By Vadim G. Korneev, Ulrich Langer

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P map in If is an are the repre- are the T °p natural can be e m b e d d e d W are both is a p r e d i c a t e Let P A in have it is a full which X UA for every such A. definition of HT The and will and equivalences. H~, then domain P X. is a m o n i c Then Up. ,Zn)) M ~+m f > M are both in and J. M ~+n Let be the diagram: M~+m+n P~+_~n > M~+n P~+m M~+m > M f Then V(H X) then ~ > M. X represents U A. limit of such that is s t r a i g h t f o r w a r d from the embedding. is a c o n j u n c t i o n X 6 0b(~) give the represents V(HX) represents (2) Hence for every there It follows and ~.

T h e connections between elementary theories and polyadic algebras have been studied by Daigneault in [5]. In this paper w e will prove the completeness t h e o r e m for elementary theories, suggested by Z a w v e r e in [141 . T h e proof is categorical, but it can be said that it follows, in a sense, the lines of the completeness proof in Henkin [81o W e will use the slightly m o r e logical category. general notion of a Aside f r o m having s o m e technical advantages, this permits an extension of the results to higher order logic.

A functor from finitely cocontinuous and finitely continuous. Also J2 R' in S T(M 2) = M 1. (Y2) is monic in > S(Y2 ) ~pn C it follows that is T is y 6~ 2 is monic, then R'(H y) Hy is an onto map is onto. Hence T(y) Yi" T: Y2 To show that lent in J~P. 4 implies that Therefore respect If Jl" is cocontinuous R'(H M2 ) Therefore to S T = Q' This to direct > J1 follows limits in is a Horn theory map. D. and both functors with preserve direct limits. D. Similarly following with respect one can show that diagram commutes.

### Approximate solution of plastic flow theory problems by Vadim G. Korneev, Ulrich Langer

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