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Inspired by a recent paper of Alain Connes and Caterina Consani which connects the geometric theory surrounding the elusive field with one element to sharply transitive group actions on finite and infinite projective spaces (“Singer actions”), we consider several fundamental problems and conjectures about Singer actions. Among other results, we show that virtually all infinite abelian groups and all (possibly infinitely generated) free groups act as Singer groups on certain projective planes, as a corollary of a general criterion. We investigate for which fields F the plane P 2 ( F ) = PG ( 2 , F ) (and more generally the space P n ( F ) = PG ( n , F ) ) admits a Singer group, and show, e.g., that for any prime p and any positive integer n > 1 , PG ( n , F p ‾ ) cannot admit Singer groups ( F p ‾ an algebraic closure of F p ). One of the main results in characteristic 0, which is a corollary of a criterion which applies to many other fields, is that PG ( m , R ) with m ≠ 0 a positive even integer, cannot admit Singer groups. [ABSTRACT FROM AUTHOR]

Convex subsets of affine spaces over the field of real numbers are described by so-called barycentric algebras. In this paper, we discuss extensions of the geometric and algebraic definitions of a convex set to the case of more general coefficient rings. In particular, we show that the principal ideal subdomains of the reals provide a good framework for such a generalization. Since the closed intervals of these subdomains play an essential role, we provide a detailed analysis of certain cases, and discuss differences from the "classical" intervals of the reals. We introduce a new concept of an algebraic closure of "geometric" convex subsets of affine spaces over the subdomains in question, and investigate their properties. We show that this closure provides a purely algebraic description of topological closures of geometric generalized convex sets. Our closure corresponds to one instance of the very general closure introduced in an earlier paper of the authors. The approach used in this paper allows to extend some results from that paper. Moreover, it provides a very simple description of the closure, with concise proofs of existence and uniqueness. [ABSTRACT FROM AUTHOR]