Geodesic
The structure of idempotent residuated chains
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 2, pp. 453-479 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green's relation $\mathcal D$ on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain some characterizations of subdirectly irreducible, simple and strictly simple idempotent residuated chains.
In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green's relation $\mathcal D$ on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain some characterizations of subdirectly irreducible, simple and strictly simple idempotent residuated chains.
Classification : 06F05, 20M10, 20M99
Keywords: idempotent residuated lattice; chain; band 
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Chen, Wei; Zhao, Xianzhong. The structure of idempotent residuated chains. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 2, pp. 453-479. http://geodesic.mathdoc.fr/item/CMJ_2009_59_2_a11/

[1] Stanovský, D.: Commutative idempotent residuated lattices. Czech. Math. J. 57 (2007), 191-200. | DOI | MR

[2] Galatos, N.: Minimal varieties of residuated lattices. Algebra Universal. 52 (2004), 215-239. | DOI | MR | Zbl

[3] Jipsen, P., Tsinakis, C.: A survey of residuated lattices. Ordered Algebraic Structures (J. Martinez, ed.), Kluwer Academic Publishers, Dordrecht (2002), 19-56. | MR | Zbl

[4] Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C.: Cancellative residuated lattices. Algebra Universal. 50 (2003), 83-106. | DOI | MR | Zbl

[5] Blount, K., Tsinakis, C.: The structure of residuated lattices. Internat. J. Algebra Comput. 13 (2003), 437-461. | DOI | MR | Zbl

[6] Burris, S., Sankappanavar, H. P.: A Course in Universal Algebra. GTM78, Springer (1981). | MR | Zbl

[7] Howie, J. M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs, New series, Vol. 12, Oxford Univ Press, New York (1995). | MR | Zbl