On nets of free polygons over a category and their categories of homomorphisms (Q1087977)

Let \(I\) be a nonempty set and \(J\subseteq I^ 2\) a reflexive and transitive relation. A family \(A=(I,J,A^ j_ i,\cdot,e_ i)\) is called a category over a graph \((I,J)\) if 1) \(\{A^ j_ i |\) \((i,j)\in J\}\) is a set of pairwise disjunct nonempty sets; 2) \(\cdot\) is a partial associative binary operation such that for any elements \(a\in A^ j_ i\) and \(b\in A^ k_ j\) we have \(a\cdot b\in A^ k_ i ;\) 3) \(e_ i\in A^ i_ i\), \(i\in I\) are elements such that \(a\cdot e_ i=a\), \(e_ i\cdot b=b\) for all \(a\in A^ i_ k\), \(b\in A_ i^{\ell}\), \((k,i),(i,\ell)\in J.\) A family \((I,M_ i)\) is called a left act over the category A if 1) \(\{M_ i |\) \(i\in I\}\) is a set of pairwise disjunct nonempty sets; 2) for any \(x\in M_ i\) and \(a\in A^ i_ j\), (j,i)\(\in J\), a multiplication \(ax\in M_ j\) is defined such that \(e_ ix=x\) and \(b(ax)=(ba)x\) for all \(b\in A^ j_ k\), \((k,j)\in J\) (there are some index-misprints in the paper). Let \(E\subseteq I\) be an nonempty subset. The family \(P=\{P_ i\), \(i\in E\), \(P_ i\subseteq M_ i\}\) is called a system of generators for M, if for any \(j\in I\), \(x\in M_ j\), there exists \(i\in E\), \((j,i)\in J\), \(y\in P_ i\), \(a\in A^ i_ j\) such that \(x=ay.\) Let \(L\subseteq K^ 2\) be a reflexive and transitive relation on a set K. The triple \(M=(K,L,M_ k)\) is said to be a net of A-acts over the graph (K,L), if to each element \(k\in K\) there corresponds an A-act \(M_ k\) and for any pair (k,\(\ell)\in L\) there exists a homomorphism \(f: M_ k\to M_{\ell}\) of A-acts. It is proved that for a net M the family \(H=(K,L,H_ k^{\ell},\cdot,d(M_ k))\) is a category, called the category of homomorphisms of the net M, where \(H_ k^{\ell}\) is the set of all homomorphisms of the A-act \(M_ k\) to the A-act \(M_{\ell}\), \((k,\ell)\in L\), and \(d(M_ k)\) the identical mapping of \(M_ k\), \(k\in K.\) The main task of the paper is an investigation of the categories of homomorphisms of nets of free A-acts. The author proves the following: Theorem. Let a category A and a net M of free A-acts \(M_ k=(M_{ki}\); \(i\in I)\) with systems of free generators \(P_ k=\{P_{ki}\); \(i\in E_ k\), \(P_{ki}\subseteq M_{ki}\}\) be given. Then its category of homomorphisms H is isomorphic to the category \(F=(K,L,F_ k^{\ell},\cdot,(d_ k,d(\tilde P_ k)))\), defined by the net \((K,L,\tilde P_ k)\) of expanded sets where \(\tilde P_ k=\cup_{i\in E_ k}P_{ki}.\) As a corollary one gets the well-known fact that if \({}_ SM\) is a free S-act with a basis \(X\), then End\((_ SM)\) is isomorphic to the wreath product on monoids \(P(X)\) and \(S\), where \(P(X)\) denotes the monoid of mappings of \(X\) into \(X\).