Embeddings of and into Nerode semirings
This paper on Nerode semirings is quite technical using background from both isols and model theory so that only a rough idea of the type of results can be given in a review. The theory of isols was invented by J. C. E. Dekker. Further work by Nerode clarified important similarities and distinctions between the theory of isols and that of the natural numbers \(\omega\). More recently certain isols X (called R.S.T. isols) were discovered which have the property that the semiring N(x) generated by x resembles the semiring of natural numbers rather closely. For example, the rings generated by such semirings have no division of zero. Furthermore it is known that such rings are isomorphic to recursive ultrapowers. The author studies embedding properties of semirings of the form n(X). Among his results are the following: 1) There exists semirings N(X) and N(Y) such that N(X) is a proper cofinal substructure of N(Y). 2) There exist semirings N(X) and N(Y) with X and Y regressive such that \(N(X)\cap N(Y)=\omega\). The paper ends with some open problems.
scientific article; zbMATH DE number 4041899
scientific article