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Title: Properties of relatively pseudocomplemented directoids (English)
Author: Chajda, Ivan
Author: Kolařík, Miroslav
Author: Švrček, Filip
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 136
Issue: 1
Year: 2011
Pages: 9-23
Summary lang: English
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Category: math
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Summary: The concept of a relatively pseudocomplemented directoid was introduced recently by the first author. It was shown that the class of relatively pseudocomplemented directoids forms a variety whose axiom system contains seven identities. The aim of this paper is three-fold. First we show that these identities are not independent and their independent subset is presented. Second, we modify the adjointness property known for relatively pseudocomplemented semilattices in the way which is suitable for relatively pseudocomplemented directoids. Hence, they can also be considered as residuated structures in a rather modified version. We also get two important congruence properties, namely congruence distributivity and $3$-permutability valid in the variety $\mathcal {V}$ of relatively pseudocomplemented directoids. Then we show some basic results connected with subdirect irreducibility in $\mathcal {V}$. Finally, we show another way how to introduce pseudocomplementation on directoids via relative pseudocomplementation. (English)
Keyword: directoid
Keyword: relatively pseudocomplemented directoid
Keyword: congruence distributivity
Keyword: $3$-permutability
Keyword: residuated structure
Keyword: adjointness property
Keyword: variety
MSC: 06A12
MSC: 06D15
MSC: 08B10
idZBL: Zbl 1224.06006
idMR: MR2807704
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Date available: 2011-03-31T11:20:10Z
Last updated: 2013-07-31
Stable URL: http://hdl.handle.net/10338.dmlcz/141445
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Reference: [1] Chajda, I.: An extension of relative pseudocomplementation to non-distributive lattices.Acta Sci. Math. (Szeged) 69 (2003), 491-496. Zbl 1048.06005, MR 2034188
Reference: [2] Chajda, I.: Pseudocomplemented directoids.Comment. Math. Univ. Carol. 49 (2008), 533-539. Zbl 1212.06005, MR 2493936
Reference: [3] Chajda, I.: Relatively pseudocomplemented directoids.Comment. Math. Univ. Carol. 50 (2009), 349-357. Zbl 1212.06004, MR 2573409
Reference: [4] Chajda, I., Eigenthaler, G., Länger, H.: Congruence Classes in Universal Algebra.Heldermann Lemgo (2003), 217. Zbl 1014.08001, MR 1985832
Reference: [5] Chajda, I., Halaš, R., Kühr, J.: Semilattice Structures.Heldermann Lemgo (2007), 228. Zbl 1117.06001, MR 2326262
Reference: [6] Hagemann, J., Mitschke, A.: On $n$-permutable congruences.Algebra Universalis 3 (1973), 8-12. Zbl 0273.08001, MR 0330010, 10.1007/BF02945100
Reference: [7] Ježek, J., Quackenbush, R.: Directoids: Algebraic models of up-directed sets.Algebra Universalis 27 (1990), 49-69. MR 1025835, 10.1007/BF01190253
Reference: [8] Jones, G. T.: Pseudo-complemented Semi-lattices.Ph.D. Thesis Univ. of California, Los Angeles (1972). MR 2622711
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