Mathematica Bohemica, Vol. 142, No. 2, pp. 125-136, 2017
Statisch pairs in atomistic posets
Alireza Vaezi, Vilas Kharat
Received November 21, 2013. First published November 24, 2016.
Abstract: We introduce statisch pairs in atomistic posets and study its relationships with some known concepts in posets such as biatomic and dual modular pairs, perspectivity and subspaces of atom space of an atomistic poset. We generalize the notion of exchange property in posets and with the help of it we prove the equivalence of dual modular, biatomic and statisch pairs in atomistic posets. Also, we prove that the set of all finite elements of a statisch poset with such property forms an ideal. $\nabla$-relation is partly studied by means of statisch pairs.
Keywords: atomistic poset; statisch pair; finite element
References:  M. F. Janowitz: On the modular relation in atomistic lattices. Fundam. Math. 66 (1970), 337-346. MR 0260635 | Zbl 0193.33901  M. F. Janowitz: Examples of statisch and finite statisch AC-lattices. Fundam. Math. 89 225-227 (1975). MR 0382095 | Zbl 0321.06009  F. Maeda, S. Maeda: Theory of Symmetric Lattices. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 173. Springer, Berlin (1970). DOI 10.1007/978-3-642-46248-1 | MR 0282889 | Zbl 0219.06002  R. S. Shewale: Modular Pairs, Forbidden Configurations and Related Aspects in Partially Ordered Sets. Ph. D. Thesis, University of Pune, Pune (INDIA) (2010).  N. K. Thakare, S. Maeda, B. N. Waphare: Modular pairs, covering property and related results in posets. J. Indian Math. Soc. (N.S.) 70 229-253 (2003). MR 2290805 | Zbl 1117.06300  A. Vaezi, V. Kharat: Completeness and compact generation in partially ordered sets. J. Math. Computer Sci. (electronic only) 1 (2016), 69-76. Avaible at http://www.isr-publications.com/jmcs/volume-16/issue-1.  B. N. Waphare: Symmetricity in Posets, Comparability Axioms in Orthomodular Lattices with Applications. Ph. D. Thesis, University of Pune, Pune, India (1992).  B. N. Waphare, V. V. Joshi: Distributive pairs in biatomic lattices. Acta. Math. Univ. Comen., New Ser. 73 (2004), 69-73. MR 2076044 | Zbl 1094.06007  B. N. Waphare, V. V. Joshi: Biatomic pairs in posets. Southeast Asian Bull. Math. 31 (2007), 789-800. MR 2363032 | Zbl 1150.06005  B. N. Waphare, V. V. Joshi: Distributive pairs in posets. Southeast Asian Bull. Math. 31 (2007), 1205-1233. MR 2386998 | Zbl 1150.06006  M. P. Wasadikar, N. K. Thakare: $p$-elements and statisch pairs in lattices. Publ. Cent. Rech. Math. Pures Sér. I 27 (1996), 1-5. Zbl 0842.06004  R. Wille: Halbkomplementäre Verbände. Math. Z. 94 (1966), 1-31. DOI 10.1007/BF01111258 | MR 0216989 | Zbl 0144.00704
Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at firstname.lastname@example.org. Subscribers of Springer need to access the articles on their site, which is http://mb.math.cas.cz/.
Affiliations: Alireza Vaezi, Department of Mathematics, Shahid Rajaee Teacher Training University, Imam Ali Highway, Tehran, 16785-136, Iran, e-mail: email@example.com; Vilas Kharat, Department of Mathematics, Savitribai Phule Pune University, Pune University Road, Ganeshkhind, Pune, Maharastra-411007, India, e-mail: firstname.lastname@example.org