Document Type : Research Paper

**Authors**

^{1}
Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju-City Jeonbuk, 561-756, Republic of Korea

^{2}
Department of Mathematics, College of Natural Science, Chonbuk National University, Jeonju-City Jeonbuk, 561-756, Republic of Korea and School of Mathematics and Science, Hebei GEO University, Shijiazhuang 050018, China

^{3}
School of Sciences, Hebei University of Science and Technology, Shijiazhuang 050018, P.R. China

**Abstract**

The aim of this paper is to extend the truth value table of

lattice-valued convergence spaces to a more general case and

then to use it to introduce and study the quantale-valued fuzzy Scott

topology in fuzzy domain theory. Let $(L,*,\varepsilon)$ be a

commutative unital quantale and let $\otimes$ be a binary operation

on $L$ which is distributive over nonempty subsets. The quadruple

$(L,*,\otimes,\varepsilon)$ is called a generalized GL-monoid if

$(L,*,\varepsilon)$ is a commutative unital quantale and the operation $*$ is

$\otimes$-semi-distributive. For generalized GL-monoid $L$ as the

truth value table, we systematically propose the stratified

$L$-generalized convergence spaces based on stratified $L$-filters,

which makes various existing lattice-valued convergence spaces as

special cases. For $L$ being a commutative unital quantale, we

define a fuzzy Scott convergence structure on $L$-fuzzy dcpos and

use it to induce a stratified $L$-topology. This is the inducing way

to the definition of quantale-valued fuzzy Scott topology, which

seems an appropriate way by some results.

lattice-valued convergence spaces to a more general case and

then to use it to introduce and study the quantale-valued fuzzy Scott

topology in fuzzy domain theory. Let $(L,*,\varepsilon)$ be a

commutative unital quantale and let $\otimes$ be a binary operation

on $L$ which is distributive over nonempty subsets. The quadruple

$(L,*,\otimes,\varepsilon)$ is called a generalized GL-monoid if

$(L,*,\varepsilon)$ is a commutative unital quantale and the operation $*$ is

$\otimes$-semi-distributive. For generalized GL-monoid $L$ as the

truth value table, we systematically propose the stratified

$L$-generalized convergence spaces based on stratified $L$-filters,

which makes various existing lattice-valued convergence spaces as

special cases. For $L$ being a commutative unital quantale, we

define a fuzzy Scott convergence structure on $L$-fuzzy dcpos and

use it to induce a stratified $L$-topology. This is the inducing way

to the definition of quantale-valued fuzzy Scott topology, which

seems an appropriate way by some results.

**Keywords**

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[8] J. M. Fang, Relationships between L-ordered convergence structures and strong L-topologies, Fuzzy Sets and Systems,

161(22) (2010), 2923–2944.

[9] J. M. Fang and Y.-L. Yue, ⊤-diagonal conditions and Continuous extension theorem, Fuzzy Sets and Systems, 321

(2017), 73–89.

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177(1) (1997), 111–138.

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their Applications to Fuzzy Subsets, Kluwer Academic Publishers, Dordrecht, 1995, pp. 53–106.

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[15] U. H¨ohle, Characterization of L-topologies by L-valued neighbourhoods, in: U. H¨ohle and S. E. Rodabaugh (Eds.),

Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, Dordrecht, 1999,

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[16] U. H¨ohle, Lattice Valued Topology and Its Applications, Springer-Verlag, New York, 2001.

[17] U. H¨ohle, A. P. ˇSostak, Axiomatic foundations of xed-basis fuzzy topology, in: U. H¨ohle and S. E. Rodabaugh

(Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, Dordrecht,

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[18] G. J¨ager, A category of L-fuzzy convergence spaces, Quaestiones Mathematicae, 24(4) (2001), 501–517.

[19] G. J¨ager, Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Systems, 156(1) (2005), 1–24.

[20] G. J¨ager, Pretopological and topological lattice-valued convergence spaces, Fuzzy Sets and Systems, 158(4) (2007),

424–435.

[21] G. J¨ager, Fischer's diagonal condition for lattice-valued convergence spaces, Quaestiones Mathematicae, 31(1)

(2008), 11–25.

[22] G. J¨ager, Lattice-valued convergence spaces and regularity, Fuzzy Sets and Systems, 159(19) (2008), 2488–2502.

[23] G. J¨ager, Compactification of lattice-valued convergence spaces, Fuzzy Sets and Systems, 161(7) (2010), 1002–1010.

