×
Zhang, Wenfeng; Xu, Xiaoquan

\( \mathcal{Z} \)-quasidistributive and \(\mathcal{Z} \)-meet-distributive posets. (English) Zbl 1483.06006

Order 37, No. 1, 103-113 (2020).
In domain theory, one fundamental result states that a poset is continuous if and only if it is quasicontinuous and meet-continuous. In this paper, the authors study two kinds of distributivity: \(Z\)-quasidistributivity and \(Z\)-meet-distributivity, which are the generalizations of quasicontinuity and meet-continuity. Here, \(Z\) is a subset system, it becomes meaningful when \(Z\) is replaced by adjectives such as “directed”, “chain”, “finite”, etc. Analogous to the above fundamental result, the authors prove that, under some conditions, a poset is \(Z\)-predistributive iff it is \(Z\)-quasidistributive and \(Z\)-meet-distributive.
In order theory, the Dedekind-MacNeille completion is the most well-known completion, which embeds a poset into a complete lattice. The order-theoretical properties which are invariant under the Dedekind-MacNeille completion are called completion-invariant. In this paper, one main result states that \(Z\)-quasidistributivity is a completion-invariant property whenever \(Z\) is completion-stable.
The way-below relation is a fundamental concept in domain theory. Replacing directed sets by \(Z\)-sets, one has the concept of \(Z\)-below. The last main result of this paper: if the \(Z\)-below relation on the subsets of a poset \(P\) has the interpolation property, then \(P\) is embeddable in a cube.
Reviewer: Zhongxi Zhang (Yantai)

MSC:

06A06 Partial orders, general
06B23 Complete lattices, completions
06D75 Other generalizations of distributive lattices

Keywords:

