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Keywords:
Boolean algebra; generalized Boolean algebra; $\frak m$-representability; lattice ordered group; generalized $MV$-algebra; radical class
Summary:
Let $\frak m$ be an infinite cardinal. We denote by $C_\frak m$ the collection of all $\frak m$-representable Boolean algebras. Further, let $C_\frak m^0$ be the collection of all generalized Boolean algebras $B$ such that for each $b\in B$, the interval $[0,b]$ of $B$ belongs to $C_\frak m$. In this paper we prove that $C_\frak m^0$ is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized $MV$-algebras.
References:
[1] Chang, C. C.: On the representation of $\alpha$-complete Boolean algebras. Trans. Amer. Math. Soc. 85 (1957), 208-218. MR 0086792 | Zbl 0080.25502
[2] Conrad, P.: $K$-radical classes of lattice ordered groups. In: Proc. Conf. Carbondale, Lecture Notes Math 848 Springer Verlag New York (1981), 186-207. MR 0613186 | Zbl 0455.06010
[3] Conrad, P., Darnel, M. R.: Subgroups and hulls of Specker lattice ordered groups. Czech. Math. J. 51 (2001), 395-413. DOI 10.1023/A:1013759300701 | MR 1844319 | Zbl 0978.06011
[4] Darnel, M.: Closure operators on radicals of lattice ordered groups. Czech. Math. J. 37 (1987), 51-64. MR 0875127
[5] Dvurečenskij, A.: Pseudo $MV$-algebras are intervals in $\ell$-groups. J. Austral. Math. Soc. 72 (2002), 427-445. DOI 10.1017/S1446788700036806 | MR 1902211
[6] Georgescu, G., Iorgulescu, A.: Pseudo $MV$-algebras: a noncommutative extension of $MV$-algebras. Proc. Fourth Int. Symp. Econ. Inf., Bucharest (1999), 961-968. MR 1730100 | Zbl 0985.06007
[7] Georgescu, G., Iorgulescu, A.: Pseudo $MV$-algebras. Multiple Valued Logic 6 (2001), 95-135. MR 1817439 | Zbl 1014.06008
[8] Jakubík, J.: Radical mappings and radical classes of lattice ordered groups. Symposia Math. 21 Academic Press New York-London (1977), 451-477. MR 0491397
[9] Jakubík, J.: Radical classes of generalized Boolean algebras. Czech. Math. J. 48 (1998), 253-268. DOI 10.1023/A:1022885303504 | MR 1624315
[10] Jakubík, J.: Radical classes of $MV$-algebras. Czech. Math. J. 49 (1999), 191-211. DOI 10.1023/A:1022428713092 | MR 1676805
[11] Jakubík, J.: Direct product decompositions of pseudo $MV$-algebras. Archivum Math. 37 (2001), 131-142. MR 1838410
[12] Jakubík, J.: Torsion classes of Specker lattice ordered groups. Czech. Math. J. 52 (2002), 469-482. DOI 10.1023/A:1021711326115 | MR 1923254
[13] Loomis, L. H.: On the representation of $\sigma$-complete Boolean algebras. Bull. Amer. Math. Soc. 53 (1947), 757-760. DOI 10.1090/S0002-9904-1947-08866-2 | MR 0021084 | Zbl 0033.01103
[14] Pierce, R. S.: Representation theorems for certain Boolean algebras. Proc. Amer. Math. Soc. 10 (1959), 42-50. DOI 10.1090/S0002-9939-1959-0106862-6 | MR 0106862 | Zbl 0091.03102
[15] Rachůnek, J.: A non-commutative generalization of $MV$-algebras. Czech. Math. J. 52 (2002), 255-273. DOI 10.1023/A:1021766309509 | MR 1905434
[16] Scott, D.: A new characterization of $\alpha$-representable Boolean algebras. Bull. Amer. Math. Soc. 61 (1955), 522-523.
[17] E. C. Smith, Jr.: A distributivity condition for Boolean algebras. Ann. Math. 64 (1956), 551-561. DOI 10.2307/1969602 | MR 0086047 | Zbl 0074.02105
[18] Sikorski, R.: On the representation of Boolean algebras as fields of set. Fund. Math. 35 (1958), 247-258. MR 0028374
[19] Sikorski, R.: Distributivity and representability. Fund. Math. 48 (1959), 95-103. MR 0109799
[20] Sikorski, R.: Boolean Algebras. Second Edition Springer Verlag Berlin-Göttingen-Heidelberg-New York (1964). Zbl 0123.01303
[21] Ton, Dao Rong: Product radical classes of $\ell$-groups. Czech. Math. J. 42 (1992), 129-142. MR 1152176
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