| Title:
|
State-homomorphisms on $MV$-algebras (English) |
| Author:
|
Jakubík, Ján |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
51 |
| Issue:
|
3 |
| Year:
|
2001 |
| Pages:
|
609-616 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
$MV$-algebra |
| Keyword:
|
state homomorphism |
| Keyword:
|
$\sigma $-closed maximal ideal |
| MSC:
|
06D35 |
| idZBL:
|
Zbl 1079.06501 |
| idMR:
|
MR1851550 |
| . |
| Date available:
|
2009-09-24T10:45:34Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127672 |
| . |
| Reference:
|
[1] F. Chovanec: States and observables on $MV$ algebras.Tatra Mt. Math. Publ. 3 (1993), 55–64. Zbl 0799.03074, MR 1278519 |
| Reference:
|
[2] P. Conrad: Lattice Ordered Groups.Tulane University, 1970. Zbl 0258.06011 |
| Reference:
|
[3] D. Gluschankof: Cyclic ordered groups and $MV$-algebras.Czechoslovak Math. J. 43 (1993), 249–263. Zbl 0795.06015, MR 1211747 |
| Reference:
|
[4] A. Goetz: On weak automorphisms and weak homomorphisms of abstract algebras.Coll. Math. 14 (1966), 163–167. MR 0184889, 10.4064/cm-14-1-163-167 |
| Reference:
|
[5] J. Jakubík: Direct product decompositions of $MV$-algebras.Czechoslovak Math. J. 44 (1994), 725–739. |
| Reference:
|
[6] J. Jakubík: On archimedean $MV$-algebras.Czechoslovak Math. J. 48 (1998), 575–582. MR 1637871, 10.1023/A:1022436113418 |
| Reference:
|
[7] J. Jakubík: Subdirect product decompositions of $MV$-algebras.Czechoslovak Math. J. 49(124) (1999), 163–173. MR 1676813, 10.1023/A:1022472528113 |
| Reference:
|
[8] D. Mundici: Interpretation of $AFC^*$-algebras in Łukasziewicz sentential calculus.J. Funct. Anal. 65 (1986), 15–53. MR 0819173, 10.1016/0022-1236(86)90015-7 |
| Reference:
|
[9] D. Mundici: Averaging the truth-value in Łukasziewicz logic.Studia Logica 55 (1995), 113–127. MR 1348840, 10.1007/BF01053035 |
| Reference:
|
[10] D. Mundici: Uncertainty measures in $MV$-algebras, and states of $AFC^*$-algebras.Notas Soc. Mat. Chile 15 (1996), 42–54. |
| Reference:
|
[11] B. Riečan: Fuzzy connectives and quantum models.In: Cybernetics and System Research 92, R. Trappl (ed.), World Scientific Publ., Singapore, 1992, pp. 335–338. |
| Reference:
|
[12] B. Riečan: On limit theorems in fuzzy quantum spaces.(Submitted). |
| Reference:
|
[13] B. Riečan and T. Neubrunn: Integral, Measure and Ordering.Kluwer Publ., Dordrecht, 1997. MR 1489521 |
| . |
