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Subdirect products of rings and distributive lattices

Published online by Cambridge University Press:  20 January 2009

Hans-J. Bandelt  and
Mario Petrich
Hans-J. Bandelt
Affiliation:
Universitäy Oldenburg, D-2900 Oldenburg, F.R. Germany
Mario Petrich
Affiliation:
Universitäy Oldenburg, D-2900 Oldenburg, F.R. Germany
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Rings and distributive lattices can both be considered as semirings with commutative regular addition. Within this framework we can consider subdirect products of rings and distributive lattices. We may also require that the semirings with these restrictions are regarded as algebras with two binary operations and the unary operation of additive inversion (within the additive subgroup of the semiring). We can also consider distributive lattices with the two binary operations and the identity mapping as the unary operation. This makes it possible to speak of the join of ring varieties and distributive lattices. We restrict the ring varieties in order that their join with distributive lattices consist only of subdirect products. In certain cases these subdirect products can be obtained via a general construction of semirings by means of rings and distributive lattices.

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 25 , Issue 2 , June 1982 , pp. 155 - 171
Copyright
Copyright © Edinburgh Mathematical Society 1982

References

REFERENCES

1.GraczyŃska, E., On the sums of double systems of some universal algebras I, II, Bull. Acad. Polon. Sci. Sèr. Sci. Math. Astronom. Phys. 23 (1975), 509–513, 1055-1058.Google Scholar
2.McAlister, D. B., Partially ordered inverse semigroups, Conference on Semigroups (Tulane, 1978).Google Scholar
3.Michler, G. und Wille, R., Die primitiven Klassen arithmetischer Ringe, Math. Z. 113 (1970), 369–372.CrossRefGoogle Scholar
4.Newman, M. H. A., A characterisation of Boolean lattices and rings, J. London Math. Soc. 16 (1941), 256–272.CrossRefGoogle Scholar
5.Petrich, M., Introduction to Semigroups (Merrill, Columbus, 1973).Google Scholar
6.Rodriquez, G., Una classe di semianelli unione di anelli con unità, Atti Sent. Mat. Fis. Univ. Modena 23 (1974), 1–21.Google Scholar
7.Werner, H. und Wille, R., Charakterisierungen der primitiven Klassen arithmetischer Ringe, Math. Z. 115 (1970), 197–200.CrossRefGoogle Scholar
8.Salil3, V. N., K teorii inversnych polukolec, Izv. Vysš. Uč. Zav Mat. 3 (1969), 52–60.Google Scholar