On some types of radical classes

Ján Jakubík

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 3, page 833-848
  • ISSN: 0011-4642

Abstract

top
Let 𝔪 be an infinite cardinal. We denote by C 𝔪 the collection of all 𝔪 -representable Boolean algebras. Further, let C 𝔪 0 be the collection of all generalized Boolean algebras B such that for each b B , the interval [ 0 , b ] of B belongs to C 𝔪 . In this paper we prove that C 𝔪 0 is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized M V -algebras.

How to cite

top

Jakubík, Ján. "On some types of radical classes." Czechoslovak Mathematical Journal 58.3 (2008): 833-848. <http://eudml.org/doc/37871>.

@article{Jakubík2008,
abstract = {Let $\mathfrak \{m\}$ be an infinite cardinal. We denote by $C_\mathfrak \{m\}$ the collection of all $\mathfrak \{m\}$-representable Boolean algebras. Further, let $C_\mathfrak \{m\}^0$ be the collection of all generalized Boolean algebras $B$ such that for each $b\in B$, the interval $[0,b]$ of $B$ belongs to $C_\mathfrak \{m\}$. In this paper we prove that $C_\mathfrak \{m\}^0$ is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized $MV$-algebras.},
author = {Jakubík, Ján},
journal = {Czechoslovak Mathematical Journal},
keywords = {Boolean algebra; generalized Boolean algebra; $\mathfrak \{m\}$-representability; lattice ordered group; generalized $MV$-algebra; radical class; Boolean algebra; generalized Boolean algebra; -representability; lattice-ordered group},
language = {eng},
number = {3},
pages = {833-848},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On some types of radical classes},
url = {http://eudml.org/doc/37871},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Jakubík, Ján
TI - On some types of radical classes
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 3
SP - 833
EP - 848
AB - Let $\mathfrak {m}$ be an infinite cardinal. We denote by $C_\mathfrak {m}$ the collection of all $\mathfrak {m}$-representable Boolean algebras. Further, let $C_\mathfrak {m}^0$ be the collection of all generalized Boolean algebras $B$ such that for each $b\in B$, the interval $[0,b]$ of $B$ belongs to $C_\mathfrak {m}$. In this paper we prove that $C_\mathfrak {m}^0$ is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized $MV$-algebras.
LA - eng
KW - Boolean algebra; generalized Boolean algebra; $\mathfrak {m}$-representability; lattice ordered group; generalized $MV$-algebra; radical class; Boolean algebra; generalized Boolean algebra; -representability; lattice-ordered group
UR - http://eudml.org/doc/37871
ER -

References

top
  1. Chang, C. C., On the representation of α -complete Boolean algebras, Trans. Amer. Math. Soc. 85 (1957), 208-218. (1957) Zbl0080.25502MR0086792
  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. (1981) Zbl0455.06010MR0613186
  3. Conrad, P., Darnel, M. R., 10.1023/A:1013759300701, Czech. Math. J. 51 (2001), 395-413. (2001) Zbl0978.06011MR1844319DOI10.1023/A:1013759300701
  4. Darnel, M., Closure operators on radicals of lattice ordered groups, Czech. Math. J. 37 (1987), 51-64. (1987) MR0875127
  5. Dvurečenskij, A., 10.1017/S1446788700036806, J. Austral. Math. Soc. 72 (2002), 427-445. (2002) MR1902211DOI10.1017/S1446788700036806
  6. Georgescu, G., Iorgulescu, A., Pseudo M V -algebras: a noncommutative extension of M V -algebras, Proc. Fourth Int. Symp. Econ. Inf., Bucharest (1999), 961-968. (1999) Zbl0985.06007MR1730100
  7. Georgescu, G., Iorgulescu, A., Pseudo M V -algebras, Multiple Valued Logic 6 (2001), 95-135. (2001) Zbl1014.06008MR1817439
  8. Jakubík, J., Radical mappings and radical classes of lattice ordered groups, Symposia Math. 21 Academic Press New York-London (1977), 451-477. (1977) MR0491397
  9. Jakubík, J., 10.1023/A:1022885303504, Czech. Math. J. 48 (1998), 253-268. (1998) MR1624315DOI10.1023/A:1022885303504
  10. Jakubík, J., 10.1023/A:1022428713092, Czech. Math. J. 49 (1999), 191-211. (1999) MR1676805DOI10.1023/A:1022428713092
  11. Jakubík, J., Direct product decompositions of pseudo M V -algebras, Archivum Math. 37 (2001), 131-142. (2001) MR1838410
  12. Jakubík, J., 10.1023/A:1021711326115, Czech. Math. J. 52 (2002), 469-482. (2002) MR1923254DOI10.1023/A:1021711326115
  13. Loomis, L. H., 10.1090/S0002-9904-1947-08866-2, Bull. Amer. Math. Soc. 53 (1947), 757-760. (1947) Zbl0033.01103MR0021084DOI10.1090/S0002-9904-1947-08866-2
  14. Pierce, R. S., 10.1090/S0002-9939-1959-0106862-6, Proc. Amer. Math. Soc. 10 (1959), 42-50. (1959) Zbl0091.03102MR0106862DOI10.1090/S0002-9939-1959-0106862-6
  15. Rachůnek, J., 10.1023/A:1021766309509, Czech. Math. J. 52 (2002), 255-273. (2002) MR1905434DOI10.1023/A:1021766309509
  16. Scott, D., A new characterization of α -representable Boolean algebras, Bull. Amer. Math. Soc. 61 (1955), 522-523. (1955) 
  17. E. C. Smith, Jr., 10.2307/1969602, Ann. Math. 64 (1956), 551-561. (1956) Zbl0074.02105MR0086047DOI10.2307/1969602
  18. Sikorski, R., On the representation of Boolean algebras as fields of set, Fund. Math. 35 (1958), 247-258. (1958) MR0028374
  19. Sikorski, R., Distributivity and representability, Fund. Math. 48 (1959), 95-103. (1959) MR0109799
  20. Sikorski, R., Boolean Algebras, Second Edition Springer Verlag Berlin-Göttingen-Heidelberg-New York (1964). (1964) Zbl0123.01303
  21. Ton, Dao Rong, Product radical classes of -groups, Czech. Math. J. 42 (1992), 129-142. (1992) MR1152176

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.