\(\Omega\)-rings, their flat and projective acts with some applications (Q2715780)

In this dissertation \(\Omega\)-rings and acts over them are studied. An \(\Omega\)-ring is a universal algebra of signature \(\Omega\) equipped with a binary associative multiplication connected with operations in \(\Omega\) by two-sided distributivity. Thus \(\Omega\)-rings are natural generalizations of rings, semirings, distributive lattices, semigroups. A right act over an \(\Omega\)-ring \(R\) is an \(\Omega\)-algebra \(A\) such that for every \(a\in A\), \(r\in R\) a multiplication \(ar\in A\) is defined such that \(a(rs)=(ar)s\) and it is connected with operations in \(\Omega\) by two-sided distributivity.NEWLINENEWLINENEWLINEThe dissertation consists of 5 chapters. The two first chapters contain preliminaries and several constructions of \(\Omega\)-rings.NEWLINENEWLINENEWLINEIn chapter 3 tensor product, flatness and projectivity of acts over \(\Omega\)-rings are considered. A generalization of the Govorov-Lazard and Stenström theorems describing strongly flat acts and a generalization of Kaplansky's theorem characterizing projective acts are given.NEWLINENEWLINENEWLINEChapter 4 deals with applications of \(\Omega\)-rings to automata theory and rewriting. Among other results here the analogue of the well-known Myhill-Nerode theorem is proved. In this chapter graph algebras are also considered.NEWLINENEWLINENEWLINEChapter 5 is devoted to projective semimodules over a semiring. For example it is shown that over any nonzero commutative, additively idempotent semiring, not all projective semimodules are free and not all strongly flat semimodules are projective.