KURATOWSKI showed that the derived set operator Z), acting on the space 2^ of closed subsets of a metric space X, is a Borel map of class exactly two and posed the problem of determining the precise classes of the higher order derivatives 0. We show that the exact classes of the higher derivatives D" are unbounded in ©i. In particular, we show that D* is not of class a and that, for limit ordin...

For digraphs D and H , a mapping f : V (D)→V (H) is a homomorphism of D to H if uv ∈ A(D) implies f(u)f(v) ∈ A(H). For a fixed digraph H , the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOMP(H). Digraphs are allowed to have loops, but not allowed to have parallel arcs. A natural optimization version of the homomorphism probl...

Let $nin mathbb{N}$. An additive map $h:Ato B$ between algebras $A$ and $B$ is called $n$-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $ain A$. We show that every $n$-Jordan homomorphism between commutative Banach algebras is a $n$-ring homomorphism when $n < 8$. For these cases, every involutive $n$-Jordan homomorphism between commutative C-algebras is norm continuous.

We prove that the class of linear context-free tree languages is not closed under inverse linear tree homomorphisms. The proof is by contradiction: we encode Dyck words into a context-free tree language and prove that its preimage under a certain linear tree homomorphism cannot be generated by any context-free tree grammar. A positive result can still be obtained: the linear monadic context-fre...

We prove that for every d ≥ 3 the homomorphism order of the class of line graphs of finite graphs with maximal degree d is universal. This means that every finite or countably infinite partially ordered set may be represented by line graphs of graphs with maximal degree d ordered by the existence of a homomorphism.

Usually we shall just call A an algebra if the field k is clear from the context. The algebra A is associative if multiplication is associative i.e. for all a, b, c ∈ A, (ab)c = a(bc), and unital if there is a multiplicative identity, i.e. an element usually denoted by 1 such that, for all a ∈ A, 1a = a1 = a. Note that, in this case, 1 = 0 ⇐⇒ A = {0}. Otherwise, the map k → A defined by t 7→ t·...

The set üS can be identified with the set of all base point preserving maps of $ into itself. SO(n)> acting on S as R with a point a t infinity, is also a set of base point preserving maps of S onto itself. This defines SO(n) C.tiS. The induced map in homotopy is called the /-homomorphism. If we allow n to go to infinity we have the stable /-homomorphism. By Bott 's results [3] 7Ty(50)=Z, i = —...

An order is dense if A < B implies A < C < B for some C. The homomorphism order of (nontrivial) graphs is known to be dense. Homomorphisms of trigraphs extend homomorphisms of graphs, and model many partitions of interest in the study of perfect graphs. We address the question of density of the homomorphism order for trigraphs. It turns out that there are gaps in the order, and we exactly chara...

A graph homomorphism is an edge preserving vertex mapping between two graphs. Locally constrained homomorphisms are those that behave well on the neighborhoods of vertices — if the neighborhood of any vertex of the source graph is mapped bijectively (injectively, surjectively) to the neighborhood of its image in the target graph, the homomorphism is called locally bijective (injective, surjecti...

in this paper we show that if q is a power of a prime p , then the projective special linear group psl(2, q) and the stabilizer of a point of the projective line have maximum sum element orders among all proper subgroups of projective general linear group pgl(2, q) for q odd and even respectively