[24] G. J¨ager, Compactness in lattice-valued function spaces, Fuzzy Sets and Systems, 161(22) (2010), 2962–2974.

[25] G. J¨ager, A one-point compactification for lattice-valued convergence spaces, Fuzzy Sets and Systems, 190 (2012),

21–31.

[26] G. J¨ager, Largest and smallest T2-compactications of lattice-valued convergence spaces, Fuzzy Sets and Systems,

190 (2012), 32–46.

[27] G. J¨ager, Stratified LMN-convergence tower spaces, Fuzzy Sets and Systems, 282 (2016), 62–73.

[28] J. L. Kelley, General Topology, Van Nostrand, Princeton, 1955.

[29] D. C. Kent, Convergence functions and their related topologies, Fundamenta Mathematicae, 54(1) (1964), 125–133.

[30] H. L. Lai, D. X. Zhang, Complete and directed complete Ω-categories, Theoretical Computer Science, 388(1-3)

(2007), 1–25.

[31] L. Q. Li, Q. Jin, On adjunctions between Lim, SL-Top, and SL-Lim, Fuzzy Sets and Systems, 182(1) (2011),

66–78.

[32] L. Q. Li, Q. Jin, On stratied L-convergence spaces: Pretopological axioms and diagonal axioms, Fuzzy Sets and

Systems, 204 (2012), 40–52.

161(22) (2010), 2923–2944.

[9] J. M. Fang and Y.-L. Yue, ⊤-diagonal conditions and Continuous extension theorem, Fuzzy Sets and Systems, 321

(2017), 73–89.

[10] B. Flag, R. Kopperman, Continuity spaces: Reconciling domains and metric spaces, Theoretical Computer Science,

177(1) (1997), 111–138.

[11] G. Gierz, et al, Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003.

[12] J. A. Goguen, L-fuzzy subsets, Journal of Mathematical Analysis and Applications, 18(1) (1967), 145–174.

[13] U. H¨ohle, Commutative, residuated l-monoids, in: U. H¨ohle and E. P. Klement (Eds.), Nonclassical Logics and

their Applications to Fuzzy Subsets, Kluwer Academic Publishers, Dordrecht, 1995, pp. 53–106.

[14] U. H¨ohle, MV-algebra valued lter theory, Quaestiones Mathematicae, 19(1-2) (1996), 23–46.

[15] U. H¨ohle, Characterization of L-topologies by L-valued neighbourhoods, in: U. H¨ohle and S. E. Rodabaugh (Eds.),

Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, Dordrecht, 1999,

pp. 389–432.

[16] U. H¨ohle, Lattice Valued Topology and Its Applications, Springer-Verlag, New York, 2001.

[17] U. H¨ohle, A. P. ˇSostak, Axiomatic foundations of xed-basis fuzzy topology, in: U. H¨ohle and S. E. Rodabaugh

(Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, Dordrecht,

1999, pp. 123–272.

[18] G. J¨ager, A category of L-fuzzy convergence spaces, Quaestiones Mathematicae, 24(4) (2001), 501–517.

[19] G. J¨ager, Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Systems, 156(1) (2005), 1–24.

[20] G. J¨ager, Pretopological and topological lattice-valued convergence spaces, Fuzzy Sets and Systems, 158(4) (2007),

424–435.

[21] G. J¨ager, Fischer's diagonal condition for lattice-valued convergence spaces, Quaestiones Mathematicae, 31(1)

(2008), 11–25.

[22] G. J¨ager, Lattice-valued convergence spaces and regularity, Fuzzy Sets and Systems, 159(19) (2008), 2488–2502.

[23] G. J¨ager, Compactification of lattice-valued convergence spaces, Fuzzy Sets and Systems, 161(7) (2010), 1002–1010.

[24] G. J¨ager, Compactness in lattice-valued function spaces, Fuzzy Sets and Systems, 161(22) (2010), 2962–2974.

[25] G. J¨ager, A one-point compactification for lattice-valued convergence spaces, Fuzzy Sets and Systems, 190 (2012),

21–31.

[26] G. J¨ager, Largest and smallest T2-compactications of lattice-valued convergence spaces, Fuzzy Sets and Systems,

190 (2012), 32–46.

[27] G. J¨ager, Stratified LMN-convergence tower spaces, Fuzzy Sets and Systems, 282 (2016), 62–73.

[28] J. L. Kelley, General Topology, Van Nostrand, Princeton, 1955.

[29] D. C. Kent, Convergence functions and their related topologies, Fundamenta Mathematicae, 54(1) (1964), 125–133.

