On the shape of solution sets of systems of (functional) equations

Solution sets of systems of linear equations over fields are characterized as being affine subspaces. But what can we say about the “shape” of the set of all solutions of other systems of equations? We study solution sets over arbitrary algebraic structures, and we give a necessary condition for a set of n-tuples to be the set of solutions of a system of equations in n unknowns over a given algebra. In the case of Boolean equations we obtain a complete characterization, and we also characterize solution sets of systems of Boolean functional equations.     

New characterizations of the Takagi function via functional equations

We provide two new characterizations of the Takagi function as the unique bounded solution of some systems of two functional equations. The results are independent of those obtained by Kairies (Wy? Szko? Ped Krakow Rocznik Nauk Dydakt Prace Mat 196:73–82, 1998), Kairies (Aequ Math 53:207–241, 1997), Kairies (Aequ Math 58:183–191, 1999) and Kairies et al. (Rad Mat 4:361–374, 1989; Errata, Rad Mat 5:179–180, 1989).     

On von Neumann regular elements in <Emphasis Type="Italic">f</Emphasis>-rings

Let B be an Archimedean reduced f-ring. A positive element \({\omega}\) in B is said to satisfy the property \({(\ast)}\) if for every f-ring A with identity e and every \({\ell}\)-group homomorphism \({\gamma : A \rightarrow B}\) with \({\gamma(e) = \omega}\), there exists a unique \({\ell}\)-ring homomorphism \({\rho: B \rightarrow B}\) such that \({\gamma = \omega \rho}\) and \({\rho(e)^{\perp \perp} = \omega^{\perp \perp}}\). Boulabiar and Hager proved that any (positive) von Neumann regular element in B satisfies the property \({(\ast)}\) and proved that the converse holds in the C(X)-case. In this regard, they asked about this converse in the general case. Our main purpose in this note is to prove, via a counter-example, that the converse in question fails in general. In addition, we shall take the opportunity to extend the direct result obtained by Boulabiar and Hager, and to get the C(X)-case we were talking about in an easier way.     

Generating sublocales by subsets and relations: a tangle of adjunctions

Generalizing the obvious representation of a subspace \({Y \subseteq X}\) as a sublocale in Ω(X) by the congruence \({\{(U, V ) | U\cap Y = V \cap Y\}}\), one obtains the congruence \({\{(a, b) |\mathfrak{o}(a) \cap S = \mathfrak{o}(b) \cap S\}}\), first with sublocales S of a frame L, which (as it is well known) produces back the sublocale S itself, and then with general subsets \({S\subseteq L}\). The relation of such S with the sublocale produced is studied (the result is not always the sublocale generated by S). Further, we discuss in general the associated adjunctions, in particular that between relations on L and subsets of L and view the aforementioned phenomena in this perspective.     

On 3-generated lattices with a completely modular element among generators

We consider a lattice generated by three elements, one of which is completely modular. The free lattice with this property is proved to be finite. It is not modular and contains exactly 39 elements. We have also found a finite set of defining relations for the generating elements of this lattice.     

Congruence preserving functions on free monoids

A function on an algebra is congruence preserving if for any congruence, it maps congruent elements to congruent elements. We show that on a free monoid generated by at least three letters, a function from the free monoid into itself is congruence preserving if and only if it is of the form \({x \mapsto w_{0}xw_{1} \cdots w_{n-1}xw_n }\) for some finite sequence of words \({w_0,\ldots ,w_n}\). We generalize this result to functions of arbitrary arity. This shows that a free monoid with at least three generators is a (noncommutative) affine complete algebra. As far as we know, it is the first (nontrivial) case of a noncommutative affine complete algebra.     

Division closed partially ordered rings

Fuchs called a partially-ordered integral domain, say D, division closed if it has the property that whenever a > 0 and ab > 0, then b > 0. He showed that if D is a lattice-ordered division closed field, then D is totally ordered. In fact, it is known that for a lattice-ordered division ring, the following three conditions are equivalent: a) squares are positive, b) the order is total, and c) the ring is division closed. In the present article, our aim is to study \({\ell}\)-rings that possibly possess zerodivisors and focus on a natural generalization of the property of being division closed, what we call regular division closed. Our investigations lead us to the concept of a positive separating element in an \({\ell}\)-ring, which is related to the well-known concept of a positive d-element.     

On the covering radius of lattice zonotopes and its relation to view-obstructions and the lonely runner conjecture

The goal of this paper is twofold; first, we show the equivalence between certain problems in geometry, such as view-obstructions, billiard ball motions, and the estimation of covering radii of lattice zonotopes. Second, we utilize the latter interpretation and provide upper bounds of said radii by virtue of the Flatness Theorem. Our results allow us to specify how rational dependencies in the view-direction influence the obstruction parameter. These problems are similar in nature to the famous Lonely Runner Problem for which we draw analogous conclusions.