This paper characterizes the injective and projective objects in the category of directed multigraphs, or quivers. Further, the injective envelope and projective cover of any quiver in this category is constructed.

Let $R$ be a commutative ring with identity. The purpose of this paper is to introduce and study two classes of modules over $R$, called $mbox{Max}$-injective and $mbox{Max}$-strongly top modules and explore some of their basic properties. Our concern is to extend some properties of $X$-injective and strongly top modules to these classes of modules and obtain some related results.

We investigate and compare four variants of the doublepushout approach to graph transformation. Besides the traditional approach with arbitrary matching and injective right-hand morphisms, we consider three variations by employing injective matching and/or arbitrary right-hand morphisms in rules. For each of the three variations, we clarify whether the well-known commutativity theorems are stil...

Image transforms are used extensively in image processing to convert one image form into another form. These transforms are either point-operation transforms or neighborhood-operation transforms. Injective transforms, also known as reversible or one-to-one transforms, are a subset of point-operation transforms where each intensity value maps to a distinct intensity value. In this paper we prese...

We discuss cellular automata over arbitrary finitely generated groups. We call a cellular automaton post-surjective if for any pair of asymptotic configurations, every pre-image of one is asymptotic to a pre-image of the other. The well known dual concept is pre-injectivity: a cellular automaton is pre-injective if distinct asymptotic configurations have distinct images. We prove that pre-injec...

We introduce a simple tool that can be used to reduce non-injective instances of the hidden shift problem over arbitrary group to injective instances over the same group. In particular, we show that the average-case non-injective hidden shift problem admit this reduction. We show similar results for (non-injective) hidden shift problem for bent functions. We generalize the notion of influence a...

1. We just write out (i). Recall that X1 ∪ X2 = {x : x ∈ X1 or x ∈ X2}. Hence x ∈ X1 =⇒ x ∈ X1 ∪ X2, so that X1 ⊆ X1 ∪ X2. By symmetry, X1 ⊆ X1 ∪ X2. Finally, suppose that Y is a set such that X1 ⊆ Y and X2 ⊆ Y . Let x ∈ X1 ∪X2. Then x ∈ X1 or x ∈ X2. If x ∈ X1, then x ∈ Y since X1 ⊆ Y . Similarly, if x ∈ X2, then x ∈ Y . Thus in either case x ∈ Y , so that X1 ∪X2 ⊆ Y . 2. (i) Injective, not su...

In this paper we investigate and compare four variants of the double-pushout approach to graph transformation. As well as the traditional approach with arbitrary matching and injective right-hand morphisms, we consider three variations by employing injective matching and/or arbitrary right-hand morphisms in rules. We show that injective matching provides additional expressiveness in two respect...

In the category Top0 of T0-spaces and continuous maps, embeddings are just those morphisms with respect to which the Sierpiński space is Kan-injective, and the Kan-injective hull of the Sierpiński space is the category of continuous lattices and maps preserving directed suprema and arbitrary infima. In the category Loc of locales and localic maps, we give an analogous characterization of flat e...

‎‎‎in this paper‎, ‎the‎ notion of injectivity with respect to order dense embeddings in ‎‎the category of $s$-posets‎, ‎posets with a monotone action of a‎ pomonoid $s$ on them‎, ‎is studied‎. ‎we give a criterion‎, ‎like the baer condition for injectivity of modules‎, ‎or skornjakov criterion for injectivity of $s$-sets‎, ‎for the order dense injectivity‎. ‎also‎, ‎we consider such injectivit...