A directed graph G ∈ D is said to be embeddable into G ∈ D if there exists an injective graph homomorphism ϕ : G → G. We consider the embeddability ordering (D, ≤) of finite directed graphs, and prove that for every G ∈ D the set {G, G T } is definable by first-order formulas in the partially ordered set (D, ≤), where G T denotes the transpose of G. We also prove that the automorphism group of (D, ≤) is isomorphic to Z 2. Keywords First-order definability • Directed graph • Embeddability ordering This research was realized in the frames of TÁMOP 4.2.4. A/2-11-1-2012-0001 "National Excellence Program-Elaborating and operating an inland student and researcher personal support system" The project was subsidized by the European Union and co-financed by the European Social Fund.
We deal with first-order definability in the embeddability ordering (D; ≤) of finite directed graphs. A directed graph G ∈ D is said to be embeddable into G ∈ D if there exists an injective graph homomorphism ϕ : G → G. We describe the first-order definable relations of (D; ≤) using the first-order language of an enriched small category of digraphs. The description yields the main result of the author's paper [5] as a corrolary and a lot more. For example, the set of weakly connected digraphs turns out to be first-order definable in (D; ≤). Moreover, if we allow the usage of a constant, a particular digraph A, in our first-order formulas, then the full second-order language of digraphs becomes available.
Abstract. We study convex cyclic polygons, that is, inscribed n-gons. Starting from P. Schreiber's idea, published in 1993, we prove that these polygons are not constructible from their side lengths with straightedge and compass, provided n is at least five. They are non-constructible even in the particular case where they only have two different integer side lengths, provided that n = 6. To achieve this goal, we develop two tools of separate interest. First, we prove a limit theorem stating that, under reasonable conditions, geometric constructibility is preserved under taking limits. To do so, we tailor a particular case of Puiseux's classical theorem on some generalized power series, called Puiseux series, over algebraically closed fields to an analogous theorem on these series over real square root closed fields. Second, based on Hilbert's irreducibility theorem, we give a rational parameter theorem that, under reasonable conditions again, turns a non-constructibility result with a transcendental parameter into a non-constructibility result with a rational parameter. For n even and at least six, we give an elementary proof for the non-constructibility of the cyclic n-gon from its side lengths and, also, from the distances of its sides from the center of the circumscribed circle. The fact that the cyclic n-gon is constructible from these distances for n = 4 but nonconstructible for n = 3 exemplifies that some conditions of the limit theorem cannot be omitted.
In the first part of this paper we present a new family of finite bounded posets whose clones of monotone operations are not finitely generated. The proofs of these results are analogues of those in the famous paper of Tardos. Another interesting family of finite posets from the finite generability point of view is the family of locked crowns. To decide whether the clone of a locked crown where the crown is of at least six elements is finitely generated or not one needs to go beyond the scope of Tardos's proof. Although our investigations are not conclusive in this direction, they led to the results in the second part of the paper. We call a monotone operation ascending if it is greater than or equal to some projection. We prove that the clones of bounded posets are generated by certain ascending idempotent monotone operations and the 0 and 1 constant operations. A consequence of this result is that if the clone of ascending idempotent operations of a finite bounded poset is finitely generated, then its clone is finitely generated as well. We provide an example of a half bounded finite poset whose clone of ascending idempotent operations is finitely generated but whose clone is not finitely generated. Another interesting consequence of our result is that if the clone of a finite bounded poset is finitely generated, then it has a three element generating set that consists of an ascending idempotent monotone operation and the 0 and 1 constant operations. Keywords Maximal clones • Finitely generated clones • Bounded posetsThe original version of this article was revised due to a retrospective Open Access order.
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