Relatively pseudocomplemented posets

Ivan Chajda; Helmut Länger

Mathematica Bohemica (2018)

  • Volume: 143, Issue: 1, page 89-97
  • ISSN: 0862-7959

Abstract

top
We extend the notion of a relatively pseudocomplemented meet-semilattice to arbitrary posets. We show some properties of the binary operation of relative pseudocomplementation and provide some corresponding characterizations. We show that relatively pseudocomplemented posets satisfying a certain simple identity in two variables are join-semilattices. Finally, we show that every relatively pseudocomplemented poset is distributive and that the converse holds for posets satisfying the ascending chain condition and one more natural condition. Suitable examples are provided.

How to cite

top

Chajda, Ivan, and Länger, Helmut. "Relatively pseudocomplemented posets." Mathematica Bohemica 143.1 (2018): 89-97. <http://eudml.org/doc/294284>.

@article{Chajda2018,
abstract = {We extend the notion of a relatively pseudocomplemented meet-semilattice to arbitrary posets. We show some properties of the binary operation of relative pseudocomplementation and provide some corresponding characterizations. We show that relatively pseudocomplemented posets satisfying a certain simple identity in two variables are join-semilattices. Finally, we show that every relatively pseudocomplemented poset is distributive and that the converse holds for posets satisfying the ascending chain condition and one more natural condition. Suitable examples are provided.},
author = {Chajda, Ivan, Länger, Helmut},
journal = {Mathematica Bohemica},
keywords = {relatively pseudocomplemented poset; join-semilattice; distributive poset},
language = {eng},
number = {1},
pages = {89-97},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Relatively pseudocomplemented posets},
url = {http://eudml.org/doc/294284},
volume = {143},
year = {2018},
}

TY - JOUR
AU - Chajda, Ivan
AU - Länger, Helmut
TI - Relatively pseudocomplemented posets
JO - Mathematica Bohemica
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 143
IS - 1
SP - 89
EP - 97
AB - We extend the notion of a relatively pseudocomplemented meet-semilattice to arbitrary posets. We show some properties of the binary operation of relative pseudocomplementation and provide some corresponding characterizations. We show that relatively pseudocomplemented posets satisfying a certain simple identity in two variables are join-semilattices. Finally, we show that every relatively pseudocomplemented poset is distributive and that the converse holds for posets satisfying the ascending chain condition and one more natural condition. Suitable examples are provided.
LA - eng
KW - relatively pseudocomplemented poset; join-semilattice; distributive poset
UR - http://eudml.org/doc/294284
ER -

References

top
  1. Balbes, R., 10.4064/fm-78-2-119-131, Fundam. Math. 78 (1973), 119-131. (1973) Zbl0277.06001MR0319832DOI10.4064/fm-78-2-119-131
  2. Chajda, I., An extension of relative pseudocomplementation to non-distributive lattices, Acta Sci. Math. 69 (2003), 491-496. (2003) Zbl1048.06005MR2034188
  3. Chajda, I., Relatively pseudocomplemented directoids, Commentat. Math. Univ. Carol. 50 (2009), 349-357. (2009) Zbl1212.06004MR2573409
  4. Chajda, I., Pseudocomplemented and Stone posets, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 51 (2012), 29-34. (2012) Zbl1302.06001MR3060006
  5. Chajda, I., Halaš, R., Kühr, J., Semilattice Structures, Research and Exposition in Mathematics 30. Heldermann, Lemgo (2007). (2007) Zbl1117.06001MR2326262
  6. Chajda, I., Kolařík, M., Švrček, F., Properties of relatively pseudocomplemented directoids, Math. Bohem. 136 (2011), 9-23. (2011) Zbl1224.06006MR2807704
  7. Chajda, I., Länger, H., Directoids. An Algebraic Approach to Ordered Sets, Research and Exposition in Mathematics 32. Heldermann, Lemgo (2011). (2011) Zbl1254.06002MR2850357
  8. Chajda, I., Rachůnek, J., 10.1007/BF00353659, Order 5 (1989), 407-423. (1989) Zbl0674.06003MR1010389DOI10.1007/BF00353659
  9. Crulis, J., Implication in sectionally pseudocomplemented posets, Acta Sci. Math. 74 (2008), 477-491. (2008) Zbl1199.03059MR2487926
  10. Köhler, P., 10.2307/1998339, Trans. AMS 268 (1981), 103-126. (1981) Zbl0473.06003MR0628448DOI10.2307/1998339
  11. Larmerová, J., Rachůnek, J., Translations of distributive and modular ordered sets, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 27 (1988), 13-23. (1988) Zbl0693.06003MR1039879
  12. Nemitz, W. C., 10.2307/1994200, Trans. AMS 117 (1965), 128-142. (1965) Zbl0674.06003MR0176944DOI10.2307/1994200
  13. Venkatanarasimhan, P. V., 10.2307/2037746, Proc. AMS 28 (1971), 9-17. (1971) Zbl0218.06002MR0272687DOI10.2307/2037746

NotesEmbed ?

top

You must be logged in to post comments.