[30] H. L. Lai, D. X. Zhang, Complete and directed complete Ω-categories, Theoretical Computer Science, 388(1-3)

(2007), 1–25.

[31] L. Q. Li, Q. Jin, On adjunctions between Lim, SL-Top, and SL-Lim, Fuzzy Sets and Systems, 182(1) (2011),

66–78.

[32] L. Q. Li, Q. Jin, On stratied L-convergence spaces: Pretopological axioms and diagonal axioms, Fuzzy Sets and

Systems, 204 (2012), 40–52.

[33] L.- Q. Li, Q. Jin, K. Hu, On stratified L-convergence spaces: Fischer's diagonal axiom, Fuzzy Sets and Systems,

267 (2015), 31–40.

[34] R. Lowen, Convergence in fuzzy topological spaces, Applied General Topology, 10(2) (1979), 147–160.

[35] L. X. Lu, W. Yao, Scott convergence of nets and filters and the Scott topology in posets, Fuzzy Systems and

Mathematics, 23(2) (2009), 38–40 (In Chinese).

[36] D. Orpen, G. J¨ager, Lattice-valued convergence spaces: Extending the lattice context, Fuzzy Sets and Systems, 190

(2012), 1–20.

[37] B. Pang, On (L;M)-fuzzy convergence spaces, Fuzzy Sets and Systems, 238 (2014), 46–70.

[38] P. M. Pu, Y. M. Liu, Fuzzy topology I. Neighborhood structure of a fuzzy point and Moore-Smith convergence,

Journal of Mathematical Analysis and Applications, 76(2) (1980), 571–599.

[39] S. E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies, Chapter 2 in: U. H¨ohle

and S. E. Rodabaugh (Eds), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of

Fuzzy Sets Series, Volume 3, Kluwer Academic Publishers, Boston/Dordrecht/London, 1999, pp. 91–116.

[40] K. I. Rosenthal, Quantales and Their Applications, Longman House, Burnt Mill, Harlow, 1990.

[41] J. J. M. M. Rutten, Elements of generalized ultrametric domain theory, Theoretical Computer Science, 170(1-2)

(1996), 349–381.

[42] D. S. Scott, Outline of a mathematical theory of computation, The 4th Annual Princeton Conference on Information

Sciences and Systems, Princeton University Press, Princeton, NJ, 1970, pp. 169–176.

[43] D. S. Scott, Continuous lattices, Topos, Algebraic Geometry and Logic, Lecture Notes in Mathematics, 274,

Springer-Verlag, Berlin, 1972, pp. 97–136.

[44] K. R. Wagner, Solving Recursive Domain Equations with Enriched Categories, Ph.D Thesis, School of Computer

Science, Carnegie-Mellon University, Pittsburgh, 1994.

[45] W. Yao, On many-valued L-fuzzy convergence spaces, Fuzzy Sets and Systems, 159(19) (2008), 2503–2519.

[46] W. Yao, On L-fuzzifying convergence spaces, Iranian Journal of Fuzzy Systems, 86(1) (2009), 63–80.

[47] W. Yao, Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed-complete poset, Fuzzy Sets and

Systems, 161(7) (2010), 983–987.

[48] W. Yao, Net-theoretical L-generalized convergence spaces, Iranian Journal of Fuzzy Systems, 8(2) (2011), 121–131.

[49] W. Yao, Moore-Smith convergence in (L;M)-fuzzy topology, Fuzzy Sets and Systems, 190 (2012), 47–62.

[50] W. Yao, A categorical isomorphism between injective fuzzy T0-spaces and fuzzy continuous lattices, IEEE Transactions

on Fuzzy Systems, 24(1) (2016), 131–139.

[51] W. Yao, Y. Li, Fuzzy domains based on a preidempotent unital commutative quantale, Journal of Hebei University

of Science and Technology, 34(2) (2013), 119–124. (In Chinese)

[52] W. Yao, L. X. Lu, Correction to On many-valued stratied L-fuzzy convergence spaces", Fuzzy Sets and Systems,

161(7) (2010), 1033–1038.

[53] W. Yao, B. Zhao, A duality between fuzzy domains and strongly completely distributive L-ordered sets, Iranian

Journal of Fuzzy Systems, 11(4) (2014), 23–43.

[54] W. Yao, F. G. Shi, Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topology on fuzzy directed complete

posets, Fuzzy Sets and Systems, 173(1) (2011), 60–80.

