On coconnected algebras
Let \(A\) be an object in a category \({\mathcal A}\) with finite products and let \(n\geq 2\) be finite. For the \(n\)-fold power \(A^n\) of \(A\), let \(\{\pi^{(n)}_i; i\in n\}\) be the collection of its product projections. \(A\) is said to be \(n\)-coconnected if every \({\mathcal A}\)-morphism \(f: A^n\to A\) has a decomposition \(f= h\circ\pi^{(n)}_i\) for some \(h\in \text{End }A\) and \(i\in n= \{0,\dots, n-1\}\). 2-coconnected objects are called coconnected. \(A\) is said to be fully coconnected if \(A\) is \(n\)-coconnected for every \(n\). In the paper it is shown that coconnectedness presents no obstacle to monoid representation by endomorphisms of algebras from certain universal varieties. It is proved that for every monoid \(M\) there exist arbitrarily large \((0,1)\)-lattices \(A_0\), \(A_1\) with \(\text{End }A_0\cong \text{End }A_1\cong M\) such that \(A_0\) is not coconnected while \(A_1\) is fully coconnected, and that there is an arithmetical variety with this property. Further, the authors show that the variety \(\text{Alg}(1,1)\) of all algebras with two unary operations is also of this kind, and that coconnected monounary algebras are determined by their endomorphism monoids. A partial answer to the problem of whether every 3-coconnected algebra is fully coconnected is given.
scientific article; zbMATH DE number 1895981
scientific article