Atlas: Partially Ordered Families of Laminated Nearrings of $N(R)$ by P. R. Misra

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The Eleventh Summer Conference on General Topology and Applications
August 10-13, 1995
University of Southern Maine
Gorham, ME, USA

Organizers
J. Baumgartner, D. Briggs, J. deBakker, B. Flagg, G. Gruenhage, M. Guay, Y. Kong, R. Kopperman, S. Shore, J. Rutten, J. Vaughan

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Partially Ordered Families of Laminated Nearrings of N(R)
by
P. R. Misra
Staten Island
Coauthors: K. D. Magill

Let \alpha be a continuous function from a Hausdorff topological group G into itself. S(G)_\alpha denotes the semigroup of all continuous functions from G into G where the product fg is defined by fg = f o \alpha o g. The group operation on G, denoted +, induces a group operation on S(G)_\alpha : (f+g)(x) = f(x) + g(x), for all x in G, and thus we get a near-ring N(G)_\alpha whose multiplicative semigroup is S(G)_\alpha. This near-ring is referred to as a laminated nearring with laminating function \alpha. When the function \alpha is the identity map, N(G)_\alpha is denoted simply as N(G).

In this talk we will restrict ourselves to the topological group R, the set of all reals under ordinary addition, and the laminators will be the Generalized Odd Degree Polynomials, i.e. continuous surjections from R into R which have a finite number of extrema. We will give necessary and sufficient conditions when two different laminated nearrings corresponding to generalized odd degree polynomials are isomorphic. This is obtained by using a more general result characterizing the embeddings of laminated nearrings for a much larger class of laminators than just generalized odd degree polynomials. This characterization will also enable us to show that the collection of isomorphism classes of laminated nearrings, when the laminators are generalized odd degree polynomials, form a partially ordered set with the largest element [N(R)]. The isomorphism class [N(R)] containing N(R) is known to be singleton and any ascending chain of laminated nearrings in the partial ordering will be shown to be finite. Moreover it will be shown that any element of the partially ordered set is the greatest element in an infinite descending chain.

Date received: April 12, 1996