[55] Q. Yu, J. M. Fang, The category of ⊤-convergence spaces and its cartesian-closedness, Iranian Journal of Fuzzy

Systems, 14(3) (2017), 121–138.

[56] D. X. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems, 158(4) (2007),

349–366.

[57] Q. Y. Zhang, L. Fan, Continuity in quantitative domains, Fuzzy Sets and Systems, 154(1) (2005), 118–131.

267 (2015), 31–40.

[34] R. Lowen, Convergence in fuzzy topological spaces, Applied General Topology, 10(2) (1979), 147–160.

[35] L. X. Lu, W. Yao, Scott convergence of nets and filters and the Scott topology in posets, Fuzzy Systems and

Mathematics, 23(2) (2009), 38–40 (In Chinese).

[36] D. Orpen, G. J¨ager, Lattice-valued convergence spaces: Extending the lattice context, Fuzzy Sets and Systems, 190

(2012), 1–20.

[37] B. Pang, On (L;M)-fuzzy convergence spaces, Fuzzy Sets and Systems, 238 (2014), 46–70.

[38] P. M. Pu, Y. M. Liu, Fuzzy topology I. Neighborhood structure of a fuzzy point and Moore-Smith convergence,

Journal of Mathematical Analysis and Applications, 76(2) (1980), 571–599.

[39] S. E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies, Chapter 2 in: U. H¨ohle

and S. E. Rodabaugh (Eds), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of

Fuzzy Sets Series, Volume 3, Kluwer Academic Publishers, Boston/Dordrecht/London, 1999, pp. 91–116.

[40] K. I. Rosenthal, Quantales and Their Applications, Longman House, Burnt Mill, Harlow, 1990.

[41] J. J. M. M. Rutten, Elements of generalized ultrametric domain theory, Theoretical Computer Science, 170(1-2)

(1996), 349–381.

[42] D. S. Scott, Outline of a mathematical theory of computation, The 4th Annual Princeton Conference on Information

Sciences and Systems, Princeton University Press, Princeton, NJ, 1970, pp. 169–176.

[43] D. S. Scott, Continuous lattices, Topos, Algebraic Geometry and Logic, Lecture Notes in Mathematics, 274,

Springer-Verlag, Berlin, 1972, pp. 97–136.

[44] K. R. Wagner, Solving Recursive Domain Equations with Enriched Categories, Ph.D Thesis, School of Computer

Science, Carnegie-Mellon University, Pittsburgh, 1994.

[45] W. Yao, On many-valued L-fuzzy convergence spaces, Fuzzy Sets and Systems, 159(19) (2008), 2503–2519.

[46] W. Yao, On L-fuzzifying convergence spaces, Iranian Journal of Fuzzy Systems, 86(1) (2009), 63–80.

[47] W. Yao, Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed-complete poset, Fuzzy Sets and

Systems, 161(7) (2010), 983–987.

[48] W. Yao, Net-theoretical L-generalized convergence spaces, Iranian Journal of Fuzzy Systems, 8(2) (2011), 121–131.

[49] W. Yao, Moore-Smith convergence in (L;M)-fuzzy topology, Fuzzy Sets and Systems, 190 (2012), 47–62.

[50] W. Yao, A categorical isomorphism between injective fuzzy T0-spaces and fuzzy continuous lattices, IEEE Transactions

on Fuzzy Systems, 24(1) (2016), 131–139.

[51] W. Yao, Y. Li, Fuzzy domains based on a preidempotent unital commutative quantale, Journal of Hebei University

of Science and Technology, 34(2) (2013), 119–124. (In Chinese)

[52] W. Yao, L. X. Lu, Correction to On many-valued stratied L-fuzzy convergence spaces", Fuzzy Sets and Systems,

161(7) (2010), 1033–1038.

[53] W. Yao, B. Zhao, A duality between fuzzy domains and strongly completely distributive L-ordered sets, Iranian

Journal of Fuzzy Systems, 11(4) (2014), 23–43.

[54] W. Yao, F. G. Shi, Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topology on fuzzy directed complete

posets, Fuzzy Sets and Systems, 173(1) (2011), 60–80.

[55] Q. Yu, J. M. Fang, The category of ⊤-convergence spaces and its cartesian-closedness, Iranian Journal of Fuzzy

Systems, 14(3) (2017), 121–138.

[56] D. X. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems, 158(4) (2007),

349–366.

[57] Q. Y. Zhang, L. Fan, Continuity in quantitative domains, Fuzzy Sets and Systems, 154(1) (2005), 118–131.

May and June 2019

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