Abstract

Recently Weber proposed to define ``weakly additive" states on a Girard algebra by the additivity only on its sub-$MV$-algebras and characterized such states on the canonical Girard algebra extensions of any finite $MV$-chain. In the present paper, we take another viewpoint: the arguable sub-$MV$-algebras are replaced by suitable substructures coming from author, H\"{o}hle and Weber's own previous investigations. We propose a new notion of \emph{fit} states on a Girard algebra by the additivity on the mentioned substructures and consider such states on the ``non-effectible" Girard algebra ``$n$-extensions" (= canonical extensions when $n=1$) of $MV$-chains restricting ourselves to ones having less    than six nontrivial elements. Our fit states appear as solutions of certain inconsistent systems of linear equations. They have extensive enough domains of the additivity-in any comparable case more extensive than Weber's states have.

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