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Abstract  

LetG andH be finite abelian groups and letF be an arbitrary field. One fundamental problem is that of determining necessary and sufficient conditions for the isomorphism of the group algebrasFG andFH. No solution has appeared in the literature. Nevertheless by combining the results of Berman, Perlis and Walker, Cohen, and Deskins and providing connecting arguments a complete solution can be obtained. It is the purpose of this note to present such a solution.

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References [1] Baird , B. B. 1977 Isomorphisms between inverse semigroups of injective transformations J. Australian Math. Soc

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Abstract  

V. Gregori and S. Romaguera [17] obtained an example of a fuzzy metric space (in the sense of A. George and P. Veeramani) that is not completable, i.e. it is not isometric to a dense subspace of any complete fuzzy metric space; therefore, and contrary to the classical case, there exist quiet fuzzy quasi-metric spaces that are not bicompletable neither D-completable, via (quasi-)isometries. In this paper we show that, nevertheless, it is possible to obtain solutions to the problem of completion of fuzzy quasi-metric spaces by using quasi-uniform isomorphisms instead of (quasi-)isometries. Such solutions are deduced from a general method, given here, to obtain extension properties of fuzzy quasi-metric spaces from the corresponding ones of the classical theory of quasi-uniform and quasi-metric spaces.

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The principal application of a general theorem proved here shows that for any choice 1=m=n=p of integers there exist metric spacesX andY such that the initialk-segments of their clones of continuous maps coincide exactly whenk=m, are isomorphic exactly whenk=n, and are elementarily equivalent exactly whenk=p.

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