Geometric and topological structures related to universal algebras (Q5894926)

The universal algebras considered in this paper are represented in the form \((U,F)\), where \(U\) is the underlying set and \(F\) the set of operations. These satisfy the following conditions: 1. \(F\) contains no nullary operations; 2. For each \(u\in U\) there is at most one representation of the form \(u=f(u_1,\dots,u_n)\), up to the order of the elements, where \(f\in F\) and \(u_1\neq u\) for at least one \(i\); 3. \(U\) is generated by the elements which have no representation of the form given in condition 2 (indecomposable elements). These conditions mean of course, that the algebras in question (here called \(A\)-systems) are free, free commutative, free idempotent, or free commutative and idempotent. Because of the uniqueness of the representation of elements of \(U\) in terms of the indecomposable elements, structures imposed on the generating set of indecomposable elements can be raised to the whole of \(U\), and, in this lengthy paper, the authors explore many connections of this kind (topological, geometrical, \textit{et cetera}).