completion-invariant; normal completion; \(Z\)-initial completion; \(Z\)-predistributive poset; \(Z\)-quasidistributive poset; \(Z\)-meet-distributive poset
PDF BibTeX XML Cite
\textit{W. Zhang} and \textit{X. Xu}, Order 37, No. 1, 103--113 (2020; Zbl 1483.06006)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Banaschewski, B.; Bruns, G., Categorical characterization of the MacNeille completion, Arch. Math., 18, 369-377 (1967) · Zbl 0157.34101
[2] Baranga, A., Z-continuous posets, topological aspects, Stud. Cerc. Mat., 49, 3-16 (1997) · Zbl 0883.06007
[3] Crawley, P.; Dilworth, RP, Algebraic Theory of Lattices (1973), Englewood Cliffs: Prentice-Hall, Inc., Englewood Cliffs · Zbl 0494.06001
[4] Dedekind, R., Stetigkeit Und Irrationale Zahlen (1872), Braunschweig: Vieweg, Braunschweig · JFM 36.0242.01
[5] Erné, Marcel, A completion - invariant extension of the concept of continuous lattices, Lecture Notes in Mathematics, 45-60 (1981), Berlin, Heidelberg: Springer Berlin Heidelberg, Berlin, Heidelberg · Zbl 0467.06004
[6] Erné, Marcel, Scott convergence and scott topology in partially ordered sets II, Lecture Notes in Mathematics, 61-96 (1981), Berlin, Heidelberg: Springer Berlin Heidelberg, Berlin, Heidelberg · Zbl 0482.54023
[7] Erné, M., Distributivgesetze und die Dedekindsche schnittvervollständigung, Abh. Braun-schweig. Wiss. Ges., 33, 117-145 (1982)
[8] Erné, M.: Adjunctions and standard constructions for partially ordered sets. In: Eigenthaler, G., et al. (eds.) Contributions to General Algebra 2, Proc. Klagenfurt, 1982, pp 77-106. Wien, Hölder - Pichler - Tempsky (1983)
[9] Erné, M., Order extensions as adjoint functors, Quaest. Math., 9, 149-206 (1986) · Zbl 0602.06002
[10] Erné, M., The Dedekind-MacNeille completion as a reflector, Order, 8, 159-173 (1991) · Zbl 0738.06004
[11] Erné, M., \( \mathcal{Z} \)-continuous posets and their topological manifestation, Appl. Cat. Struct., 7, 31-70 (1999) · Zbl 0939.06005
[12] Erné, M., Infinite distributive laws versus local connectedness and compactness properties, Topol. Appl., 156, 2054-2069 (2009) · Zbl 1190.54022
[13] Erné, M., Categories of locally hypercompact spaces and quasicontinuous posets, Appl. Categ. Struct., 26, 823-854 (2018) · Zbl 1441.06003
[14] Frink, O., Ideals in partially ordered sets, Amer. Math. Monthly, 61, 223-234 (1954) · Zbl 0055.25901
[15] Gierz, G.; Hofmann, KH; Keimel, K.; Lawson, JD; Mislove, M.; Scott, DS, Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications, vol. 93 (2003), Cambridge: Cambridge University Press, Cambridge
[16] Gierz, G.; Lawson, JD; Stralka, AR, Quasicontinuous posets, Houston J. Math., 9, 191-208 (1983) · Zbl 0529.06002
[17] Harding, J.; Bezhanishvili, G., Macneille completions of Heyting algebras, Houston J. Math., 30, 937-952 (2004) · Zbl 1066.06005
[18] Heckmann, R.; Keimel, K., Quasicontinuous domains and the Smyth powerdomain, Electron. Notes Theor. Comput. Sci., 298, 215-232 (2013) · Zbl 1334.68127
[19] MacNeille, HM, Partially ordered sets, Trans. Amer. Math. Soc., 42, 416-460 (1937) · JFM 63.0833.04
[20] Menon, VG, A note on topology of Z-continuous posets, Comment. Math. Univ. Carolin., 37, 821-824 (1996) · Zbl 0888.06005
[21] Niederle, J., On infinitely distributive ordered sets, Math. Slovaca, 55, 495-502 (2005) · Zbl 1150.06002
[22] Nowak, D., Generalization of continuous posets, Trans. Amer. Math. Soc., 272, 645-667 (1982) · Zbl 0504.06003
[23] Ruan, X.; Xu, X., s_Z-quasicontinuous posets and meet s_Z-continuous posets, Topol. Appl., 230, 295-307 (2017) · Zbl 1475.06002
[24] Ruan, X.; Xu, X., A completion-invariant extension of the concept of quasi C-continuous lattices, Filomat, 31, 8, 2345-2353 (2017)
[25] Schmidt, J., Zur Kennzeichnung der Dedekind-MacNeilleschen hülle einer geordneten Menge, Arch. Math., 7, 241-249 (1956) · Zbl 0073.03801
[26] Venugopalan, P., A generalization of completely distributive lattices, Algebra Universalis, 27, 578-586 (1990) · Zbl 0722.06003
[27] Venugopalan, P., Quasicontinuous posets, Semigroup Forum, 41, 193-200 (1990) · Zbl 0703.06004
[28] Yang, J.; Xu, X., The dual of a generalized completely distributive lattice is a hypercontinuous lattice, Algebra Universalis, 63, 275-281 (2010) · Zbl 1204.06005
[29] Zhang, Z.; Li, Q., A generalization of the Dedekind-MacNeille completion, Semigroup Forum, 96, 553-564 (2018) · Zbl 1456.06006
[30] Zhang, W.; Xu, X., s_2-Quasicontinuous posets, Theor. Comp. Sci., 574, 78-85 (2015) · Zbl 1315.06007
[31] Zhang, W.; Xu, X., The σ_1-topology and λ_1-topology on s_1-quasicontinuous posets, Topol. Appl., 204, 79-89 (2016) · Zbl 1344.06008
[32] Zhang, W.; Xu, X., Frink quasicontinuous posets, Semigroup Forum, 94, 6-16 (2017) · Zbl 1397.06008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.