Embeddings of and into Nerode semirings
type: Item
instance of: scholarly article
title: Embeddings of and into Nerode semirings
zbMATH Open document ID: 0639.03047
DOI: 10.1007/BF02766171
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03D50
Mathematics Subject Classification ID: 03C57
zbMATH DE Number: 4041899
zbMATH Keywords: Nerode semirings
zbMATH Keywords: isols
zbMATH Keywords: model theory
zbMATH Keywords: recursive ultrapowers
zbMATH Keywords: embedding
zbMATH Keywords: cofinal substructure
author: Thomas G. McLaughlin
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02766171
OpenAlex ID: W2027556312
description: scientific article; zbMATH DE number 4041899