Characterizations of L-convex spaces via domain theory

Abstract We provide some new characterizations of L-convex spaces by using the way-below relation in domain theory. For this purpose, we first study the notion of domain finiteness in the lattice-valued case, and then introduce three kinds of spaces: algebraic L-closure spaces, restricted L-hull spaces, and L-entailment spaces. These three spaces are shown to be categorically isomorphic to L-convex spaces. Additionally, we introduce the notion of L-polytopes and prove that a subcollection of a dense L-convex structure is a base if and only if it contains all L-polytopes. These results indicate that in the study of the theory of (fuzzy) convex spaces, (fuzzy) domain theory has important applications.

[1]  Antonio di Nola,et al.  Fuzzy Models of First Order Languages , 1986, Math. Log. Q..

[2]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[3]  Fu-Gui Shi,et al.  Subcategories of the category of L-convex spaces , 2017, Fuzzy Sets Syst..

[4]  Bin Pang,et al.  CHARACTERIZATIONS OF L-CONVEX SPACES , 2016 .

[5]  R. Lowen.  Fuzzy topological spaces and fuzzy compactness , 1976 .

[6]  Fu-Gui Shi,et al.  $(L,M)$-fuzzy convex structures , 2017, 1702.03521.

[7]  S. Franklin.  Some Results on Order-Convexity , 1962 .

[8]  Giangiacomo Gerla,et al.  Closure systems and L-subalgebras , 1984, Inf. Sci..

[9]  K. Menger.  Untersuchungen über allgemeine Metrik , 1928 .

[10]  Giangiacomo Gerla,et al.  Lattice valued algebras. , 1987 .

[11]  Fu-Gui Shi,et al.  (L, M)-fuzzy matroids , 2009, Fuzzy Sets Syst..

[12]  R. Belohlávek.  Fuzzy Closure Operators , 2001 .

[13]  Horst Herrlich,et al.  Abstract and concrete categories , 1990 .

[14]  Wei Yao,et al.  Fuzzy Galois connections on fuzzy posets , 2009, Math. Log. Q..

[15]  Fu-Gui Shi,et al.  A new approach to the fuzzification of matroids , 2009, Fuzzy Sets Syst..

[16]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[17]  H. Komiya,et al.  Convexity on a topological space , 1981 .

[18]  Fu-Gui Shi,et al.  L-CONVEX SYSTEMS AND THE CATEGORICAL ISOMORPHISM TO SCOTT-HULL OPERATORS , 2017 .

[19]  A. S. Mashhour,et al.  Fuzzy closure spaces , 1985 .

[20]  Frank Harary,et al.  Convexity in graphs , 1981 .

[21]  Jan Pavelka,et al.  On Fuzzy Logic I Many-valued rules of inference , 1979, Math. Log. Q..

[22]  Fu-Gui Shi,et al.  Categories isomorphic to the category of $L$-fuzzy closure system spaces , 2013 .

[23]  M. van de Vel,et al.  Binary Convexities and Distributive Lattices , 1984 .

[24]  M. V. Rosa,et al.  On fuzzy topology fuzzy convexity spaces and fuzzy local convexity , 1994 .

[25]  Venkat Murali,et al.  Lattice of fuzzy subalgebras and closure systems in I x , 1991 .

[26]  T Dr.Thrivikraman,et al.  Some Problems In Algebra And Topology A Study Of Fuzzy Convexity With Special Reference To Separation Properties , 1993 .

[27]  Martin Farber,et al.  On local convexity in graphs , 1987, Discret. Math..

[28]  E. Marczewski,et al.  Independence in abstract algebras results and problems , 1966 .

[29]  Juhani Nieminen.  The ideal structure of simple ternary algebras , 1978 .

[30]  George J. Klir,et al.  Fuzzy Logic and Mathematics: A Historical Perspective , 2017 .

[31]  Jan Pavelka,et al.  On Fuzzy Logic II. Enriched residuated lattices and semantics of propositional calculi , 1979, Math. Log. Q..

[32]  Jan Pavelka,et al.  On Fuzzy Logic III. Semantical completeness of some many-valued propositional calculi , 1979, Math. Log. Q..

[33]  Radim Belohlávek,et al.  Fuzzy closure operators II: induced relations, representation, and examples , 2002, Soft Comput..

[34]  M. Vel.  On the rank of a topological convexity , 1983 .

[35]  M. Farber,et al.  Convexity in graphs and hypergraphs , 1986 .