BCK-modules were introduced as an action of a BCK-algebra over Abelian group. Homomorphisms form exact sequence which is called BCK-sequence. In this paper, we study homomorphisms BCK-modules. We show that have module structure. Moreover, sequences Hom functors are BCK-sequences.

A retraction is a homomorphism from graph G to an induced subgraph H of that the identity on . In long line research, retractions have been studied under various algorithmic settings. Recently, problem approximately counting was considered. We give complete trichotomy for complexity all square-free graphs (graphs do not contain cycle length 4). It turns out there rich and interesting class whic...

Let $G$ be a countable monoid and let $A$ an Artinian group (resp. module). $\Sigma \subset A^G$ closed subshift which is also subgroup submodule) of $A^G$. Suppose that $\Gamma$ finitely generated consisting pairwise commuting cellular automata \to \Sigma$ are homomorphisms groups modules) with binary operation given by composition maps. We show the valuation action on $\Sigma$ satisfies natur...

The Graded Classification Conjecture states that the pointed $K_{0}^{\operatorname {gr}}$ -group is a complete invariant of Leavitt path algebras finite graphs when these are considered with their natural grading by $\mathbb {Z}$ . strong version this conjecture functor full and faithful on category graded homomorphisms modulo conjugations invertible elements zero components. We show for unital...

It is known that the number of homomorphisms from a group $F$ to $G$ divisible by greatest common divisor order and exponent $F/[F,F]$. We investigate satisfying some natural conditions such as injectivity or surjectivity. The simplest nontrivial corollary our results following fact: {\it in any finite group, generating pairs $(x,y)$ $x^3=1=y^5$, multiple 15 $[G,G]\cdot\{g^{15}\;|\;g\in G\}$.

The classical Morita Theorem for rings established the equivalence of three statements, involving categorical equivalences, isomorphisms between corners finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism infinite rings. In our main result, we establish analogous statements graded homomorphisms,

In 1939, Gelfand and Kolmogorov published a paper [4], where they showed that for a (compact Hausdorff) topological space X, homomorphisms of the algebra of continuous functions C(X) to numbers are in a one-to-one correspondence with points of X. Paper [4] is famous for giving birth to the theory of normed rings. However, it is worth stressing that the Gelfand–Kolmogorov theorem may be viewed a...