# Abstract:

A partial order on a bounded lattice $L$ is called t-order if it is defined by means of the t-norm on $L$. It is obtained that for a t-norm on a bounded lattice $L$ the relation $a\preceq_{T}b$ iff $a=T(x,b)$ for some $x\in L$ is a partial order. The goal of the paper is to determine some conditions such that the new partial order induces a bounded lattice on the subset of all idempotent elements of $L$ and a complete lattice on the subset $A$ of all elements of $L$ which are the supremum of a subset of atoms.

# Keywords:

triangular norm, bounded lattice, triangular action, $\bigvee$-distributive, idempotent element

03E72, 03B52

# References:

1. G. Birkhoff: Lattice Theory. Third edition. Providience 1967.   CrossRef
2. B. De Baets and R. Mesiar: Triangular norms on product lattices. Fuzzy Sets and Systems104 (1999), 61-75.   CrossRef
3. B. De Baets and R. Mesiar: Triangular norms on the real unit square. In: Proc. 1999 EUSFLAT-EST YLF Joint Conference, Palma de Mallorca 1999, pp. 351-354.   CrossRef
4. J. Casasnovas and G. Mayor: Discrete t-norms and operations on extended multisets. Fuzzy Sets and Systems 1599 (2008), 1165-1177.   CrossRef
5. C. A. Drossos: Generalized t-norm structures. Fuzzy Sets and Systems 104 (1999), 53-59.   CrossRef
6. L. Gonzalez: A note on the infinitary action of triangular norms and conorms. Fuzzy Sets and Systems 101 (1999), 177-180.   CrossRef
7. S. Gottwald: A Treatise on Many-Valued Logics. Research Studies Press Ltd., Baldock, Hertfordshire 2001.   CrossRef
8. T. Hungerford: Algebra. Springer-Verlag 1974.   CrossRef
9. P. Hájek: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht 1998.   CrossRef
10. U. Höhle: Commutative, residuated $\ell$-monoids. In: Non-Classical Logics and Their Applications to Fuzzy Subsets: A Handbook on the Math. Foundations of Fuzzy Set Theory (U. H$\ddot{o}$hle and E. P. Klement, eds.). Kluwer, Dordrecht 1995.   CrossRef
11. S. Jenei and B. De Baets: On the direct decomposability of t-norms on product lattices. Fuzzy Sets and Systems 139 (2003), 699-707.   CrossRef
12. F. Karaçal and Y. Sa\u{g}{\i}ro\u{g}lu: Infinetely $\bigvee$-distributive t-norm on complete lattices and pseudo-complements. Fuzzy Sets and Systems 160 (2009), 32-43.   CrossRef
13. F. Karaçal and Dj. Khadjiev: $\bigvee$-distributive and infinitely $\bigvee$-distributive t-norms on complete lattice. Fuzzy Sets and Systems 151 (2005), 341-352.   CrossRef
14. F. Karaçal: On the direct decomposability of strong negations and S-implication operators on product lattices. Inform. Sci. 176 (2006), 3011-3025.   CrossRef
15. E. P.Klement: Operations on fuzzy sets-an axiomatic approach. Inform. Sci. 27 (1982), 221-232.   CrossRef
16. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000.   CrossRef
17. X. Liang and W. Pedrycz: Logic-based fuzzy networks: A study in system modeling with triangular norms and uninorms. Fuzzy Sets and Systems 160 (2009), 3475-3502.   CrossRef
18. K. C. Maes and A. Mesiarová-Zemánková: Cancellativity properties for t-norms and t-subnorms. Inform. Sci. 179 (2009), 1221-1233.   CrossRef
19. A. Mesiarová: H-transformation of t-norms. Inform. Sci. 176 (2006), 1531-1545.   CrossRef
20. H. Mitsch: A natural partial order for semigroups. Proc. Amer. Math. Soc. 97 (1986), 384-388.   CrossRef
21. S. Saminger: On ordinal sums of triangular norms on bounded lattices. Fuzzy Sets and Systems 157 (2006), 1403-1416.   CrossRef
22. S. Saminger-Platz, E. P. Klement and R. Mesiar: On extensions of triangular norms on bounded lattices. Indag. Math. 19 (2009), 135-150.   CrossRef
23. S. Samuel: Calculating the large N phase diagram in the fundamental-adjoint action lattice theory. Phys. Lett. 122 (1983), 287-289.   CrossRef
24. B. Schweizer and A. Sklar: Probabilistic Metric Spaces. Elsevier, Amsterdam 1983.   CrossRef
25. Z. Wang: T-filters of integral residuated $\ell$-monoids. Inform. Sci. 177 (2007), 887-896.   CrossRef