zbMATH — the first resource for mathematics

$$L$$-ordered and $$L$$-lattice ordered groups. (English) Zbl 1387.06005
Summary: This paper pursues an investigation on groups equipped with an $$L$$-ordered relation, where $$L$$ is a fixed complete Heyting algebra. First, by the concept of join and meet on an $$L$$-ordered set, the notion of an $$L$$-lattice (a weak $$L$$-lattice) is introduced and some related results are obtained. Then we applied them to define an $$L$$-lattice ordered group. We also introduce convex $$L$$-subgroups to construct a quotient $$L$$-ordered group. At last, a relation between the positive cone of an $$L$$-ordered group and special type of elements of $$L^G$$ is found, where $$G$$ is a group.

MSC:
 06A75 Generalizations of ordered sets 06D20 Heyting algebras (lattice-theoretic aspects) 06F15 Ordered groups
Full Text:
References:
 [1] Anderson, M.; Feil, T., Lattice-ordered groups: an introduction, (1988), D. Reidel Publishing Co. Dordrecht · Zbl 0636.06008 [2] Běhounek, L.; Bodenhofer, U.; Cintula, P., Relations in fuzzy class theory: initial steps, Fuzzy Sets Syst., 159, 1729-1772, (2008) · Zbl 1185.03078 [3] Bělohlávek, R., Fuzzy relational systems: foundations and principles, (2002), Kluwer Acad. Publ. New York · Zbl 1067.03059 [4] Bělohlávek, R., Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic, 128, 277-298, (2004) · Zbl 1060.03040 [5] Blyth, T. S., Lattices and ordered algebraic structures, (2005), Springer-Verlag London · Zbl 1073.06001 [6] Coppola, C.; Gerla, G.; Pacelli, T., Convergence and fixed points by fuzzy orders, Fuzzy Sets Syst., 159, 1178-1190, (2008) · Zbl 1176.06002 [7] Demirci, M., A theory of vague lattices based on many-valued equivalence relations. I: general representation results, Fuzzy Sets Syst., 151, 437-472, (2005) · Zbl 1067.06006 [8] Demirci, M., A theory of vague lattices based on many-valued equivalence relations. II: complete lattices, Fuzzy Sets Syst., 151, 473-489, (2005) · Zbl 1067.06007 [9] Denniston, J. T.; Melton, A.; Rodabaugh, S. E.; Solovyov, S. A., Lattice-valued preordered sets as lattice-valued topological systems, Fuzzy Sets Syst., 259, 89-110, (2015) · Zbl 1360.54012 [10] Fan, L., A new approach to quantitative domain theory, Electron. Notes Theor. Comput. Sci., 45, 77-87, (2001) · Zbl 1260.68217 [11] Fuentes-González, R., Down and up operators associated to fuzzy relations and t-norms: a definition of fuzzy semi-ideals, Fuzzy Sets Syst., 117, 377-389, (2001) · Zbl 0965.03065 [12] Höhle, U.; Blanchard, N., Partial ordering in $$L$$-underdeterminate sets, Inform. Sci., 35, 133-144, (1985) · Zbl 0576.06004 [13] Johnstone, P. T., Stone spaces, (1982), Cambridge University Press Cambridge · Zbl 0499.54001 [14] Kim, Y. C., Join-meet approximation operators and fuzzy preorders, J. Intell. Fuzzy Syst., 28, 1089-1097, (2015) · Zbl 1351.06001 [15] Klir, G. J.; Yuan, B., Fuzzy sets and fuzzy logic: theory and applications, (1995), Prentice-Hall Upper Saddle River · Zbl 0915.03001 [16] H. Lai, D. Zhang, Closedness of the category of liminf complete fuzzy orders, Fuzzy Sets Syst. (2014) (in press). http://dx.doi.org/10.1016/j.fss.2014.10.031. [17] Liu, M.; Zhao, B., Two Cartesian closed subcategories of fuzzy domains, Fuzzy Sets Syst., 238, 102-112, (2014) · Zbl 1315.18011 [18] Mordeson, J. N.; Malik, D. S., Fuzzy commutative algebra, (1998), World Scientific Publishing Co. Pte. Ltd. · Zbl 1026.13002 [19] Ovchinnikov, S. V., Structure of fuzzy binary relations, Fuzzy Sets Syst., 6, 169-195, (1981) · Zbl 0464.04004 [20] Venugopalan, P., Fuzzy ordered sets, Fuzzy Sets Syst., 46, 221-226, (1992) · Zbl 0765.06008 [21] Wang, K.; Zhao, B., Join-completions of $$L$$-ordered sets, Fuzzy Sets Syst., 199, 92-107, (2012) · Zbl 1261.06010 [22] Xie, W.; Zhang, Q.; Fand, L., The Dedekind-macneille completions for fuzzy posets, Fuzzy Sets Syst., 160, 2292-2316, (2009) · Zbl 1185.06003 [23] Yao, W., Quantitative domains via fuzzy sets: part I: continuity of fuzzy directed complete posets, Fuzzy Sets Syst., 161, 973-987, (2010) · Zbl 1193.06007 [24] Yao, W.; Shi, F. G., Quantitative domains via fuzzy sets: part II: fuzzy Scott topology on fuzzy directed-complete posets, Fuzzy Sets Syst., 173, 60-80, (2011) · Zbl 1234.06007 [25] Yao, W., An approach to fuzzy frames via fuzzy posets, Fuzzy Sets Syst., 166, 75-89, (2011) · Zbl 1239.06006 [26] Yao, W.; Lu, L. X., Fuzzy Galois connections on fuzzy posets, Math. Log. Quart., 55, 105-112, (2009) · Zbl 1172.06001 [27] Zadeh, L. A., Similarity relations and fuzzy orderings, Inform. Sci., 3, 177-200, (1971) · Zbl 0218.02058 [28] Zhang, Q. Y.; Fan, L., Continuity in quantitative domains, Fuzzy Sets Syst., 154, 118-131, (2005) · Zbl 1080.06007 [29] Zhang, D. X.; Liu, Y. M., $$L$$-fuzzy version of stones representation theorem for distributive lattices, Fuzzy Sets Syst., 76, 259-270, (1995) · Zbl 0852.54008 [30] Zhang, Q. Y.; Xie, W. X., Section-retraction-pairs between fuzzy domains, Fuzzy Sets Syst., 158, 99-114, (2007) · Zbl 1116.06006 [31] Zhang, Q. Y.; Xie, W. X.; Fan, L., Fuzzy complete lattices, Fuzzy Sets Syst., 160, 2275-2291, (2009) · Zbl 1183.06004
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.