Matematicky ustav SAV, Gresakova 6, 040 01 Kosice, Slovakia, e-mail: musavke@mail.saske.sk
Abstract: Riecan [12] and Chovanec [1] investigated states in $MV$-algebras. Earlier, Riecan [11] had dealt with analogous ideas in $D$-posets. In the monograph of Riecan 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 $\Cal A$ be an $MV$-algebra which is defined on a set $A$ with $\card A>1$. In the present paper we show that there exists a one-to-one correspondence between the system of all state-homomorphisms on $\Cal A$ and the system of all $\sigma$-closed maximal ideals of $\Cal A$. 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.
Keywords: $MV$-algebra, state homomorphism, $\sigma$-closed maximal ideal
Classification (MSC 2000): 06D35
Full text available as PDF (smallest), as compressed PostScript (.ps.gz) or as raw PostScript (.ps).
Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade.
To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer (formerly Kluwer) need to access the articles on their site, which is http://www.springeronline.com/10587.
