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State-homomorphisms on $MV$-algebras
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 609-616 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Riečan [12] and Chovanec [1] investigated states in $MV$-algebras. Earlier, Riečan [11] had dealt with analogous ideas in $D$-posets. In the monograph of Riečan and Neubrunn [13] (Chapter 9) the notion of state is applied in the theory of probability on $MV$-algebras. We remark that a different definition of a state in an $MV$-algebra has been applied by Mundici [9], [10] (namely, the condition (iii) from Definition 1.1 above was not included in his definition of a state; in other words, only finite additivity was assumed). Below we work with the definition from [13]; but, in order to avoid terminological problems we use the term “state-homomorphism” (instead of “state”). The author is indebted to the referee for his suggestion concerning terminology. Let $\mathcal A$ be an $MV$-algebra which is defined on a set $A$ with $\mathop {\mathrm card}A>1$. In the present paper we show that there exists a one-to-one correspondence between the system of all state-homomorphisms on $\mathcal A$ and the system of all $\sigma $-closed maximal ideals of $\mathcal A$. For $MV$-algebras we apply the notation and the definitions as in Gluschankof [3]. The relations between $MV$-algebras and abelian lattice ordered groups (cf. Mundici [8]) are substantially used in the present paper.
Riečan [12] and Chovanec [1] investigated states in $MV$-algebras. Earlier, Riečan [11] had dealt with analogous ideas in $D$-posets. In the monograph of Riečan and Neubrunn [13] (Chapter 9) the notion of state is applied in the theory of probability on $MV$-algebras. We remark that a different definition of a state in an $MV$-algebra has been applied by Mundici [9], [10] (namely, the condition (iii) from Definition 1.1 above was not included in his definition of a state; in other words, only finite additivity was assumed). Below we work with the definition from [13]; but, in order to avoid terminological problems we use the term “state-homomorphism” (instead of “state”). The author is indebted to the referee for his suggestion concerning terminology. Let $\mathcal A$ be an $MV$-algebra which is defined on a set $A$ with $\mathop {\mathrm card}A>1$. In the present paper we show that there exists a one-to-one correspondence between the system of all state-homomorphisms on $\mathcal A$ and the system of all $\sigma $-closed maximal ideals of $\mathcal A$. For $MV$-algebras we apply the notation and the definitions as in Gluschankof [3]. The relations between $MV$-algebras and abelian lattice ordered groups (cf. Mundici [8]) are substantially used in the present paper.
Classification : 06D35
Keywords: $MV$-algebra; state homomorphism; $\sigma $-closed maximal ideal 
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Jakubík, Ján. State-homomorphisms on $MV$-algebras. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 609-616. http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a11/

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