Geodesic
Properties of relatively pseudocomplemented directoids
Mathematica Bohemica, Tome 136 (2011) no. 1, pp. 9-23

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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.
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.
DOI : 10.21136/MB.2011.141445
Classification : 06A12, 06D15, 08B10
Keywords: directoid; relatively pseudocomplemented directoid; congruence distributivity; $3$-permutability; residuated structure; adjointness property; variety 
Chajda, Ivan; Kolařík, Miroslav; Švrček, Filip. Properties of relatively pseudocomplemented directoids. Mathematica Bohemica, Tome 136 (2011) no. 1, pp. 9-23. doi: 10.21136/MB.2011.141445
@article{10_21136_MB_2011_141445,
     author = {Chajda, Ivan and Kola\v{r}{\'\i}k, Miroslav and \v{S}vr\v{c}ek, Filip},
     title = {Properties of relatively pseudocomplemented directoids},
     journal = {Mathematica Bohemica},
     pages = {9--23},
     year = {2011},
     volume = {136},
     number = {1},
     doi = {10.21136/MB.2011.141445},
     mrnumber = {2807704},
     zbl = {1224.06006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2011.141445/}
}
TY  - JOUR
AU  - Chajda, Ivan
AU  - Kolařík, Miroslav
AU  - Švrček, Filip
TI  - Properties of relatively pseudocomplemented directoids
JO  - Mathematica Bohemica
PY  - 2011
SP  - 9
EP  - 23
VL  - 136
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2011.141445/
DO  - 10.21136/MB.2011.141445
LA  - en
ID  - 10_21136_MB_2011_141445
ER  - 
%0 Journal Article
%A Chajda, Ivan
%A Kolařík, Miroslav
%A Švrček, Filip
%T Properties of relatively pseudocomplemented directoids
%J Mathematica Bohemica
%D 2011
%P 9-23
%V 136
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2011.141445/
%R 10.21136/MB.2011.141445
%G en
%F 10_21136_MB_2011_141445

[1] Chajda, I.: An extension of relative pseudocomplementation to non-distributive lattices. Acta Sci. Math. (Szeged) 69 (2003), 491-496. | MR | Zbl

[2] Chajda, I.: Pseudocomplemented directoids. Comment. Math. Univ. Carol. 49 (2008), 533-539. | MR | Zbl

[3] Chajda, I.: Relatively pseudocomplemented directoids. Comment. Math. Univ. Carol. 50 (2009), 349-357. | MR | Zbl

[4] Chajda, I., Eigenthaler, G., Länger, H.: Congruence Classes in Universal Algebra. Heldermann Lemgo (2003), 217. | MR | Zbl

[5] Chajda, I., Halaš, R., Kühr, J.: Semilattice Structures. Heldermann Lemgo (2007), 228. | MR | Zbl

[6] Hagemann, J., Mitschke, A.: On $n$-permutable congruences. Algebra Universalis 3 (1973), 8-12. | DOI | MR | Zbl

[7] Ježek, J., Quackenbush, R.: Directoids: Algebraic models of up-directed sets. Algebra Universalis 27 (1990), 49-69. | DOI | MR

[8] Jones, G. T.: Pseudo-complemented Semi-lattices. Ph.D. Thesis Univ. of California, Los Angeles (1972). | MR

Cité par Sources :