Solution. Let α : Z/nZ → Z/mnZ be defined by α(1) = m. Since 〈1〉 = Z/nZ, this determines the entire homomorphism. Let x ∈ kerα. Then mx ≡ 0, which implies mn | mx so n | x. But since 0 ≤ x < n, we must have x = 0. Therefore the kernel of α is trivial, so α is an injection. Since m | mn, by the fundamental theorem on cyclic groups Z/mnZ has a subgroup of order m isomorphic to Z/mZ. Let β : Z/mnZ...

An (m,n)-colored-mixed graph G=(V,A1,A2,⋯,Am,E1,E2,⋯,En) is a having m colors of arcs and n edges. We do not allow two or edges to have the same endpoints. A homomorphism from an G another H morphism φ:V(G)→V(H) such that each edge (resp. arc) mapped color (and orientation). T said be Pg(m,n)-universal if every in Pg(m,n) (the planar graphs with girth at least g) admits T. show exist for 2m+n≥3...

the starting point of this paper is given by priestley’s papers, where a theory of representation of distributive lattices is presented. the purpose of this paper is to develop a representation theory of fuzzy distributive lattices in the finite case. in this way, some results of priestley’s papers are extended. in the main theorem, we show that the category of finite fuzzy priestley space...

This is a continuation of the paper by the same author published in this journal, v. 19 (2007), no. 6. This paper is a continuation of the paper [4] and makes a whole with it. We retain the notation, the definitions, and the assumptions of [4], and we continue the numeration of sections, theorems and lemmas. The proofs of Theorems 2–4 in §6 of [4] were in fact missing; it was said only that one...

For every pair of nite connected graphs F and H, and every integer k, we construct a universal graph U with the following properties: 1. There is a homomorphism : U ! H, but no homomorphism from F to U . 2. For every graph G with maximal degree no more than k having a homomorphism h : G! H, but no homomorphism from F to G, there is a homomorphism : G! U , such that h = . Particularly, this solv...

The goal of this paper is to develop some applications of the Lefschetz fixed point theory techniques, already available in dynamics, in control theory. A dynamical system on a manifold M is a map f : M → M. The current state, x ∈ M, of the system determines the next state, f(x), and its equilibria are the fixed points of f, f(x) = x. More generally, one deals with coincidences of a pair of map...