‎from hilbert's theorem of zeroes‎, ‎and from noether's ideal theory‎, ‎birkhoff derived certain algebraic concepts (as explained by tholen) that have a dual significance in general toposes‎, ‎similar to their role in the original examples of algebraic geometry‎. ‎i will describe a simple example that illustrates some of the aspects of this relationship‎. the dualization from algebra ...

A retraction from a structure P to its substructure Q is a homomorphism from P onto Q that is the identity on Q. We present an algebraic condition which completely characterises all posets and all reflexive graphs Q with the following property: the class of all posets or reflexive graphs, respectively, that admit a retraction onto Q is first-order definable.

The strong isometric dimension of a reflexive graph is related to its injective hull: both deal with embedding reflexive graphs in the strong product of paths. We give several upper and lower bounds for the strong isometric dimension of general graphs; the exact strong isometric dimension for cycles and hypercubes; and the isometric dimension for trees is found to within a factor of two.

If G is a simple graph (a non-oriented graph without loops or multiple edges), its (0, 1)-adjacency matrix A is symmetric and roots of the characteristic polynomial PG (λ) = det (λI −A) (the eigenvalues of G, making up its spectrum) are all real numbers, for which we assume their non-increasing order: λ1 ≥ λ2 ≥ · · · ≥ λn. In a connected graph for the largest eigenvalue λ1 (the index of the gra...

A reflexive graph is a simple undirected graph where a loop has been added at each vertex. If G and H are reflexive graphs and U ⊆ V (H), then a vertex map f : U → V (G) is called nonexpansive if for every two vertices x, y ∈ U , the distance between f(x) and f(y) in G is at most that between x and y in H . A reflexive graph G is said to have the extension property (EP) if for every reflexive g...

‎in this article we discuss examples of fractal $omega$-operads‎. ‎thus we show that there is an $omega$-operadic approach to explain existence of‎ ‎the globular set of globular setsfootnote{globular sets are also called $omega$-graphs by the french school.}‎, ‎the reflexive globular set of reflexive globular sets‎, ‎the $omega$-magma of $omega$-magmas‎, ‎and also the reflexive $omega$-magma of...