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On a common abstraction of Boolean rings and lattice ordered groups II

Published online by Cambridge University Press:  09 April 2009

V. V. Rama Rao
V. V. Rama Rao
Affiliation:
Department of Mathematics (Arts College) Andhra University Waltair (India)
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In an earlier paper, the author has obtained a solution [8] to Birkhoff's problem No. 105 [1]: Is there a common abstraction which includes Boolean algebras (Rings) and lattice ordered groups as special cases? The solution actually turns out to be the direct product of a Boolean ring and a lattice ordered group. Birkhoff's problem has also been solved by Swamy, Wyler, and Nakano by presenting respectively i) Dually Residuated Lattice Ordered Semigroups (D.R.1. semigroups) [2, 3, 4, 5], ii) Clans [6], and iii) Multirings [7].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 2 , March 1972 , pp. 185 - 195
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Birkhoff, G., Lattice Theory (Amer. Math. Soc. Col. Pub XXV, 1948).Google Scholar
[2]Swamy, K. L. N., ‘Dually Residuated Lattice Ordered Semigroups’, Math. Ann. 159, 105–114.CrossRefGoogle Scholar
[3]Swamy, K. L. N., ‘Dually Residuated Lattice Ordered Semigroups II’, Math. Ann. 160, 64–74.CrossRefGoogle Scholar
[4]Swamy, K. L. N., ‘Dually Residuated Lattice Ordered Semigroup III’, Math. Ann. 167, 71–74.CrossRefGoogle Scholar
[5]Swamy, K. L. N., Dually Residuared Lattice Ordered Semigroups (Doctoral Thesis, Andhra University, 196–).Google Scholar
[6]Wyler, O., ‘Clans’, Comp. Math. 17, 172–189.Google Scholar
[7]Nakano, T., ‘Rings and Partly Ordered Systems’, Math. Z. 99. 355–376.CrossRefGoogle Scholar
[8]Rao, V. V. Rama, ‘On a Common Abstraction of Booleanrings and Lattice Ordered groups 1’, Monat. Für. Math. 73, 411–421 (1969).CrossRefGoogle Scholar
[9]Pitcher, Everett and Smiley, M. F., ‘Transitivities of Betweenness’, T. Amer. Math. Soc. 52. (1942), 95–114.Google Scholar