Symmetric relations (undirected graphs) with given semigroups

Abstract

The present note is closely related to the papers [1] and [2] by the authors. As the results of them are used here, we preserve the definitions and notation. Our aim is to prove the following theorem:Let S 1 be a semigroup with a unity element, cardinality of S i beeing less than the first inaccessible cardinal. Then there exists a symmetric relation R on a set X such that G(R, X) under composition is isomorphic with S 1.Using "graph" language the theorem asserts that there is an lmdirected graph with the given semigroup S i of all endomorphisms.

Automorphism groups of posets and lattices with a given subset of fixed points

“…A verification of these properties from the explicit form of the full embedding into the category of undirected graphs [10] is straightforward and we omit it here, except perhaps for pointing out that (4) is needed to prove (14).…”

“…Note also that homomorphisms of objects in 2R and their images in R correspond in a one-to-one fashion explicitly described by the assignment off* tof In particular, images of nonlsomorphic objects are nonisomorphic, and automorphism groups are preserved. As can easily be seen, if an object of 2 R has no fixed points then neither has its image in R. A full embedding of the category of directed graphs into that of undirected graphs described in [10] or in [12] has a form that enables us to use the present graphs and convert them into undirected graphs having all properties listed in Lemma 5 below.…”

A stronger form of the theorem on the existence of a rigid binary relation on any set

“…It is known ( [1], [2], [4]) that for each infinite cardinal number κ there exists a rigid symmetric relation R ⊆ κ × κ. Let Φ denote the family of all relations S ⊆ κ × κ which satisfy:…”

The lattice of quasivarieties of undirected graphs

“…On the other hand, by Hedrlín and Pultr [5], there is a full and faithful functor F : G a −→ G u which we now specify.…”