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.
Highlights
We propose a new notion of 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
We propose a new notion of states on these algebras
In this paper we have proposed a new notion of fit states on a Girard algebra opposing to Weber’s concept of weakly additive states which is based on additivity only for all sub-MV-algebras
Summary
Hohle and Weber (1997) proposed the following notion of an additivity of a state on a Girard algebra (Q; ℘, ℘−) (with the dual multiplication ℘ viewed as an addition and its dual residual ℘−): an isotonic map st: Q → [0, 1] satisfying boundary conditions is additive whenever x℘y ℘x. Weber (2010) proved the uniqueness of an additive state on a finite non-Boolean MV-chain, say Wm, having m − 1 (with m ≥ 2) non-trivial elements and proceeded to extend this state from Wm to its “canonical” non-MV-Girard algebra extension, say WmN1 The latter is defined as the set of all pairs (a, b) of elements a, b of Wm with a ≤ b equipped with a certain Girard algebra structure. Vol 6, No 4; 2014 of Q) is a mapping st: Q → [0, 1] such that st( ) = 1, and st(x℘ ̇ y) = st(x) + st(y), whenever the partial product x℘ ̇ y is defined in Q-it is nothing but the additive state in the sense of (1) It appeared that all MV-algebras are effectible and that principal non-trivial examples of effectible non-MV-Girard algebras are only mentioned canonical extensions W2N1 and W3N1 of MV-algebras W2 and W3, respectively. The author prefers to speak of multiplicativity property of states to be discussed on closer examination
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