State-homomorphisms on $MV$-algebras

Jakubík, Ján. "State-homomorphisms on $MV$-algebras." Czechoslovak Mathematical Journal 51.3 (2001): 609-616. <http://eudml.org/doc/30658>.

@article{Jakubík2001,
abstract = {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 $\mathop \{\mathrm \{c\}ard\}A>1$. 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.},
author = {Jakubík, Ján},
journal = {Czechoslovak Mathematical Journal},
keywords = {$MV$-algebra; state homomorphism; $\sigma $-closed maximal ideal; -algebra; state homomorphism; -closed maximal ideal},
language = {eng},
number = {3},
pages = {609-616},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {State-homomorphisms on $MV$-algebras},
url = {http://eudml.org/doc/30658},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Jakubík, Ján
TI - State-homomorphisms on $MV$-algebras
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 3
SP - 609
EP - 616
AB - 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 $\mathop {\mathrm {c}ard}A>1$. 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.
LA - eng
KW - $MV$-algebra; state homomorphism; $\sigma $-closed maximal ideal; -algebra; state homomorphism; -closed maximal ideal
UR - http://eudml.org/doc/30658
ER -