Document Type : Research Paper

**Authors**

Department of Mathematics, Ocean University of China, 238 Songling Road, 266100, Qingdao, P.R. China

**Abstract**

In this paper, it is shown that the category of stratified $L$-generalized convergence spaces is monoidal closed if the underlying truth-value table $L$ is a complete residuated lattice. In particular, if the underlying truth-value table $L$ is a complete Heyting Algebra, the Cartesian closedness of the category is recaptured by our result.

**Keywords**

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(S.E. Rodabaugh, E.P. Klement, U. Höhle, eds.), Kluwer, Dordrecht, 14 (1992), 153–176.

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New York, 5 (1998), 251–266.

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(2012), 1–20.

[11] J. M. Fang, Y. Yue, ⊤-Diagonal conditions and continuous extension theorem, Fuzzy Sets and Systems, 321 (2017),

73–89.

[12] P. V. Flores, R. N. Mohapatra, G. Richardson, Lattice-valued spaces: Fuzzy convergence, Fuzzy Sets and Systems,

157 (2006), 2706–2714.

[13] J. Gutiérrez García, M. A. De Prada Vicente, A. P. Šostak, A unified approach to the concept of fuzzy L-uniform

space, Topological and Algebraic Structures in Fuzzy Sets (S.E. Rodabaugh, E.P. Klement Eds.), Kluwer Academic

Publishers, 20 (2003), 81–114.

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

[15] U. Höhle, Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Mathematica, 38 (1982), 289–323.

[16] U. Höhle, A. P. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. Höhle, S. E. Rodabaugh (Eds.),

Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory, Kluwer Academic Publishers, Boston, 3 (1999),

123–272.

[17] G. Jäger, A category of L-fuzzy convergence spaces, Quaestiones Mathematicae, 24 (2001), 501–507.

[18] G. Jäger, Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Systems, 156 (2005), 1–24.

[19] G. Jäger, Pretopological and topological lattice-valued convergence spaces, Fuzzy Sets and Systems, 158 (2007),

424–435.

[20] G. Jäger, Fischer’s diagonal condition for lattice-valued convergence spaces, Quaestiones Mathematicae, 31 (2008),

11–25.

[21] G. Jäger, Lattice-valued convergence spaces and regularity, Fuzzy Sets and Systems, 159 (2008), 2488–2502.

[22] G. Jäger, Diagonal conditions for lattice-valued uniform convergence spaces, Fuzzy Sets and Systems, 210 (2013),

39–53.

[23] G. Jäger, M. H. Burton, Stratified L-uniform convergence spaces, Quaestiones Mathematicae, 28 (2005), 11–36.

[24] F. W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. del Seminario Matematico e Fisico

di Milano, 43 (1973), 135–166.

[25] H. Lai, D. Zhang, Closedness of the category of liminf complete fuzzy orders, Fuzzy Sets and Systems, 282 (2016),

86–98.

[26] L. Li, D. Zhang, On the relationship between limit spaces, many valued topological spaces, and many valued preorders,

Fuzzy Sets and Systems, 160 (2009), 1204–1217.

[27] L. Li, Q. Jin, On adjunctions between Lim, SL-Top, and SL-Lim, Fuzzy Sets and Systems, 182 (2011), 66–78.

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

Systems, 204 (2012), 40–52.

[29] L. Li, Q. Jin, p-Topologicalness and p-regularity for lattice-valued convergence spaces, Fuzzy Sets and Systems, 238

(2014), 26–45.

[30] E. Lowen, R. Lowen, P. Wuyts, The categorical topology approach to fuzzy topology and fuzzy convergence, Fuzzy

Sets and Systems, 40 (1991), 347–373.

[31] E. Lowen, R. Lowen, A topological universe extension of FTS, In:Applications of category theory to fuzzy sets,

(S.E. Rodabaugh, E.P. Klement, U. Höhle, eds.), Kluwer, Dordrecht, 14 (1992), 153–176.

[32] S. MacLane, Categories for the working mathematician, Second Edition, Graduate Texts in Mathematics, Springer,

New York, 5 (1998), 251–266.

[33] D. Orpen, G. Jäger, Lattice-valued convergence spaces: Extending the lattice context, Fuzzy Sets and Systems, 190

(2012), 1–20.

[34] B. Pang, J. M. Fang, L-fuzzy Q-convergence structures, Fuzzy Sets and Systems, 182 (2011), 53–65.

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

[36] B. Pang, The category of stratified L-filter spaces, Fuzzy Sets and Systems, 247 (2014), 108–126.

[37] B. Pang, Degrees of separation properties in stratified L-generalized convergence spaces using residual implication,

Filomat, 31(20) (2017), 6293-6305.

[38] B. Pang, Stratified L-ordered filter spaces, Quaestiones Mathematicae, 40(5) (2017), 661-678.

[39] G. Preuss, Foundations of topology, Kluwer Academic Publishers, Dordrecht, Boston, London 2002.

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

[41] W. Yao, On L-fuzzifying convergence spaces, Iranian Journal of Fuzzy Systems, 6(1) (2009), 63–80.

[42] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems, 158 (2007), 349–366.

[43] L. A. Zadeh, Fuzzy sets, Information Control, 8(3) (1965), 338–353.

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

[36] B. Pang, The category of stratified L-filter spaces, Fuzzy Sets and Systems, 247 (2014), 108–126.

[37] B. Pang, Degrees of separation properties in stratified L-generalized convergence spaces using residual implication,

Filomat, 31(20) (2017), 6293-6305.

[38] B. Pang, Stratified L-ordered filter spaces, Quaestiones Mathematicae, 40(5) (2017), 661-678.

[39] G. Preuss, Foundations of topology, Kluwer Academic Publishers, Dordrecht, Boston, London 2002.

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

[41] W. Yao, On L-fuzzifying convergence spaces, Iranian Journal of Fuzzy Systems, 6(1) (2009), 63–80.

[42] D. Zhang, An enriched category approach to many valued topology, Fuzzy Sets and Systems, 158 (2007), 349–366.

[43] L. A. Zadeh, Fuzzy sets, Information Control, 8(3) (1965), 338–353.

September and October 2019

Pages 139-153