×

zbMATH — the first resource for mathematics

Radical classes of MV-algebras. (English) Zbl 0953.06012
For defining the notion of MV-algebra, several (equivalent) systems of axioms have been used in the literature. In the present paper it is shown that an MV-algebra can be characterized as a bounded distributive lattice with a partial binary operation (partial addition) satisfying certain axioms. In this context, the notion of a substructure of an MV-algebra is defined in a natural way. A nonempty class \(Y\) of MV-algebras which is closed with respect to isomorphisms will be called a radical class if the following conditions are satisfied: (i) whenever \(\mathcal A_1\in Y\) and \(\mathcal A_2\) is a substructure of \(\mathcal A_1\), then \(\mathcal A_2\in Y\); (ii) if \(\mathcal B\) is an MV-algebra and \(\mathcal A_1,\mathcal A_2,\dots , \mathcal A_n\) are substructures of \(\mathcal B\) such that \(\mathcal A_i\in Y\) for \(i=1,2,\dots ,n\), then \(\mathcal A_1\vee \mathcal A_2\vee \dots \vee \mathcal A_n\) belongs to \(Y\). Radical classes of lattice ordered groups have been investigated in several papers. Let \(\mathcal R_a\) and \(\mathcal R_m\) be the collection of all radical classes of abelian lattice ordered groups or the collection of all radical classes of MV-algebras, respectively. Both \(\mathcal R_a\) and \(\mathcal R_m\) are partially ordered by the class-theoretical inclusion. It is proved that the partially ordered collections \(\mathcal R_a\) and \(\mathcal R_m\) are isomorphic.
Reviewer: P.Němec (Praha)

MSC:
06D35 MV-algebras
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] C. C. Chang: Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88 (1958), 467-490. · Zbl 0084.00704
[2] R. Cignoli: Complete and atomic algebras of the infinite valued Łukasiewicz logic. Studia Logica 50 (1991), 375-384. · Zbl 0753.03026
[3] P. Conrad: \(K\)-radical classes of lattice ordered groups. Proc. Conf. Carbondale, Lecture Notes Math. 848, 1981, pp. 186-207. · Zbl 0455.06010
[4] M. Darnel: Closure operations on radicals of lattice ordered groups. Czechoslovak Math. J. 37(112) (1987), 51-64. · Zbl 0624.06022
[5] D. Gluschankof: Cyclic ordered groups and \(MV\)-algebras. Czechoslovak Math. J. 43(118) (1993), 249-263. · Zbl 0795.06015
[6] J. Jakubík: Radical mappings and radical classes of lattice ordered groups. Symposia Math. 21, Academic Press, New York-London, 1977, pp. 451-477.
[7] J. Jakubík: Products of radical classes of lattice ordered groups. Acta Math. Univ. Comenianae 39 (1980), 31-41.
[8] J. Jakubík: On \(K\)-radicals of lattice ordered groups. Czechoslovak Math. J. 33(108) (1983), 149-163.
[9] J. Jakubík: Direct product decompositions of \(MV\)-algebras. Czechoslovak Math. J. 44(119) (1994), 725-739. · Zbl 0821.06011
[10] J. Jakubík: On complete \(MV\)-algebras. Czechoslovak Math. J. 45(120) (1995), 473-480. · Zbl 0841.06010
[11] J. Jakubík: On archimedean \(MV\)-algebras. Czechoslovak Math. J. 48(123) (1998), 575-582. · Zbl 0951.06011
[12] J. Jakubík: Radical classes of generalized Boolean algebras. Czechoslovak Math. J. 48(123) (1998), 253-268. · Zbl 0952.06017
[13] N. Ya. Medvedev: On the lattice of radicals of a finitely generated \(\ell \)-group. Math. Slovaca 33 (1983), 185-188. · Zbl 0513.06009
[14] D. Mundici: Interpretation of \(AFC^*\)-algebras in Łukasiewicz sentential calculus. Journ. Functional Anal. 65 (1986), 15-63. · Zbl 0597.46059
[15] Dao-Rong Ton: Product radical classes of \(\ell \)-groups. Czechoslovak Math. J. 43(108) (1992), 129-142. · Zbl 0773.06019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.