Discussiones
Mathematicae General Algebra and Applications 20(1) (2000) 87-95
DOI: https://doi.org/10.7151/dmgaa.1008
RING-LIKE OPERATIONS IN PSEUDOCOMPLEMENTED SEMILATTICES
Ivan Chajda Department of Algebra and Geometry, Palacký
University Olomouc |
Helmut Länger Technische Universität Wien, Institut für Algebra
und Computermathematik |
Abstract
Ring-like operations are introduced in pseudocomplemented semilattices in such a way that in the case of Boolean pseudocomplemented semilattices one obtains the corresponding Boolean ring operations. Properties of these ring-like operations are derived and a characterization of Boolean pseudocomplemented semilattices in terms of these operations is given. Finally, ideals in the ring-like structures are defined and characterized.
Keywords: pseudocomplemented semilattice, Boolean algebra, Boolean ring, distributivity, linear equation, ideal, congruence kernel.
1991 Mathematics Subject Classification: 06A12, 08C10, 06E20, 16Y99.
References
| [1] | I. Chajda, Pseudosemirings induced by ortholattices, Czechoslovak Math. J. 46 (121) (1996), 405-411. |
| [2] | G. Dorfer, A. Dvurecenskij and H. Länger, Symmetric difference in orthomodular lattices, Math. Slovaca 46 (1996), 435-444. |
| [3] | D. Dorninger, H. Länger and M. Maczy\'nski, The logic induced by a system of homomorphisms and its various algebraic characterizations, Demonstratio Math. 30 (1997), 215-232. |
| [4] | O. Frink, Pseudo-complements in semi-lattices, Duke Math. J. 29 (1962), 505-514. |
| [5] | H. Länger, Generalizations of the correspondence between Boolean algebras and Boolean rings to orthomodular lattices, Tatra Mt. Math. Publ. 15 (1998), 97-105. |
| [6] | A.I. Mal'cev, On the general theory of algebraic systems (Russian), Mat. Sb. 35 (1954), 3-20. |
Received 21 September 1998
Revised 7 June 1999
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