in so doing “revisits” the pushout invariance of the authors as it was discussed in a paper of some fifteen years ago.

Weakly nonlinear hypothesis on parameters in nonlinear regression models can be tested by methods of linear statistical models in some cases. Certain conditions must be satisfied. The aim of the paper is to find them for models without/with constraints on parameters.